A process implemented by a computer, a computer system and apparatus for determining the properties of an underground formation, a non-transient computer-readable medium, and a method for manufacturing a geophysical data product.

Abstract

Methods and systems described herein are directed to determining properties of a subterranean formation using an acoustic wave-equation in terms of a velocity model and a vector reflectivity model of the subterranean formation. The acoustic wave equation may be used with simultaneous inversion to simultaneously build velocity and pre-stack reflectivity in the form of angle gathers of a subterranean formation. The velocity and angle gathers may be employed for quantitative interpretation. The velocity and vector reflectivity may be employed to determine relative impedance and relative density of the subterranean formation for prospectivity assessment. The acoustic wave equation may extend into the angle domain to build angle gathers of the subterranean formation with enhanced resolution and amplitude fidelity. The velocity, angle gathers, and vector reflectivity reveal the structure and lithology of features of the subterranean formation and may reveal the presence of oil and natural gas reservoirs.

Description

54 A process implemented by a computer, computer system, or device for determining properties. OF AN UNDERGROUND FORMATION, NON-TRANSIENTIAL COMPUTER-READABLE MEDIUM, AND A METHOD FOR MANUFACTURING A GEOPHYSICAL DATA PRODUCT CROSS-REFERENCE TO RELATED ORDERS

[001] This application claims the benefit of Provisional Application No. Application No. 63 / 452.777, filed on March 17, 2023, and Non-Provisional Application No. 18 / 600.002, filed on March 8, 2024, are incorporated by reference herein as if fully set forth in this document. FUNDAMENTALS

[002] Marine seismology companies invest heavily in the development of marine seismic survey equipment and seismic data processing techniques to obtain accurate, high-resolution images of subsurface formations located beneath a body of water. These images are used, for example, to determine the structural features of subsurface formations, to discover oil and natural gas reservoirs, and to monitor oil and natural gas reservoirs during production. A typical marine seismic survey is conducted with one or more research vessels towing a seismic source and numerous seismographic cables across the body of water. The research vessel contains seismic acquisition equipment such as navigation control, seismic source control, seismic receiver control, and recording equipment. A seismic source control manages the activation of one or more seismic sources at selected times or locations.A seismic source can be an impulsive source consisting of an array of air cannons or sparkers that are activated to produce pulses of acoustic energy. Alternatively, a seismic source can be a marine vibrator that emits energy. Petition 870250102319, dated 07 / 11 / 2025, p. 6 / 133 / 54 acoustic energy over a longer period of time. The acoustic energy generated by a seismic source spreads in all directions. A portion of the acoustic energy travels through the body of water and enters an underground formation to propagate as sound waves within the underground formation. At each interface between different types of liquid, rock, and sediment, a portion of the sound wave is refracted, a portion is transmitted, and another portion is reflected in the body of water to propagate as a reflected wave field toward the water surface. Seismographic cables are elongated, cable-like structures, spaced and towed by a research vessel in the direction the research vessel is traveling and are generally arranged substantially parallel to each other.Each seismographic cable contains many seismic receivers or sensors that detect pressure wave fields and / or particle motion from sound waves. The seismographic cables collectively form a seismic data acquisition surface that records wave fields as seismic data on the recording equipment. Alternatively, a seismic data acquisition surface can be created by deploying the receivers on the bottom of the water body and directly on or near the surface of the subsurface formation. The recorded pressure wave fields and / or particle motion are processed to generate and display images of the subsurface formation, allowing geoscientists to identify potential oil and natural gas reservoirs and monitor producing oil and natural gas reservoirs. DESCRIPTION OF THE DRAWINGS

[003] Figure 1A shows a side elevation view of an example of a seismic data acquisition system.

[004] Figure 1B shows a top view of an example of a seismic data acquisition system.

[005] Figure 2 shows a side elevation view of a system Petition 870250102319, dated 07 / 11 / 2025, p. 7 / 133 / 54 regarding the acquisition of seismic data with an enlarged view of a receiver.

[006] Figure 3 shows a side elevation view of different examples of ray paths that acoustic energy travels between a seismographic cable source and receiver.

[007] Figure 4 shows an example of grouping of common shots of example traces of reflected wave fields measured by receivers located along a seismographic cable shown in Figure 3.

[008] Figure 5 shows a process for constructing vector velocities and reflectivities of subsurface formation from a pressure wave field recorded in a marine seismic survey of the subsurface formation according to an example implementation.

[009] Figure 6 is a flow diagram illustrating an example implementation of the “iterative full waveform inversion (“FWI”) procedure to simultaneously construct a high-resolution velocity model and a vector reflectivity model of the subsurface formation” referenced in Figure 5.

[0010] Figure 7 shows a high-level representation of the iterative FWI performed in Figure 6.

[0011] Figure 8A shows a cross-correlation kernel plot produced for a model consisting of a single homogeneous layer overlaying a half-space.

[0012] Figure 8B shows an ISIC impedance core plot produced with a dynamically weighted impedance sensitivity core for the same model as Figure 8A.

[0013] Figure 8C shows an ISIC velocity core graph produced with a dynamically weighted velocity sensitivity core for the same model as Figure 8A.

[0014] Figure 9 shows a vertical plane cross-section of Petition 870250102319, dated 07 / 11 / 2025, page 8 / 133 / 54 examples of vector reflectivity and pressure gradients for a pressure wave field propagating through a vertical cross-section of a body of water and an underground formation.

[0015] Figure 10 shows a three-dimensional geometric representation of the scattering angle and azimuth angle for an image point in a subsurface in an underground formation.

[0016] Figure 11 is a flow diagram of a process for creating a grouping of angles of a subsurface formation from field data of pressure waves recorded in multiple marine survey shots.

[0017] Figure 12 shows a simultaneous inversion process to construct a velocity model and a vector reflectivity prestacking of a subsurface formation from a pressure wave field recorded in a marine seismic survey of the subsurface formation.

[0018] Figure 13 is a flow diagram that illustrates an example of implementing the “simultaneous speed and reflectivity pre-stacking inversion” procedure referenced in Figure 12.

[0019] Figure 14 is a flow diagram that illustrates an example of implementing the “simultaneous velocity inversion and pre-stacking reflectivity” procedure referenced in Figure 12 based on an acoustic wave equation extended to the angle domain.

[0020] Figure 15 shows an example of a computer system that executes an efficient process to perform a method of generating a velocity model and a pre-stacking vector reflectivity.

[0021] Figures 16A to 16F show the output of the simultaneous inversion application for the same cross-section in line (xz plane) of the Salar Basin.

[0022] Figure 16A shows that simultaneous inversion improved and Petition 870250102319, dated 07 / 11 / 2025, page 9 / 133 / 54 significantly increased the speed resolution.

[0023] Figure 16B shows the final full stacking reflectivity model.

[0024] Figure 16C shows the inverted angle groupings.

[0025] Figure 16D shows a relative density model calculated from the final velocity (Figure 16A) and the impedance calculated from the full stack reflectivity model (Figure 16B).

[0026] In Figure 16E, close-angle stacking is shown.

[0027] Figure 16F shows the far-angle stacking.

[0028] Figures 17A to 17F show depth slices or xy-plane sections of the Salar Basin at a reservoir level around 5,000 m deep.

[0029] Figure 17A shows the reversed velocity model at the reservoir level.

[0030] Figure 17B shows the reflectivity of full stacking at the reservoir level.

[0031] Figure 17C shows the accumulated velocity updates superimposed on the reflectivity image.

[0032] Figure 17D shows the relative density model derived from velocity and relative impedance calculated from full stack reflectivity.

[0033] Figures 17E and 17F show reflectivity as a function of angles.

[0034] Figure 17E shows the reflectivity of near-stacking at the reservoir level.

[0035] Figure 17F shows the reflectivity of distant stacking at the reservoir level. DETAILED DESCRIPTION Petition 870250102319, dated 07 / 11 / 2025, page 10 / 133 / 54

[0036] Obtaining seismic images of a subsurface formation is the primary goal of seismic data processing. Seismic images in the form of sections and volumes are used to detect subsurface targets of scientific interest or petroleum deposits. Targets can be the morphology of rock strata or faults and the distribution of various types of fluids, such as oil and natural gas deposits, in the rocks. However, interpreting seismic images is challenging because typical seismic images are responses to variations in subsurface impedance, and these responses are band-limited, carry artifacts and noise, and are not exclusively related to the targeted properties. These challenges are among the main reasons for the development of seismic attributes.Seismic attributes are quantitative measures of seismic characteristics of a subsurface formation that are extracted from seismic data recorded in a survey of the subsurface formation. Processing seismic data into seismic attributes involves ways to make seismic data more interpretable or to reduce the negative impacts of bandwidth, artifacts, noise, and non-uniqueness in seismic images. Seismic attributes are analyzed by geoscientists to enhance information that may be more subtle in a traditional seismic image of the subsurface formation, leading to a better geological or geophysical interpretation of the seismic data. Seismic attributes are widely used in seismic exploration and play a vital role in identifying oil and natural gas reservoirs.

[0037] The processes and systems described in this document are geared toward an enhanced approach for simultaneous velocity- and angle-dependent reflectivity inversion. The processes are executed with a vector-reflectivity-based acoustic wave equation to obtain seismic attributes in the form of high-resolution, accurate velocity models, reflectivity images, and pre-set angle clusters. Petition 870250102319, dated 07 / 11 / 2025, page 11 / 133 / 54 stacking. The acoustic wave equation allows for a precise and efficient simulation of the transmitted and reflected components of acoustic waves propagating within the underground formation. In particular, the acoustic wave equation can be used with full waveform inversion (“FWI”) to construct accurate and high-resolution velocity and vector reflectivity of the underground formation and can be used with least squares reverse time migration (“LSRTM”) to construct an angle-dependent vector reflectivity of the underground formation. This approach is based on extracting geometric information from the scalar product between the vector reflectivity and the gradient of the pressure wave field in the acoustic wave equation. The angle attribute information obtained is used to construct pre-stacking angle groupings.The processes and systems described in this document can be implemented using one of two data domain workflows for simultaneous pre-stack inversion: The first workflow is based on stack demigration, and the second workflow is based on pre-stack demigration. The least-squares inversion process improves resolution and compensates for incomplete acquisitions and variable illumination, thus providing higher resolution and enhanced amplitudes for amplitude versus angle (“AVA”) analysis.

[0038] As explained above, seismic attributes play an important role in hydrocarbon exploration, identifying potential prospects. To obtain velocity models and reflectivity images, seismic inversion has been the traditional approach, followed by attribute calculations that aid in interpretation. Seismic inversion has typically been used to derive seismic attributes such as velocity, density, and vector reflectivity, which can be used to aid in the interpretation of seismic images. Velocity, density, and strata reflectivity are examples of attributes that show the composition. Petition 870250102319, dated 07 / 11 / 2025, p. 12 / 133 / 54 of the strata within an underground formation. Oil and natural gas reservoirs, for example, are typically located in layers of sandstone, clastic rocks, and carbonates such as limestone. Seismic velocities and reflectivities are used by geoscientists to identify the composition of the layers in an image of an underground formation. For example, shales have seismic velocities in a range of about 0.9 to 2.5 km / s, oil has seismic velocities in a range of about 1.2 to 1.25 km / s, sandstones have seismic velocities in a range of about 2.0 to 6.0 km / s, and granite and basalt have seismic velocities in a range of about 4.5 to 6.0 km / s. (See, for example, Figure 5.5 in Exploration Seismology, 2nd Ed., R.E. Sheriff and L.P. Geldart, Cambridge University Press, 1995).Geoscientists in the oil and gas industry carefully examine images, velocity models, and / or vector reflectivity of an underground formation and use these models to identify rock interfaces between layers and features that potentially contain oil and natural gas reservoirs. Without accurate seismic images or associated vector reflectivity and velocity models of underground formations, geoscientists would have to resort to randomly drilling test wells in the hope of finding an oil and natural gas reservoir.

[0039] Seismic image formation based on wave equations is a two-step process for generating images, velocity models, and / or vector reflectivity of a subsurface formation from seismic data recorded in a survey. In the first step, an acoustic wave equation is used to forward propagate a wave field from the source and reverse propagate reflection events recorded in the seismic data. In the second step, an image formation condition is used to cross-correlate the propagated wave fields and thus obtain an image that shows the detailed structural properties or attributes of the subsurface formation. The acoustic wave equation employed Petition 870250102319, dated 07 / 11 / 2025, page 13 / 133 / 54, in step one, models the propagation of acoustic waves in the underground formation and is traditionally expressed in terms of a seismic velocity model. The seismic velocity model is a three-dimensional (“3D”) map of the seismic velocities associated with the layers of the underground formation.

[0040] LSRTM is an iterative seismic imaging process performed in the data domain to update and improve an image or vector reflectivity model of the subsurface formation at each iteration. The iterative process minimizes a difference between reflection events recorded at the receiver locations during the survey and reflection events that are simulated during the direct propagation of the source wavefield and is completed when the resulting image or vector reflectivity minimizes the difference. However, the velocity models typically used in iterative LSRTM do not represent all the impedance contrasts of subsurface formations that simulate reflection events. Thus, a first-order approximation of Born theory is used to generate these reflections. The corresponding wave equation is an approximation and does not generate all events in the recorded seismic data.Furthermore, in each iteration, two different wave equations are solved during the forward and reverse propagation.

[0041] FWI is an iterative process similar to LSRTM, except that instead of updating the reflectivity of a subsurface formation, FWI improves the resolution of a velocity model of the subsurface formation. Conventional FWI does not require reflection events, and refraction events are sufficient to improve the velocity model when refraction events are available. However, the maximum penetration depth of refraction events is limited by the maximum source-receiver displacement of the marine survey. By using reflection events in FWI, the depth limitation is removed, and it is possible to correctly update the velocity model to a maximum depth where Petition 870250102319, dated 07 / 11 / 2025, page 14 / 133 / 54 reflection events are generated at the boundaries of the underground formations. Furthermore, vector reflectivity can be updated at each iteration, as the velocity model is improved.

[0042] In reflection-based FWI, a smooth velocity model is generally used, and most reflection events cannot be simulated from such a model. Thus, a density model is used in some approaches. However, building accurate density models of a subsurface formation is challenging and expensive because the process requires interpretation and integration of wells, which in some cases is not possible. When wells are available, density models can also be inaccurate far from the actual well locations. Other reflection-based FWI approaches use reflectivity (or imaging) and first-order Born theory to generate reflection events. To generate the complete wavefield, two different wave equations are used in each modeling realization.

[0043] On the other hand, the processes described in this document are geared toward an innovative simultaneous inversion workflow based on a vector reflectivity procedure (“Determining properties of a subterranean formation using an acoustic wave equation with a reflectivity parameterization,” Whitmore et al., U.S. Publication No. 2021 / 0103065) in which the velocity and vector reflectivity of a subterranean formation are iteratively estimated in a single frame (“Simultaneous inversion of velocity and reflectivity,” Yang, Y., et al. First International Meeting for Applied Geoscience & Energy, Expanded Abstracts, 2022). This approach is equivalent to performing FWI and LSRTM, which provide high-fidelity velocity and vector reflectivity for quantitative interpretation (“QI”). Simultaneous inversion processes and systems that produce stacked 3D velocity and vector reflectivity are described in U.S. Publication No. 2021 / 0103065. Petition 870250102319, dated 07 / 11 / 2025, p. 15 / 133 / 54

[0044] In this description, the simultaneous inversion workflow is expanded, updating stacked 3D velocity and vector reflectivity models by incorporating pre-stacking reflectivity dependent on the reflection angle and azimuth to obtain four-dimensional (“4D”) and / or five-dimensional (“5D”) velocity and angle-dependent vector reflectivity. The extended simultaneous inversion workflow incorporates geometric information extracted from the dot product between vector reflectivity and the gradient of the pressure wave field in the acoustic wave equation, allowing the calculation of the reflection angle between the incident pressure wave field and the vector reflectivity, thus facilitating the construction of angle groupings. The velocity model and angle groupings are iteratively updated via simultaneous inversion, leading to improved model resolution and compensating for incomplete acquisitions and variations in illumination. Marine Seismic Survey

[0045] Figures 1A to 1B show a side elevation view and a top view, respectively, of an example of a marine seismic data acquisition system comprising an exploration seismology research vessel 102 and a source 104. A seismic data acquisition system is not limited to one source, as shown in Figures 1A to 1B. In practice, the number of sources can vary from a single source towed by a research vessel to multiple sources towed by different research vessels. The body of water can be, for example, an ocean, a sea, or any portion thereof. In this example, the research vessel 102 tows six seismographic cables 106 to 111 below the free surface of a body of water. Each seismographic cable is connected at one end to the research vessel 102 by means of a seismographic cable data transmission cable. A data acquisition surface is not limited to six seismographic cables, as Petition 870250102319, dated 07 / 11 / 2025, p. 16 / 133 / 54 shown in Figure 1B. In practice, the number of seismographic cables used to form a data acquisition surface can vary from just one seismographic cable to 20 or more seismographic cables. Source 104 can be an impulsive source, such as an array of air guns or sparkers, or source 104 can be a vibrational source, such as one or more marine vibrators. Additional research vessels (not shown) can be used to tow additional sources.

[0046] Figure 1A includes an xz plane 114, and Figure 1B includes an xy plane 116, of a Cartesian coordinate system having three orthogonal geometric spatial coordinate axes labeled x, y, and z. The coordinate system specifies orientations and coordinate locations within the body of water. The geometric x-axis specifies the position of a point in a direction parallel to the length of the seismographic cables or the direction of the research vessel and is called the “in line” direction. The geometric y-axis specifies the position of a point in a direction perpendicular to the geometric x-axis substantially parallel to the free surface 112 and is referred to as the “cross line” direction. The geometric z-axis, also known as the geometric “depth” axis, specifies the position of a point in a direction perpendicular to the xy plane (i.e., perpendicular to the free surface 112) with the positive z-direction pointing downward, away from the free surface 112.

[0047] Seismographic cables can be towed to form a horizontal, flat seismic data acquisition surface relative to the free surface 112. However, in practice, seismographic cables may vary slightly due to active ocean currents and weather conditions. A seismic data acquisition surface is not limited to the parallel seismographic cables shown in Figures 1A and 1B. In other implementations, seismographic cables can be towed with a progressively greater separation of seismographic cables in the direction of the line. Petition 870250102319, dated 07 / 11 / 2025, page 17 / 133 / 54 transversally towards greater distances from the research vessel 102 in a process called “seismographic cable fanning”. Seismographic cable fanning spreads the seismographic cables further and further apart as the distance from the research vessel increases in the linear direction. Seismographic cable fanning can improve coverage at distances far from the source / receiver without compromising seismic data resolution or quality, and can also increase acquisition efficiency by reducing seismic data fill. In other implementations, seismographic cables can be towed with a downward slope as the distance from the research vessel increases.

[0048] Seismographic cables 106 to 111 are typically long cables containing power and data transmission lines coupled to receivers (represented by shaded rectangles), such as receiver 118, which are spaced along the length of each seismographic cable. The data transmission lines couple receivers to seismic data acquisition equipment, computers, and data storage devices located aboard the research vessel 102. The depth of the seismographic cable below the free surface 112 can be estimated at various locations along the seismographic cables using depth measuring devices attached to the seismographic cables. For example, the depth measuring devices may measure hydrostatic pressure or use acoustic distance measurements.Depth measuring devices can be integrated with depth controllers, such as paravanes or water kites, which control and maintain the depth and position of the seismographic cables while the seismographic cables are towed through the body of water. Depth measuring devices are typically placed at intervals (e.g., intervals of about 300 meters in some implementations) along each cable. Petition 870250102319, dated 07 / 11 / 2025, p. 18 / 133 / 54 seismographic. Note that, in other implementations, buoys may be attached to seismographic cables and used to maintain the orientation and depth of the seismographic cables below the free surface 112.

[0049] In Figure 1A, curve 122, the formation surface, represents a top surface of the subsurface formation 120 located at the bottom of the water body. The subsurface formation 120 may have subsurface layers of sediments and rocks. Curves 124, 126, and 128 represent interfaces between subsurface layers of different compositions. A shaded region 130, bounded at the top by curve 132 and at the bottom by curve 134, represents a subsurface hydrocarbon deposit, such as oil and natural gas, whose depth and positional coordinates can be determined, at least in part, by the processes and systems described in this document. As the research vessel 102 moves over the subsurface formation 120, the source 104 is activated (i.e., triggered or fired) to produce an acoustic signal.Figure 1A shows an acoustic signal expanding outward from source 104 as a pressure wave field 136 represented by semicircles of increasing radius centered on source 104. The wavefronts expanding outward from the source may be spherical, but are shown in vertical planar cross-section in Figure 1A. The outward and downward expanding portion of the pressure wave field 136 and any portion of the pressure wave field 136 reflected from the free surface 112 are called the “source wave field”. The source wave field eventually reaches the formation surface 122 of the underground formation 120, at which point the source wave field may be partially reflected from the formation surface 122 and partially refracted downward into the underground formation 120, becoming elastic waves within the underground formation 120.In other words, in the body of water, the acoustic signal mainly comprises compressive pressure waves, or P-waves, while in the subsurface formation 120, the waves include P-waves and... Petition 870250102319, dated 07 / 11 / 2025, page 19 / 133 / 54 transverse waves, or S-waves. Within the subterranean formation 120, at each interface between different types of materials or at discontinuities in density or in one or more of several other physical characteristics or parameters, downward propagating waves can be partially reflected and partially refracted. As a result, each point on the formation surface 122 and each point on the interfaces 124, 126, and 128 can be a reflector or reflection point that becomes a potential secondary point source from which the energy of acoustic and elastic waves, respectively, can emanate upwards towards the receivers 118 in response to the acoustic signal generated by the source 104 and downward propagating elastic waves generated by the pressure impulse.As shown in Figure 1A, waves of significant amplitude can generally be reflected from points on or near the surface of formation 122, such as reflection point 138, and from reflection points at interfaces in or very near underground formation 120, such as reflection points 140 and 142. The upward expanding waves reflected from underground formation 120 are collectively the “reflected wave field”.

[0050] The waves that make up the reflected wavefield can generally be reflected at different times within a range of times after the initial source wavefield. A point on the formation surface 122, such as reflection point 138, may receive a pressure disturbance from the source wavefield more quickly than a point within the underground formation 120, such as reflection points 140 and 142. Similarly, a reflection point on the formation surface 122 directly below source 104 may receive the pressure disturbance earlier than a more distant reflection point on the formation surface 122. Thus, the times at which waves are reflected from various reflection points within the underground formation 120 may be related to the distance, in three-dimensional space, from the reflection points of the activated source 104. Petition 870250102319, dated 07 / 11 / 2025, page 20 / 133 / 54

[0051] Acoustic and elastic waves can travel at different speeds within different materials, as well as within the same material under different pressures. Therefore, the travel times of the source wavefield and the reflected wavefield are functions of the distance from the source 104, as well as the materials and physical characteristics of the materials through which the wavefields travel. Furthermore, the expanding wavefronts of the wavefields can be altered as the wavefronts cross interfaces and as the speed of sound varies in the medium traversed by the wavefront.The superposition of reflected waves from within the subterranean formation 120 in response to the source wavefield can be a generally complicated wavefield that includes information about the shapes, sizes, and material characteristics of the subterranean formation 120, including information about the shapes, sizes, and locations of the various reflective features within the subterranean formation 120 of interest to geoscientists.

[0052] Each 118 receiver can be a multicomponent sensor, including particle motion sensors and a pressure sensor. A pressure sensor detects variations in water pressure over time. The term “particle motion sensor” refers to a sensor that detects particle displacement, particle velocity, or particle acceleration over time. Each pressure sensor and particle motion sensor may include an analog-to-digital converter that converts time-dependent analog signals into distinct time series consisting of consecutively measured values, called “amplitudes,” separated in time by a sampling rate. The time series data generated by a pressure or particle motion sensor is called a “trace,” which may consist of thousands of samples collected at a typical sampling rate of about 1 to 5 samples per millisecond.A trace is a record of acoustic energy, such as the acoustic energy in a formation. Petition 870250102319, dated 07 / 11 / 2025, page 21 / 133 17 / 54 underground in response to the wave field of the source that passes from source 104 to the underground formation, where a portion of the acoustic energy is reflected and / or refracted and finally detected by a sensor. In general, each trace is an ordered set of distinct amplitudes of pressure or particle motion sensors, dependent on time and space, denoted by: (1) where represents a trace of pressure, particle displacement, particle velocity or particle acceleration data; t represents time; are the Cartesian coordinates) and a receiver 118; χ5 are the Cartesian coordinates of source 104; A represents pressure, particle displacement, particle velocity, or particle acceleration amplitude; is the k-th sampling time; and M is the number of time samples in the trace.

[0053] The x-coordinate location of each receiver can be determined from global position information obtained from one or more global positioning devices located along the seismographic cables, research vessel, and buoys, and from the known geometry and arrangement of the seismographic cables and receivers. The x-coordinate location of source 104 can also be obtained from one or more global positioning devices located at each source and from the known geometry and arrangement of the source elements of source 104. The source and receiver coordinates define an acquisition geometry for recording seismic data. In the following discussion, the source coordinate location is omitted. Each trace also includes a trace header not shown in Equation (1) that identifies the specific receiver that generated the Petition 870250102319, dated 07 / 11 / 2025, page 22 / 133 18 / 54 trace, the receiver and GPS spatial coordinates of the source, and may include the time sampling rate and the number of time samples.

[0054] Figure 2 shows an enlarged side elevation view 202 of the receiver 118. In this example, the enlarged view 202 shows that the receiver 118 is a multicomponent sensor comprising a pressure sensor 204 and a particle motion sensor 206. The pressure sensor may be, for example, a hydrophone. Each pressure sensor is a non-directional sensor that measures changes in hydrostatic pressure over time to produce a pressure data trace denoted by FJ. The particle motion sensors can respond to water motion. The particle motion sensors are directional sensors that detect the motion of particles (i.e., displacement, velocity, or acceleration) in a specific direction and can respond to this directional displacement of the particles, particle velocity, or particle acceleration.A particle motion sensor that measures particle displacement produces a particle displacement data trace denoted by where the vector represents the direction along which the particle displacement is measured. A particle motion sensor that measures particle velocity (i.e., particle velocity sensor) generates a particle velocity data trace denoted by . A particle motion sensor that measures particle acceleration (i.e., accelerometer) generates a particle acceleration data trace denoted by . The data generated by one type of particle motion sensor can be converted to another type. For example, particle displacement data can be differentiated to obtain particle velocity data, and particle acceleration data can be integrated to obtain particle velocity data.

[0055] The term “particle motion data” refers to data Petition 870250102319, dated 07 / 11 / 2025, page 23 / 133 19 / 54 Particle displacement, particle velocity wave field data, or particle acceleration data. The term “seismic data” refers to pressure wave field data and / or particle motion data. Pressure wave field data may also be called “pressure wave field”. Particle displacement data represents a particle displacement wave field, particle velocity wave field data represents a particle velocity wave field, and particle acceleration data represents a particle acceleration wave field. Particle displacement, velocity, and acceleration wave field data are correspondingly called particle displacement, velocity, and acceleration wave fields.

[0056] Particle motion sensors are normally oriented so that particle motion is measured in the vertical direction (i.e., φ = case where Ue rtj £ is called the vertical displacement wave field, Λ is called the vertical velocity wave field and Λ is called the vertical acceleration wave field). Alternatively, each receiver 118 may include two additional particle motion sensors that measure particle motion in two other directions, ε / L2, which are orthogonal to ε1 (i.e., ε / ε = ft, ε = U where Ue £ is the dot product) and orthogonal to each other (i.e., ε = U). In other words, each receiver 118 may include one pressure sensor and three particle motion sensors that measure particle motion in three orthogonal directions.For example, in addition to having a particle motion sensor that measures the velocity of particles in the z direction to provide each receiver, it may include a particle motion sensor that measures the wave field in the linear direction to obtain the linear velocity wave field, ,t), and a particle motion sensor that measures the wave field in the a direction. Petition 870250102319, dated 07 / 11 / 2025, page 24 / 133 20 / 54 cross line to obtain the cross line velocity wave field. In certain implementations, the receivers may be only pressure sensors, and in other implementations, the receivers may be only particle motion sensors. The three orthogonal velocity data sets form a velocity vector—^y'

[0057] Seismographic cables 106 to 111 and research vessel 102 may include electronic detection components and data processing facilities that allow the seismic data generated by each receiver to be correlated with the source location 104, absolute positions on the free surface 112, and absolute three-dimensional positions relative to an arbitrary three-dimensional coordinate system. The seismic data may be stored in the receiver and / or may be transmitted along the seismographic cables on data transmission cables to the research vessel 102, where the seismic data may be stored in data storage devices located on board the research vessel 102 and / or transmitted ashore to a seismic data processing facility.

[0058] As explained above, the reflected wave field typically arrives first at receivers located closest to the sources. The distance from the sources to a receiver is called the “source-receiver offset” or simply “offset.” A larger offset generally results in a longer arrival time delay. Traces are classified according to different source and receiver locations and are collected to form “clusters” that can be further processed using various seismic data processing techniques to obtain information about the structure of the subsurface formation. Traces can be classified into different domains, such as a common shot domain, a common receiver domain, a common receiver station domain, and a common midpoint domain. A Petition 870250102319, dated 07 / 11 / 2025, page 25 / 133 / 54 A collection of traits classified in the common firing domain is called a common firing grouping. A collection of traits classified in the common receiver domain is called a common receiver grouping.

[0059] The portion of the acoustic signal that is reflected off the water body from the subsurface formation and travels directly to the receivers is called the primary reflected wavefield or simply “primary”. Other portions of the acoustic signal that are reflected off the water body may be reflected many times between the free surface and the interfaces within the subsurface formation before reaching the receivers. These multiple reflected wavefields are simply called “multiples”. Still other portions of the acoustic signal may create head waves and dip waves within the subsurface formation before being reflected off the water body. Head waves are created when a portion of the acoustic signal traveling down through a low-velocity layer strikes a higher-velocity layer at the critical angle.Head waves travel in the higher-velocity layer parallel to an interface between layers before being reflected upward toward the formation's surface. Dip waves are created when a portion of the acoustic signal travels within a progressively compacted layer, creating a velocity gradient in which velocities increase with depth. Dip waves are continuously refracted along curved ray paths that turn upward toward the surface. The deepest point along the curved ray path is called the "inflection point".

[0060] Figure 3 shows a side elevation view of different examples of ray paths that acoustic energy travels between source 104 and a receiver 302 of seismographic cable 108. Directional arrows with different patterns 304 to 307 represent ray paths of different portions of an acoustic signal generated by source 104. Ray paths 304 Petition 870250102319, dated 07 / 11 / 2025, p. 26 / 133 / 54 to 306 represent different portions of acoustic energy that interact with the subsurface formation 120. Ray path 307 represents a portion of the acoustic energy that travels directly to receiver 302. Ray paths 305 and 306 represent acoustic energy paths that strike interface 124 and surface 122 at critical angles 308 and 309, respectively, creating headwaves that travel along ray paths 310 and 311 adjacent to interface 124 and surface 122. Ray paths 310 and 311 represent headwave paths traveling within the high-velocity layer overlain by a low-velocity layer. Ray paths 312 and 313 represent upward reflections of the acoustic energy of the head waves to receiver 302.Ray paths 314 and 315 represent different portions of the acoustic energy traveling along ray paths 305 and 306, respectively, which are reflected upwards from interface 124 and surface 122 to receiver 302. Ray path 304 reaches interface 126 of a progressively compacted layer 316, creating a vertical velocity gradient in which velocities increase with increasing depth. Curved ray path 317 represents a continuously refracted path of a dipping wave that reaches an inflection point 318 within layer 316 and is directed upwards towards receiver 302 along ray path 319. Ray path 320 represents a portion of the acoustic energy traveling along ray path 304 that is reflected upwards from interface 126 to receiver 302.Deeper penetrating acoustic energy (not shown) tends to be reflected back to surface 112, but may reach surface 112 too far away to be registered by the receivers.

[0061] Figure 4 shows an example of a common shot cluster of 400 example traces of reflected wave fields measured by receivers located along seismographic cable 108 shown in Figure Petition 870250102319, dated 07 / 11 / 2025, page 27 / 133 / 54 3. The vertical geometric axis 401 represents time. The horizontal geometric axis 402 represents channels or trace numbers, with trace “1” representing a trace of seismic data generated by a receiver located closer to the source 104 than trace “10” representing a trace of seismic data generated by a receiver located farther from the source 104. Pulses represent reflection events from an interface or surface. The distance along a trace from time zero to the location of a pulse represents the travel time of the acoustic energy output from the source 104 to an interface or surface and, eventually, to a receiver located along the seismographic cable 108. Lines with different patterns are added to represent wave fields that correspond to the example reflection events represented by corresponding ray paths with different patterns illustrated in Figure 3.For example, the pulses located along trace 6 correspond to reflection events reaching receiver 302, as represented by lines with different patterns in Figure 3. The pulses located along dashed curve 404 represent a portion of the acoustic signal generated by source 104 that travels directly to the receivers. The pulses located along dashed curve 406 represent pressure changes corresponding to acoustic energy reflected upwards from the formation surface 122, as represented by ray path 315 in Figure 3. The pulses located along dashed line 408 represent pressure changes created by head waves traveling just below surface 122, as represented by ray paths 311 and 313 in Figure 3.The pulses located along the dashed curve 410 represent pressure changes that correspond to acoustic energy reflected upwards from interface 124, as represented by ray path 314 in Figure 3. The dashed line 412 represents pressure changes created by head waves traveling just below interface 124, as shown. Petition 870250102319, dated 07 / 11 / 2025, page 28 / 133 / 54 by ray paths 310 and 312 in Figure 3. The pulses located along curve 414 represent pressure changes that correspond to acoustic energy reflected upwards from interface 126, as represented by ray path 320 in Figure 3. The pulses located along curve 416 represent a dip wave field created by a portion of the acoustic signal that is directed upwards from a vertical velocity gradient of layer 316, as represented by ray paths 317 and 319 in Figure 3.Note that, for the sake of simplicity in illustration and discussion, the example traces in Figure 4 record only a small number of reflected wave fields and do not represent other reflections and various types of random and coherent noise that are typically recorded during a marine seismic survey, such as gunshot noise, ripple noise, barnacle noise, vibration of seismographic cables, and bird noise.

[0062] Subsurface formations can also be surveyed using seafloor seismic techniques. In one implementation, these techniques can be performed with ocean-floor cables (“OBCs”) placed on or near the seabed. OBCs are similar to the towed seismographic cables described above, as OBCs include spaced receivers, such as pressure and / or particle motion sensors placed together, deployed approximately every 25 to 50 meters. In other implementations, ocean-floor nodes (“OBNs”) can be deployed along the formation surface. Each node may have pressure and / or particle motion sensors placed together. OBCs and OBNs can be electronically connected to an anchored recording vessel that provides power, instrument command, and control of the pressure and / or vertical velocity waveform field sent to the recording equipment located on board the vessel.Traces of seismic data recorded using seismographic cables, as described above, OBCs or OBNs. Petition 870250102319, dated 07 / 11 / 2025, page 29 / 133 25 / 54 can be processed as described below. Acoustic Wave Equation Parameterized in terms of Velocity and Vector Reflectivity

[0063] The equation for a density-varying acoustic wave in terms of velocity and density is given by ϋ^υίχ,ί) , / 1\ ——--VÁUJpü; V —I =(2) dt* \p(x) / where Λ is an observation point with Cartesian coordinates y,z) in a three-dimensional space; faAVd is the pressure wave field; It is the seismic velocity; AWéa density; 0 is the wave field of the source; and V is the gradient operator.

[0064] The collection of observation points forms the domain of the three-dimensional image. The acoustic wave impedance is a product of the seismic velocity and the density: = (3)

[0065] Using Equation (3) to substitute density into Equation (2) gives the acoustic wave equation in terms of seismic velocity and impedance as follows: ——--PwzaxjV- 1 I =MX, d (4)

[0066] Equation (4) can be expanded to obtain yp(X,t) - - y . -f \ . —--l·' U.)vmú vpU / J + / LUjzUJV Wx,tJ = Mx,t) (5)

[0067] Vector reflectivity is defined as =(6) Petition 870250102319, dated 07 / 11 / 2025, p. 30 / 133 26 / 54 where is the in-line reflectivity component; μ is the cross-line reflectivity component; and MU is the depth or vertical reflectivity component.

[0068] The acoustic wave equation in Equation (4) can be rewritten in terms of velocity and vector reflectivity as follows: A2(χ) j = òv.e (7)

[0069] The solution of Equation (7) is a complete pressure wave field for sharp reflection events (i.e., large slopes). The time and space derivative operator on the left-hand side of Equation (7) models the propagation of seismic waves in time and space through various materials of the subsurface formation based on the seismic velocities FXJ and vector reflectivity Z of the various materials.

[0070] The parameterized acoustic wave equation in Equation (7) provides advantages over traditional acoustic wave equations used in seismic velocity modeling and imaging: (1) The parameterized acoustic wave equation does not require the construction of a high-velocity density and / or contrast model of the subsurface formation to simulate reflection events used for iterative velocity modeling from reflections, such as FWI, and inversion-based imaging, such as LSRTM. Reflection events can be used to update velocity at depths beyond the penetration depth of transmitted waves in FWI and to refine the initial reflectivity used in both procedures. (2) The parameterized acoustic wave equation allows the generation of vector reflectivity with only a smooth velocity model.(3) The use of the parameterized acoustic wave equation to determine vector velocity and reflectivity models. Petition 870250102319, dated 07 / 11 / 2025, page 31 / 133 27 / 54 in FWI and LSRTM is computationally more efficient than traditional FWI and LSRTM, which use a first-order Bom approximation to perturbation theory.

[0071] The acoustic wave equation in Equation (7) does not require a high-velocity density or contrast field to compute simulated reflections in the modeled data. Instead, the acoustic wave equation relies on a velocity model and reflectivity (or image), which are available from earlier steps in the velocity model construction and imaging process. An acoustic wave traveling through an underground formation has a seismic velocity and a vector reflectivity at each observation point Λ. The seismic velocity represents the acoustic wave properties of a medium in terms of the speed at which acoustic waves travel within an underground formation. Each vector reflectivity component is the normalized change in impedance in a specific direction. For example, in a medium with horizontal layers, the vertical component of reflectivity is equivalent to the reflection coefficient.Seismic velocity and vector reflectivity depend on the observation point1 because the composition of the medium varies from observation point to observation point. An observation point can represent a point located along a surface of an underground formation or represent a point along an interface between two different types of rock, sediment, or fluid within the underground formation. An observation point can also represent a point within a layer of fluid or solid with a homogeneous composition.

[0072] With the parameterization in Equation (7), velocity and reflectivity are model parameters that eliminate the need to construct a density model. A velocity core of inverse scattering image-forming condition (“ISIC”) and a core of Petition 870250102319, dated 07 / 11 / 2025, page 32 / 133 / 54 ISIC impedances obtained through inverse scattering theory are combined with the acoustic wave equation to form the basis for the simultaneous velocity and reflectivity inversion described below. Simultaneous Inversion of Velocity and Stacked Reflectivity

[0073] The processes and systems described below are aimed at generating velocity and vector reflectivity of a subsurface formation from a pressure wave field recorded in a marine survey of the subsurface formation using simultaneous inversion. The velocity and vector reflectivity models are obtained with iterative FWI using the acoustic wave equation given by Equation (7) and can be used to identify features that correspond to oil and natural gas reservoirs. The velocity model alone can be used in depth migration to improve the resolution of an image of the subsurface formation.

[0074] Figure 5 shows a process for constructing velocity and vector reflectivity models of a subsurface formation from a pressure wave field recorded in a marine seismic survey of a subsurface formation. Each block represents a different module of machine-readable instructions implemented by computer, stored in one or more data storage devices and executed using one or more processors of a computer system. The construction of the velocity model may include additional modules or certain modules may be omitted or executed in a different order, depending on how the recorded seismic data are collected, the conditions under which the recorded seismic data are collected, and the depth of the water body above the subsurface formation.

[0075] In Figure 5, block 501 represents the recovery of a pressure wave field recorded in a marine survey of a subsurface formation, as described above with reference to Figures 1A to 3. Blocks 502 to 505 describe computational operations that pre Petition 870250102319, dated 07 / 11 / 2025, page 33 / 133 29 / 54 condition the pressure wave field data. In block 502, acquisition noise in the pressure wave field, such as ripple noise and barnacle noise, is attenuated. In block 503, the pressure wave field is corrected for receiver motion created by seismographic cables moving in the water body. In block 504, an iterative “full waveform inversion (“FWI”) implementation procedure is performed to simultaneously construct a high-resolution velocity model and a vector reflectivity f of the subsurface formation.” An example of implementing the iterative “full waveform inversion (“FWI”) implementation procedure to simultaneously construct a model of 1 / High-resolution velocity f and vector reflectivity f model of the underground formation” is described below with reference to Figure 6. In block 505, the acoustic wave impedance model Z and the density model p of the underground formation are calculated. In block 506, the velocity model and vector reflectivity are used to identify structural and lithological features in the underground formation that correspond to oil and gas reservoirs. The velocity and vector reflectivity of an underground formation are displayed on monitors, projected onto screens, or displayed using other visual display devices.In some cases, velocity and vector reflectivity can be used to identify the composition of features and layers within an underground formation, such as oil and natural gas accumulations, and can be used for pre-drilling pore pressure prediction, allowing geoscientists to take measures to mitigate the risks and hazards associated with drilling in high-pressure oil reservoirs. Velocity and vector reflectivity, and images generated from the pressure wave field recorded at different stages of oil or natural gas extraction from a reservoir, can be used by geoscientists to monitor oil and natural gas extraction from the reservoir over time. Petition 870250102319, dated 07 / 11 / 2025, page 34 / 133 30 / 54

[0076] Figure 6 is a flow diagram illustrating an example of implementing the “iterative full waveform inversion (“FWI”) procedure to simultaneously construct a high-resolution velocity model f and a vector reflectivity / subsurface formation model” referenced in block 504 of Figure 5. In block 601, an initial smooth velocity model, denoted by , is received as input. The initial velocity model may have been generated from previous velocity model construction and imaging processes. In block 602, traces of a recorded pressure wave field, denoted by , which were obtained as described above with reference to Figures 1A to 3, are received as input. The iterative FWI is executed by the computational operations represented by blocks 603 to 608. The iterative FWI produces a final velocity model denoted by and a final vector reflectivity f in block 609.

[0077] Figure 7 shows a high-level representation of the iterative FWI performed by blocks 603 to 608 of Figure 6. Consider an example of a three-dimensional synthetic medium. Block 700 represents the initial uniform reflectivity model of the medium, and block 702 represents the initial velocity model of different layers of the medium. For simplicity, the velocity model 702 comprises eight isotropic layers. In this example, each layer has a uniform thickness in the z-direction and represents a homogeneous layer of fluid or solid within the model 702. For example, the top layer 704 could represent a body of water, and layers located below the top water layer 704 could represent velocities of different layers of rock, sediment, or fluid within the subsurface formation volume. Each layer of the model 702 has an associated initial seismic velocity.The 702 velocity model is a representative initial velocity model of an underground formation and, for ease of illustration, has only eight layers with seismic velocities. Petition 870250102319, dated 07 / 11 / 2025, page 35 / 133 31 / 54 and corresponding reflectivity. The seismic velocities of the layers in the velocity model are denoted by s, where the superscript “0” identifies the seismic velocities of the initial velocity model 'I' and the subscript 4=Â —corresponds to the eight layers of the synthetic medium 700. In other implementations, the number of layers may be greater or less than eight. The initial reflectivity model 700 can be initialized with zero values ​​for the reflectivities at the image points. In practice, the velocities of a real underground formation may be different at each point. In Figure 7, the iterative FWI process 706 is represented by the directional arrows 708 and 710. The initial velocity model 702 and the initial reflectivity model 700 are inserted into the iterative FWI 706. Each iteration of the iterative FWI 706 updates the velocities at image points in the velocity model 702 and updates the reflectivities at image points in the reflectivity model 700.The velocities and reflectivities at each image point generated after each iteration of iterative FWI 706 are denoted by and respectively, where j is a non-negative integer used to denote the j-th iteration of iterative FWI 706. Figure 7 shows an example of a final velocity model 712 and a final reflectivity model 714 after completion of the final iteration of iterative FWI 706. For illustrative purposes, the final velocity model 712 is composed of eight layers of different velocities denoted by , where the subscript __1 is q = 1, and the final reflectivity model 714 is composed of seven n layers of different reflectivities denoted by , where the subscript is F = . As shown in Figure 7, the velocity model 712 of iterative FWI 706 and the reflectivity model 714 approximate the velocities. and reflectivities of materials within the actual underground formation.

[0078] Returning to Figure 6, each iteration of iterative FWI begins with block 603. In block 603, direct modeling is performed to calculate a synthetic pressure wave field at each subsurface point as Petition 870250102319, dated 07 / 11 / 2025, page 36 / 133 32 / 54 D17(Y tj a function of time, ' based on the j-th velocity model IZ / 1 / je in the j-th vector reflectivity J. Direct modeling can be performed with a finite differentiation method, a pseudoanalytical method, a pseudospectral method, a finite element method, a method of IZ spectral elements or a finite volume method. The J and J models are then updated simultaneously in block 608. The synthetic pressure data traces 610 at the receiver locations are denoted by ' . In block 603, for the j-th iteration, direct modeling is performed using Equation (7) as follows: -----5' UJV [x)Vp~Jix, t)j + Vpf = dyU, tj where ixt)1 is a synthetic seismic datum at the observation point Λ and time t in the synthetic medium; and is a wave field from the source at the observation point x and time t in the synthetic medium.

[0079] An acoustic wave propagates in a medium by compressing and decompressing the medium so that a small volume of the material oscillates in the direction in which the acoustic wave is traveling. The synthetic pressure wave field is the pressure wave field at the observation point Λ in the medium at time t and is determined uniquely by the acoustic wave equation in Equation (8). The source wave field is the source wave field generated by source 104 and can be obtained from near-field pressure measurements recorded using hydrophones located near source 104 at the time source 104 is activated or by modeling the source array. Direct modeling with Equation (8) in block 603 can be performed with a finite differentiation method, a pseudospectral method, a pseudoanalytical method, a finite element method, a spectral element method, or a method of Petition 870250102319, dated 07 / 11 / 2025, page 37 / 133 33 / 54 finite volumes to obtain the synthetic pressure wave field ' 610 at each receiver location in the subsurface formation. The synthetic pressure wave field obtained using direct modeling is a function of the velocity model, the vector reflectivity model, and the source wave field: py (x\ d = f Kj, s ix-, d) (9) where represents a direct modeling operator.

[0080] In certain implementations, font 104 can be considered a point source represented as follows: XÁd = — xdóVJ (10) where Á) is a time function of the source.

[0081] In this case, the synthetic pressure wave field obtained using direct modeling is a function of the velocity model, the vector reflectivity model, and the time function of the source: p^'\x'rt) = π (Π)

[0082] In block 604, a residual can be calculated for each receiver coordinate and time sample as follows: =Pj ~ P^n^k) (12) where n = 1, / V g0íncqcej0receptor; and K = 1, M is a time sample index.

[0083] The residual AFiA is the difference between the synthetic seismic data trace *7ne and the recorded pressure wave field trace Aw for each of the N receivers and for each time sample. In block 605, a residual magnitude is calculated for the j-th iteration as follows: JV M ~ y ^^^11 (13) 71-1 7=1 where II H2 is an L2 norm. Petition 870250102319, dated 07 / 11 / 2025, p. 38 / 133 34 / 54

[0084] The iterative FWI, as represented in Figure 6, for when the residual magnitude satisfies the following condition: Ψ] < * (14) where is a residual magnitude limit.

[0085] Output 609 comprises the final velocity model, which is the j-th velocity model 'j', and the final reflectivity A, which is the jjLJ-th final reflectivity J, of the final iteration of the iterative FWI.

[0086] In block 606, the adjoint migration is performed using the Equation (8) in reverse time with the source term replaced by the superposition of the residual wave field determined at each receiver location in Equation (12) as follows: (y(x)V) + ZVÃxlÃjUJ- V) (15) where is the backpropagated residual wave field; and T is the maximum recording time for the pressure wave field.

[0087] In block 607, the ISIC velocity core is calculated by KJVuj = tip. ϋ^Λχ,ΐ) —--JM οι ΰΐ — IV2ii, t) fx, t) t) &t t (16) where is the sampling rate; / 1.^1 is a lighting term; £ a field of migrated residual waves; and t)eW2{x,t) are dynamic velocity weights.

[0088] The ISIC speed core can substantially reduce or Petition 870250102319, dated 07 / 11 / 2025, p. 39 / 133 35 / 54 eliminate short wavelength components of the FWI gradient and enhance macro velocity features. In Equation (17), the migrated residual wave field (Λ A) is obtained by time reversal of the backpropagated residual wave field. The illumination term is A*)=ΣγΙΜχΖ)Ι2Δϊ at each pointΛ. The dynamic weights are designed to optimally suppress the small-scale components of the property updates. The velocity dynamic weights are calculated by minimization as follows: 1¾. ix, tj = ai giniii r A (x^py'1u, d vy (x, d A d -i- (1 — r) —2--2----- di di w2ix,t) = i where r is a test weight and —r— 1.

[0089] Similarly, the ISIC core impedance is calculated by 1 1V ---— ίv. t)1------ZKx) dl di Γ ΔΓ (17) + 2 1¾ U, tj VpA u, tJ VQj U, t) Át r where ΆχΛ)eΙΆΑ)sã0weightes dinâmica de impedância que são diferentes de edescritados na equation (16).

[0090] The ISIC impedance core can substantially reduce or eliminate long wavelength components of the FWI gradient and increase the short wavelengths associated with the impedance. In Equation (17), the migrated residual wave field is obtained by the same time reversal as the backpropagated residual wave field. The illumination term is AxJ = Σγ|5IX,tjem cajap0nt0x which is the same as used in Equation (16). The dynamic weights are designed to optimally suppress the large-scale components of the updates. Petition 870250102319, dated 07 / 11 / 2025, page 40 / 133 36 / 54 property. The dynamic impedance weights are calculated by minimization as follows: W3íx, tj = áigiiiiõ “ ix, tj t) + [1 — rj — --2---dL di = 1 - M4(xfd where r is a test weight and:

[0091] In block 608, the seismic velocity at each observation point Λ of the velocity model Ό is updated as follows: = y UÜ + dvKJvW (18) where is a constant called the “speed step length”.

[0092] Similarly, vector reflectivity is updated simultaneously as follows: = K / W -l-dR\JKzW (19) where is a constant called the “reflectivity step length”.

[0093] The ISIC velocity core in Equation (16) improves updates of the long wavelength components of the velocity model, which cannot be achieved with a gradient obtained using conventional FWI. Since the long wavelength components of the velocity model are updated with improved accuracy in subsequent FWI iterations, a conventional FWI gradient can be used to correctly position the short wavelength features of the velocity model ^J, thus further increasing the resolution of the updated velocity model output A of block 608.

[0094] Velocity and reflectivity are model parameters in the parameterized acoustic wave equation (i.e., Equation (7)), which eliminates the need to construct a density model. The Petition 870250102319, dated 07 / 11 / 2025, page 41 / 133 37 / 54 velocity and impedance cores ISIC in the corresponding Equations (16) and (17) are combined with the acoustic wave equation to form the basis for the simultaneous velocity and reflectivity inversion described below.

[0095] Figure 8A shows a plot of a conventional cross-correlation kernel produced for a model consisting of a single homogeneous layer overlying a halfspace. The locations of a source and a receiver are correspondingly denoted by S and R. The cross-correlation kernel includes low wavenumber (i.e., large-scale) components 802 and high wavenumber (i.e., small-scale) components 804 that are related to a wavelength λ of an acoustic wave (i.e., ^ / ^). The low wavenumber components 802 are the result of the cross-correlation of the descending wave fields and the backscattering produced by an interface. The low wavenumber components 802 correspond to the low wavenumber features of the velocity model. The high wavenumber components 804 can be called migration isochrones and correspond to specular reflections of the vector reflectivity.

[0096] Figure 8B shows an ISIC impedance core plot produced with a dynamically weighted impedance sensitivity core for the same model as Figure 8A. The impedance sensitivity core corresponds to Equation (17). The image illustrates that the high wavenumber 804 components are preserved while the low wavenumber 802 components are suppressed.

[0097] Figure 8C shows an ISIC velocity core plot produced with a dynamically weighted velocity sensitivity core for the same model as Figure 8A. The core velocity corresponds to Equation (16). The image illustrates that the high wavenumber 804 components present in Figure 8A are suppressed while the low wavenumber 802 components are preserved. Petition 870250102319, dated 07 / 11 / 2025, page 42 / 133 38 / 54

[0098] The workflow described below is similar to performing FWI and LSRTM, where velocity and reflectivity are updated simultaneously in each iteration. Iterative inversion compensates for incomplete acquisitions and variable illumination in a subsurface formation to provide true amplitude ground reflectivity. The final velocity model *7 and the final vector reflectivity can be used to calculate additional ground attributes, such as the acoustic wave impedance model ^xJ and the density model f^xj of the subsurface formation, which are used to evaluate the subsurface formation in search of potential oil and natural gas reserves.As described above with reference to Equation (6), the final vector reflectivity Kf^x^ is related to the relative acoustic impedance. Returning to Figure 5, in block 505, the relative acoustic impedance model ^xJ of the underground formation can be calculated from the path integration of the final vector reflectivity as follows: z L,yJ = *< J 2(rj dL) (20) Vx. / and the relative density model can be calculated by pfr J = =---—----------L(21) where Léo is the path integral that starts at a point on the surface and extends to a depth in the subsurface.

[0099] In block 505, any one or more of the following attributes can be used to identify compositions of the various features and layers within an image of the subsurface formation: the ultimate velocity model, the ultimate vector reflectivity model, the relative acoustic wave impedance model, and the relative density model. For example, one or more of the following attributes can be used to identify compositions of the various features and layers within an image of the subsurface formation: the ultimate velocity model, the ultimate vector reflectivity model, the relative acoustic wave impedance model, and the relative density model. Petition 870250102319, dated 07 / 11 / 2025, page 43 / 133 39 / 54 Identify deposits, such as natural gas and water, and identify the different types of rocks, porous materials, and sediments that may be present in the layers of the underground formation. Vector reflectivity (fy^x) provides information about the shape of the different interfaces of underground formations and can be a strong indication of potential structures that may be oil and natural gas reservoirs. The velocity model can also be used to determine the pressure within an oil deposit, allowing petroleum engineers to reduce the risks and hazards of drilling in a high-pressure oil deposit.

[00100] Reverse time migration by least squares can be applied to the recorded pressure wave field using the velocity model obtained in block 608 to improve the resolution of a vector reflectivity of the underground formation. The image or vector reflectivity of the underground formation can be displayed on a monitor or other display device to provide a visual representation of underground formation structures and features. The underground formation image can be a two-dimensional visual representation of a cross-section of the underground formation. Alternatively, the underground formation image can be a three-dimensional visual representation of the underground formation. Constructing an Angle Map from the Acoustic Wave Equation

[00101] The construction of angle groupings involves calculating an angle map denoted by where f(x) is the angle of reflection at an observation point x from a source location *, and Ψ^χ) is the azimuth angle of the observation point x from the source at x. The acoustic wave equation in Equation (7) provides the elements to calculate the angle of reflection &{χϊ} and the azimuth angle Ψ^χ) of the angle map for each observation point from the source location x. The angle of reflection between the incident pressure wave and the normal direction of the reflector at an observation point Λ is given by the scalar product between the vector reflectivity and Petition 870250102319, dated 07 / 11 / 2025, page 44 / 133 40 / 54 the gradient of the pressure wave field Vpt xj is in the following form: (22a>

[00102] The azimuth angle is calculated from the horizontal components of the vector reflectivity: Ψ^χ) = aTctan\--^\ (22b)

[00103] Figure 9 shows a vertical planar cross-section of a water body 902 and an underground formation 904. The horizontal line 906 represents the top surface of formation 904. Solid curves, such as solid curves 908 to 912, represent interfaces between media with different compositions. The dashed curves 914, 915, and 916 represent a pressure wave field emanating from a source (not shown) located in the water body 902 at a given time. Figure 9 shows pressure wave 914 striking bottom 906. A portion of the pressure wave is reflected upward from bottom 906 as pressure wave 916 and a portion of the pressure wave travels into the subterranean formation 904 as pressure wave 915. Dashed directional arrows, like dashed directional arrows 918 to 920, represent the pressure gradient of the pressure wave field 914 to 916.For illustrative purposes, solid arrows, such as solid arrows 922 to 924, represent the normal to the reflectivity image at points along the interfaces of the subsurface formation 904. In reality, the normals are located at all points on the interfaces and at all points in the subsurface. But for ease of illustration, the normals are shown at selected points on the interfaces. At each point in the subsurface, the angle between each normal and each gradient is the scattering angle at each time step for the shot illustrated in Figure 9.

[00104] Figure 10 shows a three-dimensional geometric representation of the reflection angle and the azimuth angle for an example of an observation point in the subsurface of a formation. Petition 870250102319, dated 07 / 11 / 2025, page 45 / 133 41 / 54 underground. In Figure 10, a curved surface 1002 represents an interface between two layers of different compositions within an underground formation. A tangent (i.e., sloped) plane 1004 is centered at observation point Λ1006 and is tangent to the interface 1002 at observation point Λ1006. The solid directional arrow 1008 represents the direction of propagation of the pressure gradient of the incident pressure wave field emanating from a source 1010 located at *Se reaches the observation point 1006. The dashed directional arrow 1012 represents the vector reflectivity ΑΧ of the pressure wave field reflected from the observation point 1006. The vector reflectivity K{x) £ is oriented perpendicular to the tangent (i.e., inclination) plane 1004 at the observation point 1006. The angle of reflection ^*7 between the pressure gradient 1008 and the vector reflectivity K{x) 1012 is calculated according to Equation (22a).The azimuth angle is calculated according to Equation (22b) and represents the projection of reflectivity. 1012 in the horizontal xy plane 1014 and is oriented in the x-direction in a line. The directional arrow 1016 corresponds to the direction in a line (x-direction) and is inside the xy plane 1014. The directional arrow 1018 represents the projection of the reflectivity KIaJ 1012 onto the horizontal xy plane 1014.

[00105] Figure 11 is a flow diagram of a process for creating angle groupings of a subsurface formation from pressure wave field data recorded in multiple shots of a marine survey. Each block represents a different module of machine-readable instructions implemented by a computer, stored in one or more data storage devices and executed using one or more processors of a computer system. A loop that begins with block 1101 repeats the computational operations represented by blocks 1104 to 1108 for each recorded pressure wave field that is recorded for each of the S number of shots in the marine survey. The Petition 870250102319, dated 07 / 11 / 2025, page 46 / 133 42 / 54 block 1102 represents a velocity model ^(xJ) and a vector reflectivity of the subsurface formation. Block 1103 represents a recorded pressure wave field registered for the s-th shot in the marine survey. In block 1104, direct modeling is performed with the velocity model and vector reflectivity using Equation (7) to generate a synthetic pressure wave field IxYJ as follows: ----—--V j + Zl / ztjj (23) = 5'íx, tj

[00106] In block 1105, a residual wave field is calculated for each receiver coordinate and time sample as follows: L^Xm J=P7íx^^J Ps^Xm^J (24) where 1,A is the receptor index; and 1, ...rM £umínc[icedeamostra de tempo.

[00107] In block 1106, adjoint migration is performed with the velocity model ^x) and the vector reflectivity ^xJ in reverse time with the source term replaced by the residual wave field as follows: 0“ — - - - , Λ — - VW7 (VCdV)-t- V) VÍYT- t) (25) where V^x^—is the backpropagated residual wave field; and T is the maximum recording time for the pressure wave field.

[00108] In block 1107, the ISIC is calculated. In one implementation, the ISIC is the ISIC velocity core given by: Petition 870250102319, dated 07 / 11 / 2025, p. 47 / 133 43 / 54 / MO = 1 dpiyn(x,t) ύψίκ,ΐϋ —- Τ^Τ^τ) Μ4^,υ—------Al £ r — H6 (x,tj Vpjy“ (x, tj \QÍx, t)àt t (26) where the dynamic velocity weights are calculated as follows: ^(χ,ί) = — rV^fxJVp^UGtJ VQ{x,t) + (1 — r)--1 dt dt ) W2(.x,t) = 1 - ^(χΛ)

[00109] In another implementation, the following expression is used to construct the angle groupings: Kz(x,xs(v, φ)) 1 \ . ^P, ix,t) ô(jÂX,t) = — 177777 ) W^X, tj —1--2----At zíw ΕΜΔΰΐtfr ut (27) ix, tj [x, t) vy;ir, t) At where the dynamic impedance weights are calculated as follows: fr3fx,tj = argiuiii|?Vzlxjvp-5u,t) Vv(x,tj . yyLxA) ) + (1 - r)----------- dt di ) = 1 - tj

[00110] The ISIC of Equation (26) is applied to calculate the gradient used for velocity, and Equation (27) is applied to calculate the angle groupings ^(χ,χ (&,φ)) once the angle map is calculated for a shot generated at the source location. There are several ways to construct the angle groupings. For example, in one implementation, the angle groupings can be calculated from the maximum amplitude for all image times calculated by Equation Petition 870250102319, dated 07 / 11 / 2025, p. 48 / 133 44 / 54 27, since the angles are calculated for each point in the image. For the first iteration, Equation 27 assumes that the residual wave field consists only of the data. In subsequent iterations, updates of the initial angle groupings are calculated from the actual residual wave field, as shown in Equation 25. In block 1108, the vector reflectivity output from the previous iteration in block 1104 is used to calculate an angle map Μ{ϋ,φ) as described above with reference to Equations (22a) to (22b). The operations represented by blocks 1104 to 1108 produce an angle grouping x1109 for the s-th shot. In decision block 1110, when an angle grouping has been calculated for each of the S shots, control flows to block 1111. In block 1111, the angle groupings for each shot are combined to form a set of global angle groupings W ? (y,çp)),K�^χ,χ (ϋ,φ))) for the entire underground formation region. Simultaneous Velocity and Reflectivity Inversion Pre-Stacking with Angle Clustering Output

[00111] RTM is a migration technique for imaging a subsurface formation from seismic data with complex seismic wave phenomena because RTM can handle combinations of structural tilt with high velocity contrasts, which are common conditions in salt basins and other geological formations with complex structures and velocity distributions. However, even with an accurate velocity model of the subsurface formation, RTM alone still produces an approximation of the true reflectivity of the subsurface formation. Furthermore, RTM alone does not compensate for the limitations associated with seismic data acquisition and variable acoustic illumination under complex overburden, such as salts or carbonates. In contrast, the simultaneous inversion performed as illustrated in Figure 11 overcomes these problems. Petition 870250102319, dated 07 / 11 / 2025, page 49 / 133 45 / 54 that RTM or other conventional migration methods fail to solve and often produces images with fewer artifacts, higher resolution, and more accurate amplitudes than conventional migration methods. In particular, in addition to obtaining a more accurate velocity field in each iteration, simultaneous inversion, as illustrated in Figure 11, performs image formation as an inverse problem with an updated vector reflectivity, thus resulting in an image of a subsurface formation that is closer to the actual reflectivity of the subsurface formation. Furthermore, pre-stacking reflectivity in the form of angle clusters, as described above, can provide important quantitative information that is useful for more accurate interpretation. In addition to inverted angle clusters, it is possible to calculate a number of other attributes used in AVO analysis.

[00112] Figure 12 shows a simultaneous inversion process to construct a velocity model and a prestacking vector reflectivity of a subsurface formation from a pressure wave field recorded in a marine seismic survey of the subsurface formation. Each block represents a different module of machine-readable instructions implemented by computer, stored in one or more data storage devices and executed using one or more processors of a computer system. The construction of the vector reflectivity may include additional modules or certain modules may be omitted or executed in a different order, depending on how the recorded seismic data are collected, the conditions under which the recorded seismic data are collected, and the depth of the water body above the subsurface formation.

[00113] In Figure 12, blocks 1201 to 1203 correspond to the same preconditioning operations represented by blocks 501 to 503 described above with reference to Figure 5. In block 1204, a Petition 870250102319, dated 07 / 11 / 2025, p. 50 / 133 46 / 54 procedure of “performing simultaneous velocity inversion and pre-stacking reflectivity”. An example implementation of the “performing simultaneous velocity inversion and pre-stacking reflectivity” procedure is described below with reference to Figure 13. In block 1205, the velocity model and angle grouping output of the simultaneous inversion process from block 1204 are attributes that can be used to identify underground formation features such as oil and natural gas deposits. The final inverted velocity model and pre-stacking angle groupings can be used for amplitude versus angle analysis and other quantitative interpretation processes of the underground formation.

[00114] Figure 13 is a flow diagram illustrating the implementation of the “simultaneous velocity inversion and pre-stack reflectivity” procedure mentioned in block 1204 of Figure 12. In the example in Figure 13, the process is performed in the data domain, which comprises time, the locations of the shot coordinates, and the locations of the receiver coordinates (i.e., t,x,x). In block 1301, an initial velocity model W is received as input. The initial velocity model may be composed of simple approximations of seismic velocities of the subsurface formation, as described above with reference to Figure 7. In block 1302, traces of the recorded pressure wave field obtained in block 1201 of Figure 12 are received as input. The simultaneous inversion is an iterative process executed in computational operations represented by blocks 1303 to 1308. Each simultaneous inversion iteration begins with block 1303.In block 1303, direct modeling is performed to calculate the synthetic wavefield composed of synthetic pressure data traces at receiver locations denoted by X'. The direct modeling is based on the initial velocity model Weuma j-th vector reflectivity κΑχ} updated. Petition 870250102319, dated 07 / 11 / 2025, page 51 / 133 47 / 54 in block 1309. Direct modeling is performed with Equation (7) based on the parameterization to determine a synthetic pressure wave field —--uj v ( u uj vpy u, d) + ot XJ / = {X, t) WX ÜÓ - VpSjynÍV, tj ) (28)

[00115] The source wave field is the source wave field generated by source 104 and can be obtained from near-field pressure wave field measurements recorded using hydrophones located near source 104 or can be calculated from modeling as described above with reference to Figure 6. Direct modeling with Equation (28) on block 1303 can be performed with a finite differentiation method, a pseudoanalytical method, a pseudospectral method, a finite element method, a spectral element method, or a finite volume method to obtain the synthetic pressure wave field to*3*^** [ y* 1 —r Π ' ' in block 1310 at each receiver coordinate location in the subsurface formation. In block 1304, a residual is calculated for each receiver coordinate and time sample, as described above with reference to Equation (12). In block 1305, a residual magnitude is calculated for the j-th iteration using Equation (13). In block 1306, adjoint migration is performed using Equation (28) in reverse time, where the source term is replaced by the superposition of the residual wave field determined at each receiver location: - Foix)V· ^0(x)V) -b ΖΙ / χχΧξΧΐ V) Qjix. í - t) (29)

[00116] In block 1307, an ISIC core velocity is calculated as described above with reference to Equation (16) and an ISIC core impedance ^z is calculated as described above with Petition 870250102319, dated 07 / 11 / 2025, page 52 / 133 48 / 54 reference to Equation (17). In block 1308, an angle map Μ{ϋ,φ) is calculated as described above with reference to Equation (22) and Equation 23. In block 1309, the seismic velocity at each observation point Λ in the velocity model Ό is updated as follows: = I7-W -i- (30) where is a constant called “speed step length”.

[00117] Angle groupings are updated as follows: φ)) -+- d tpj) (31) where is a constant called the “reflectivity step length”.

[00118] When the residual magnitude satisfies the condition in block 1305, the iterative simultaneous inversion process stops and the final velocity model (i.e., W) and the final angle groupings obtained in the most recent execution of the model update operation performed in block 1309 are output 1312 and control returns to Figure 12. Equation (30) is calculated in each iteration with the updated velocity from the previous iteration. The final inverted velocity model is the final velocity result of the complete inversion problem.

[00119] In an alternative implementation, the acoustic wave equation in Equation (7) is modified to extend to the angle domain as follows: (7 £7 L ít __L __L . — / 7 — ÚÍxJV· p)||||Vptx,tJ || lus 0 .7(52)

[00120] The angle is the reflection angle determined according to Equation (22). The parameter 11^(-^,^ II is extracted from the angle groupings and the corresponding angle and azimuth map for each shot. Returning to Figure 12, in block 1204, a procedure is performed of Petition 870250102319, dated 07 / 11 / 2025, page 53 / 133 49 / 54 “implementation of simultaneous speed reversal and pre-stacking reflectivity”. The implementation of the “implementation of simultaneous speed reversal and pre-stacking reflectivity” procedure is described below with reference to Figure 14.

[00121] Figure 14 is a flow diagram illustrating an example of implementing the “simultaneous velocity inversion and pre-stacking reflectivity” procedure referenced in block 1204 of Figure 12 based on the acoustic wave equation extended to the angle domain, as distinguished by Equation (32). In Figure 14, the computational operations represented by blocks 1404, 1407, and 1408 are the same as the computational operations represented by blocks 1304, 1307, and 1308 in Figure 13. In block 1403, direct modeling is performed to calculate the synthetic wave field composed of synthetic pressure data traces (χ' tj) at receiver locations denoted by ' . The direct modeling is based on the initial velocity model ''□W and an updated j-th vector reflectivity in block 1409. The direct modeling is performed with Equation (32) based on the parameterization to determine a synthetic pressure wave field: d^p^1* (x,t)A —--mjv- (nxjvpyxvnv J 7(33) -I- 2 UJ ||Ã;(x, 8r^IIIK u, t)|| CUi & = òy t)

[00122] Direct modeling with Equation (33) in block 1403 can be performed with a finite differentiation method, a pseudoanalytical method, a pseudospectral method, a finite element method, a spectral element method, or a finite volume method I t) to obtain the synthetic pressure wave field r / ' in block 1410 at each receiver coordinate location x in the subsurface formation. In block 1406, adjoint migration is performed using Equation (32) in reverse time, in which the source term is given by the superposition of the field Petition 870250102319, dated 07 / 11 / 2025, page 54 / 133 50 / 54 of residual waves determined at each receiver location, as described in Equation (13): ---1-^2--VWV yV UJ vüfx, i - tj J + ZixJ || / ?;A, &, φ) || || VQ / x,T - Õ II quJ 1*^7 = 'j W ti H- 1 where it is calculated, for example, from equation 22.

[00123] In block 1409, the seismic velocity at each observation point * in the velocity model is updated as follows: Vi CA = w + avKfa) (35) where is a constant called “speed step length”.

[00124] Vector reflectivity is updated simultaneously as follows: φ) = + ά^K^χ,θ,φ) (36) where is a constant called the “reflectivity step length”.

[00125] When the residual magnitude $1 satisfies the condition in block 1405, the iterative simultaneous inversion process for and the final velocity model ΆA (i.e., P / W—and the final angle grouping (i.e., φ) = AV Ά) obtained in the most recent execution of the model update operation performed in block 1409 are output 1412 and control resumes to Figure 12.

[00126] Figure 15 shows an example of a computer system that executes an efficient process to generate a velocity model and a grouping of angles. The internal components of many small, medium, and large computer systems, as well as specialized processor-based storage systems, can be described in relation to this generalized architecture, although each system Petition 870250102319, dated 07 / 11 / 2025, page 55 / 133 / 54 may present many additional components, subsystems and similar parallel systems with architectures similar to this generalized architecture. The computer system contains one or more central processing units (“CPUs”) 1502 to 1505, one or more electronic memories 1508 interconnected with the CPUs by a CPU / memory subsystem bus 1510 or multiple buses, a first bridge 1512 that interconnects the CPU / memory subsystem bus 1510 with additional buses 1514 and 1516, or other types of high-speed interconnection media, including multiple high-speed serial interconnections.Serial buses or interconnections, in turn, connect CPUs and memory with specialized processors, such as a graphics processor 1518, and with one or more additional bridges 1520, which are interconnected with high-speed serial links or with multiple controllers 1522 to 1527, such as the controller 1527, which provide access to various different types of computer-readable media, such as computer-readable media 1528, electronic displays, input devices, and other computing components, subcomponents, and resources. Electronic displays, including visual display screens, audio speakers, and other output interfaces, and input devices, including mice, keyboards, touch screens, and other input interfaces, together constitute input and output interfaces that allow the computer system to interact with human users.A 1528 computer-readable medium is a data storage device and may include, for example, electronic memory, optical or magnetic disk drive, USB drive, flash memory, and other data storage devices. A 1528 computer-readable medium can be used to store machine-readable instructions that encode the computational processes described above and can be used to store encoded data during storage operations, from which the encoded data can be retrieved. Petition 870250102319, dated 07 / 11 / 2025, page 56 / 133 / 54 during read operations, by computer systems, data storage systems and peripheral devices.

[00127] The processes and systems described in this document can be used to form a geophysical data product indicative of certain properties of a subsurface formation. The geophysical data product can be manufactured using the processes and systems described in this document to generate geophysical data and store the geophysical data in computer-readable media 1528. The geophysical data product includes geophysical data such as pressure wave field data, particle motion data, particle velocity data, particle acceleration data, and upward and downward pressure wave field data.The geophysical data product also includes multidimensional (i.e., angle groupings) data of the subsurface formation and seismic volumes, such as velocity models, vector reflectivity, partially stacked vector reflectivity by angle bands, relative impedance models, and relative density models of a subsurface formation calculated from the use of the processes and systems described in this document. The geophysical data product can be produced offshore (i.e., by equipment on the research vessel 102) or onshore (i.e., at an onshore computing facility), or both.

[00128] The simultaneous inversion workflow described above was applied to recorded pressure wave field data acquired from a marine survey of the Salar Basin, located in southeastern Newfoundland and Labrador, Canada. The marine survey comprised 16 cables with a seismographic cable separation of 100 meters and a seismographic cable length of 8 km. The primary objective of the survey was to construct a detailed, high-resolution velocity model while further defining the target fan system. The goal was to refine the velocity model and provide a reflectivity-dependent model. Petition 870250102319, dated 07 / 11 / 2025, page 57 / 133 / 54 reliable pre-stacking angle (i.e., angle groupings) for subsequent interpretation analysis. Success in achieving these objectives contributed to reducing the risks of exploration activities in the Salar Basin. In the inversion process, a maximum frequency of 40 Hz and a tomographic velocity model were used as the initial velocity model.

[00129] Figures 16A to 16F show the output of the simultaneous inversion application for the same cross-section in line (xz plane) of the Salar Basin. Simultaneous inversion significantly improved and increased the velocity resolution, as shown in Figure 16A. Figure 16B shows the final full stacking reflectivity model. Figure 16C shows the inverted angle groupings.Figure 16D shows a relative density model calculated from the final velocity (Figure 16A) and the impedance calculated from the full stack reflectivity model (Figure 16B). Figures 16E and 16F show the near-angle and far-angle stacks, respectively. These partial stacks can be used to calculate other attributes for quantitative interpretation at the reservoir level.

[00130] Figures 17A to 17F show depth slices or xy-plane sections of the same outlets described in Figures 16A to 16F of the Salar Basin at a reservoir level around 5,000 m depth. Figure 17A shows the inverted velocity model at the reservoir level. Figure 17B shows the full stack reflectivity at the reservoir level. Figure 17C shows the accumulated velocity updates overlaid on the reflectivity image. Figure 17D shows the relative density model derived from the velocity and relative impedance calculated from the full stack reflectivity. Notably, the prospecting zone showed a decrease in both velocity and density. Figures 17E and 17F show the reflectivity as a function of angles. In particular, Figure 17E shows the stack reflectivity. Petition 870250102319, dated 07 / 11 / 2025, page 58 / 133 / 54 near the reservoir level. Figure 17F shows the reflectivity of distant stacking at the reservoir level. The results highlight clear evidence of the presence of AVO, providing valuable information for estimating reservoir attributes.

[00131] It is recognized that the preceding description of the embodiments described is provided to enable any person skilled in the art to make or use the present description. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the description. Thus, the present description is not intended to be limited to the embodiments shown herein, but should have the broadest scope consistent with the principles and new features described herein. Petition 870250102319, dated 07 / 11 / 2025, p. 59 / 133

Claims

1 / 9 CLAIMS 1. A computer-implemented process for determining properties of a subsurface formation located beneath a body of water, characterized in that it uses a pressure wave field recorded during a marine survey of the subsurface formation, the improvement comprising: iteratively determining a velocity model and angle groupings of the subsurface formation based on the recorded pressure wave field and using an acoustic wave equation that models acoustic wave fields and depends on velocities and reflectivity of materials comprising the subsurface formation; and using the angle groupings to identify feature properties in the subsurface formation.

2. Process according to claim 1, characterized in that the iterative determination of the velocity model and angle groupings of the underground formation comprises iterative determination of the velocity model, an angle map and vector reflectivity of the underground formation based on the pressure wave field, the acoustic wave equation and an initial velocity model.

3. Process according to claim 1 or 2, characterized in that the iterative determination of the velocity model and angle groupings of the underground formation comprises: providing an initial velocity model of the underground formation; and iteratively, directly modeling a synthetic pressure wave field based on the recorded pressure wave field, the acoustic wave equation, the velocity model and a vector reflectivity model, Petition 870250102319, dated 07 / 11 / 2025, p.60 / 133 2 / 9 determine a residual between the synthetic pressure wave field and the recorded pressure wave field, use the acoustic wave equation to perform adjoint migration and obtain a migrated residual wave field based on the residual and a backpropagated residual wave field, determine inverse scattering image formation condition (“ISIC”) cores for velocity and ISIC impedance core based on the synthetic pressure wave field and the backpropagated residual wave field, simultaneously update the velocity model based on the ISIC velocity core, determine an angle map of the underground formation, and generate the velocity model and vector reflectivity when the residual is less than a residual magnitude threshold.

4. A process according to any one of claims 1 to 3, characterized in that the iterative determination of the velocity model and angle groupings of the subsurface formation comprises parameterization of the acoustic wave equation in terms of velocity and reflectivity.

5. Process according to any one of claims 1 to 4, characterized in that it further comprises: calculating an acoustic wave impedance model of the underground formation based on the vector reflectivity model; calculating a relative density model of the underground formation based on the acoustic wave impedance model and the velocity model; and using at least one of the angle grouping, the velocity model, the vector reflectivity, the relative acoustic wave impedance model and the density model to identify properties of the underground formation.

6. A computer system for determining the properties of a subsurface formation from a pressure wave field recorded in a marine seismic survey of the subsurface formation, characterized in that the system comprises: one or more processors; one or more data storage devices; and machine-readable instructions stored in the one or more data storage devices that, when executed using the one or more processors, control the system to perform operations comprising: simultaneously determining a velocity model and angle groupings of the subsurface formation based on the recorded pressure wave field and an acoustic wave equation that models acoustic wave fields and depends on velocities and reflectivity of materials comprising the subsurface formation; and using at least one of the velocity model and angle groupings to display the structure and lithology of subsurface formation features.

7. Computer system according to claim 6, characterized in that the simultaneous determination of the velocity model and vector reflectivity of the subterranean formation comprises iterative determination of the velocity model and vector reflectivity of the subterranean formation based on the pressure wave field, the acoustic wave equation and an initial velocity model.

8. Computer system according to claim 6 or 7, characterized in that the determination of the groundwater formation velocity model comprises: Petition 870250102319, dated 07 / 11 / 2025, page 62 / 133 4 / 9 providing an initial groundwater formation velocity model; and iteratively updating the groundwater formation velocity and vector reflectivity model in a simultaneous strategy by directly modeling a synthetic pressure wave field based on the recorded pressure wave field, the acoustic wave equation, the velocity model and the vector reflectivity, determining a residual between the synthetic pressure wave field and the recorded pressure wave field, using the acoustic wave equation to perform adjoint migration and obtaining a migrated residual wave field based on the residual and a backpropagated residual wave field,Determine inverse scattering image formation condition (“ISIC”) cores for velocity and impedance based on synthetic and recorded pressure wave fields and backpropagated residual wave field, determine an angle map of the underground formation using the direct source wave field and vector reflectivity, and simultaneously update the velocity model based on the ISIC velocity core or a conventional FWI gradient and the vector reflectivity based on the ISIC impedance core.

9. Computer system according to any one of claims 6 to 8, characterized in that the simultaneous determination of velocity and vector reflectivities of the subsurface formation comprises parameterization of the acoustic wave equation in terms of velocity and reflectivity in the image domain.

10. Computer system according to any one of claims 6 to 9, characterized in that the simultaneous determination of velocity and vector reflectivity models of the underground formation Petition 870250102319, dated 07 / 11 / 2025, page 63 / 133 5 / 9 comprises parameterization of the acoustic wave equation in terms of velocity and reflectivity in the angle domain.

11. Apparatus for determining properties of a subsurface formation from a recorded pressure wave field obtained in a marine seismic survey of the subsurface formation, characterized in that the apparatus comprises: means for determining a grouping of angles of the subsurface formation based on the recorded pressure wave field and an acoustic wave equation that models acoustic wave fields and depends on velocities and reflectivity of different materials comprising the subsurface formation; and means for displaying at least one of the angle groupings, velocity model, vector reflectivity model, in a display device, thus showing properties of the subsurface formation.

12. Apparatus according to claim 11, characterized in that the means for determining the angle grouping of the underground formation iteratively determine the velocity model and the vector reflectivity model of the underground formation based on the pressure wave field, the acoustic wave equation and an initial velocity model.

13. Apparatus according to claim 11 or 12, characterized in that the means for determining the angle grouping of the underground formation: provide an initial velocity model of the underground formation; and iteratively, perform direct modeling to obtain a synthetic pressure wave field based on the pressure wave field recorded in Petition 870250102319, dated 07 / 11 / 2025, p.64 / 133 6 / 9 Acoustic wave equation, in the velocity model and vector reflectivity, determine a residual between the synthetic pressure wave field and the recorded pressure wave field, use the acoustic wave equation to perform adjoint migration and obtain a migrated residual wave field based on the residual and a backpropagated residual wave field, determine inverse scattering image formation condition (“ISIC”) cores for velocity and reflectivity based on the synthetic pressure wave field and the migrated residual, determine ISIC cores for velocity and impedance based on the synthetic pressure wave field and the migrated residual wave field, determine an angle map of the underground formation using the direct source wave field and vector reflectivity, and simultaneously update the velocity model based on the ISIC velocity core, the vector reflectivity based on the ISIC impedance core, and the angle map.

14. Apparatus according to any one of claims 11 to 13, characterized in that means for determining vector velocities and reflectivities of the subsurface formation parameterize the acoustic wave equation in terms of velocity and reflectivity.

15. Non-transient computer-readable medium, characterized by being encoded with machine-readable instructions to enable one or more processors of a computer system to determine properties of a subsurface formation by performing operations comprising: simultaneously determining a velocity model and an angle grouping of the subsurface formation based on the recorded pressure wave field and using an acoustic wave equation that models acoustic wave fields and depends on velocities and reflectivity of materials comprising the subsurface formation; determining an image of the subsurface formation based on the pressure wave field and a velocity model; and using the image, the velocity model, the angle map and the vector reflectivity to identify composition and lithology of features in the subsurface formation.

16. Means according to claim 15, characterized in that the simultaneous determination of the velocity model and vector reflectivity of the subsurface formation comprises iterative determination of the velocity model and vector reflectivity of the subsurface formation based on the pressure wave field, the acoustic wave equation and an initial velocity model.

17. Means according to claim 15 or 16, characterized in that the simultaneous determination of the velocity model and the vector reflectivity model of the underground formation comprises: providing an initial velocity model of the underground formation; and iteratively, directly modeling a synthetic pressure wave field based on the recorded pressure wave field, the acoustic wave equation, the velocity model and the vector reflectivity, determining a residual between the synthetic pressure wave field and the recorded pressure wave field, using the acoustic wave equation to perform adjoint migration and obtaining a migrated residual wave field based on the residual and a backpropagated residual wave field, Petition 870250102319, dated 07 / 11 / 2025, p.66 / 133 8 / 9 determine inverse scattering image formation condition (“ISIC”) cores for velocity and impedance based on the synthetic pressure wave field and the backpropagated residual wave field, simultaneously update the velocity model based on the ISIC velocity core and the vector reflectivity based on the ISIC impedance core, determine an angle map of the underground formation using the direct source wave field and the vector reflectivity, and generate the velocity model and the vector reflectivity model when the residual is less than a residual magnitude threshold.

18. Means according to any one of claims 15 to 17, characterized in that the simultaneous determination of the velocity model and the vector reflectivity of the subsurface formation comprises parameterization of the acoustic wave equation in terms of velocity and reflectivity.

19. Means according to any one of claims 15 to 18, characterized in that it further comprises: calculating an acoustic wave impedance model of the underground formation based on vector reflectivity; calculating a density model of the underground formation based on the acoustic wave impedance model and the velocity model; calculating an image of the underground formation based on the velocity model and the recorded pressure wave field; and using at least one of the image, the velocity model, the angle clustering, the vector reflectivity, the acoustic wave impedance model, and the density model to identify properties of the underground formation. Petition 870250102319, dated 07 / 11 / 2025, p. 67 / 133 9 / 9 20. A method for fabricating a geophysical data product, characterized in that the method comprises: simultaneously determining a velocity model and an angle grouping of the subsurface formation based on a recorded pressure wave field and using an acoustic wave equation that models acoustic wave fields and depends on velocities and reflectivity of materials comprising the subsurface formation; calculating an acoustic wave impedance model of the subsurface formation based on vector reflectivity; calculating a density model of the subsurface formation based on the acoustic wave impedance model and the velocity model; calculating an image of the subsurface formation based on the velocity model and the recorded pressure wave field; and storing the image, velocity model, angle groupings, vector reflectivity, acoustic wave impedance, and density model in a computer-readable medium.Petition 870250102319, dated 07 / 11 / 2025, pp. 68 / 133.

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