Systems, methods, and computer-readable media for building model using SIFRIAN inversion to generate image of measured medium

Abstract

Systems, methods, and computer-readable media for imaging a measured medium using full waveform inversion are provided. The full waveform inversion uses the Sifrian functional to leverage the Hessian information, and updates (575) the model by augmenting and combining (555, 560) data derived from the Sifrian functional at the time of equilibrium. The Sifrian inversion (550) produces a high resolution image of the measured medium, which is typically seen only with full Hessian inversion, and can produce such an image without the need for the computing power of a supercomputer or very long computing time.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims priority to U.S. application serial number 17 / 806,299 filed on June 10, 2022, which was issued as U.S. Patent No. 11,551,146 on January 10, 2023, and also claims priority to U.S. application serial number 18 / 067,299 filed on December 19, 2022, which is a continuation of U.S. application serial number 17 / 806,299, the contents of both of which are incorporated herein by reference. Technical Field

[0003] The present disclosure relates to systems, methods, and computer-readable media for generating waveform images, and more particularly to using Sifrian inversion to update a model that infers properties of a propagation medium, thereby generating a waveform image. Background Art

[0004] Many different imaging techniques are known and used in a variety of industries. It is desirable that images generated using these techniques be as accurate and deterministic as possible. To meet these goals, computationally expensive techniques are used to build models that accurately incorporate the properties of the propagation medium. One way to obtain high-resolution results involves estimating the inverse of the Hessian to appropriately scale gradient-based updates. The Hessian is a square matrix of second-order partial derivatives of a scalar field. However, computing and storing the Hessian is computationally expensive and is not feasible for real-world applications. The Hessian scales on the order of N. 2 , N is the number of inversion parameters. For relatively small inversion tasks, N is usually in the order of millions to tens of millions. In addition, the Hessian is often degenerate, and its inversion has N even with regularization. 3 computational cost. In addition, the Hessian often contains negative eigenvalues, which undermines the cost minimization process. As a result, many traditional methods that wish to use the Hessian require the use of truncation or approximation methods to generate an approximate scaling matrix representing the Hessian or its inverse. Images generated using such approximate scaling matrices are easy to compute and implement, but may lack sufficient resolution. For example, when the approximate scaling matrix is ​​used to model subsurface formations, the large amount of approximation may produce a scaling that performs better than identity, and the resulting depth resolution may not enable users of the image to identify items of interest, such as oil, gas and other natural resources or general anomalies.

[0005] Therefore, it is necessary to develop better imaging techniques and equipment to fully utilize the Hessian information to study the structure of the propagation medium being measured. Summary of the invention

[0006] In some embodiments, a method for waveform imaging is disclosed herein. A computing system receives observation data from a measured medium through a plurality of sensors. The computing system generates a model or an image of the measured medium. The generation includes obtaining a model, generating synthetic data using an available model, comparing the synthetic data with the observed data, determining that the observed data and the synthetic data are not within a matching threshold, and based on the determination, iteratively updating the model using Sifrian inversion until the synthetic data matches the observed data within the matching threshold. The iterative update includes augmenting the wavefield equations, adjoint wavefields, and gradient definitions of the available model by adding a first augmented variable, a second augmented variable, and a third augmented variable to each of the wavefield equations, adjoint wavefields, and gradient definitions, wherein the cost function that quantifies the mismatch between the synthetic data and the observed data is strictly omitted, and using the first augmented variable, the second augmented variable, and the third augmented variable to combine the augmented wavefield function, the augmented adjoint wavefield, and the augmented gradient definition to create a Sifrian functional, and performing an updated variational perturbation and characterization update. The model is updated based on the output from the Sifrian functional. The computing system generates a final model or a final image of the measured medium.

[0007] In some embodiments, a device for waveform imaging and model building is disclosed herein. The device includes an interface and a data processing unit. The interface is configured to accommodate an initial model and observation data of a measured medium. The data processing unit is connected to the interface. The data processing unit is configured to generate synthetic data. The data processing unit is further configured to compare the synthetic data with the observation data. The data processing unit is further configured to determine that the observation data and the synthetic data are not within a matching threshold. The data processing unit is further configured to iteratively update the initial model using Sifrian inversion based on the determination until the synthetic data matches the observation data within the matching threshold. The iterative update includes augmenting the wavefield equations, adjoint wavefields, and gradient definitions of the initial model by adding a first augmented variable, a second augmented variable, and a third augmented variable to each of the wavefield equations, adjoint wavefields, and gradient definitions, wherein a cost function that quantifies the mismatch between the synthetic data and the observation data is omitted, and the augmented wavefield function, the augmented adjoint wavefield, and the augmented gradient definition are combined using the first augmented variable, the second augmented variable, and the third augmented variable to create a Sifrian functional, and perform variational characterization, suppression, and analysis of second-order updates. The data processing unit is further configured to output a model based on iteratively updating the initial model until the synthesized data matches the observed data within a matching threshold. The data processing unit is further configured to generate an image or model of the measured medium.

[0008] In some embodiments, a method for waveform imaging and model building is disclosed herein. The method includes collecting observation data of a measured medium from a plurality of sensors. The method further includes obtaining an initial model and synthetic data derived from the initial model. The method further includes iteratively updating the model until the synthetic data matches the observed data within a matching threshold. The model includes a wavefield equation, an adjoint wavefield, and a gradient definition. Iteratively updating the model includes using Sifrian inversion. Sifrian augments the wavefield equation, the adjoint wavefield, and the gradient definition by adding corresponding first, second, and third augmented variables to each of the wavefield equation, the adjoint wavefield, and the gradient definition, wherein a cost function that quantifies the mismatch between the synthetic data and the observed data is omitted. Sifrian combines the augmented wavefield function, the augmented adjoint wavefield, and the augmented gradient definition; and performs a variational characterization of a second-order update. The method further includes generating an image of the measured medium using the updated model, wherein the image of the measured medium can be used to locate objects suitable for further exploration.

[0009] This overview is provided to summarize some example embodiments, thereby providing a basic understanding of some aspects of the subject matter described in this document. Therefore, it should be understood that the features described in this overview are merely examples and should not be interpreted as narrowing the scope or spirit of the subject matter described herein in any way. Unless otherwise stated, features described in the context of one example may be combined or used with features described in the context of one or more other examples. Other features, aspects, and advantages of the subject matter described herein will become apparent from the following detailed description, drawings, and claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] This patent or application file contains at least one drawing printed in color. Copies of the patent or patent application publication in color will be provided by the Office upon request and payment of the necessary fee.

[0011] In order to be able to understand the above-mentioned features of the present disclosure in detail, the present disclosure briefly summarized above may be described in more detail with reference to the embodiments, some of which are shown in the accompanying drawings. However, it should be noted that the accompanying drawings only illustrate typical embodiments of the present disclosure and are therefore not to be considered as limiting the scope thereof, as the present disclosure may allow other equally effective embodiments.

[0012] Figure 1 An example receiver network is shown according to an example embodiment.

[0013] Figure 2 is a block diagram illustrating a computing environment according to an example embodiment.

[0014] 3 is a schematic diagram of data acquisition of a measured medium to generate an image of the medium from observation data according to an example embodiment.

[0015] FIG. 4 shows a schematic diagram of a conventional process for building a model for generating an image of a measured medium according to an example embodiment.

[0016] FIG. 5 shows a schematic diagram of a process for building a model for generating an image of a measured medium according to an example embodiment.

[0017] Figure 6 is a flow chart illustrating a method of generating an image of a measured medium according to an example embodiment.

[0018] Figure 7 to Fig. 9 Illustrative results comparing an image generated using Sifrian inversion according to an example embodiment with an image generated using conventional techniques are shown.

[0019] Fig.10 is a block diagram of a special purpose computer system according to an example embodiment.

[0020] The above and other aspects of the present disclosure, its nature and various features will become more apparent from the following detailed description considered in conjunction with the accompanying drawings, wherein like reference numerals refer to the same parts throughout. DETAILED DESCRIPTION

[0021] Disclosed herein are systems, methods, and computer-readable media for using a waveform imaging method in the frequency domain that allows for the complete and efficient use of second-order derivatives to infer propagation medium properties of a measured medium. A Lagrangian-like function referred to herein as Sifrian can be used to fully exploit the full second-order Hessian tensor to perform inversion and infer wavefield properties of the measured medium. In some embodiments, a model can be updated using Sifrian inversion and intrinsic simplification, which can be used to generate high-resolution images of the measured medium. Sifrian inversion can be used to resolve image resolution issues observed when conventional methods are used to process data acquired for a measured propagation medium.

[0022] In some embodiments, a method for waveform imaging detection is provided. The method may include collecting observed data of a measured medium from a plurality of sensors, obtaining an initial model and synthetic data derived from the initial model, and iteratively updating the model until the synthetic data matches the observed data within a small predetermined threshold. The model and the inverse problem can include wavefield equations, cost functions, and gradient definitions. The model can be iteratively updated based on a Sifrian function, which augments the wavefield equations, adjoint wavefields, and gradient definitions by adding corresponding first, second, and third augmented variables to each of the wavefield equations, adjoint wavefields, and gradient definitions, wherein the cost function that quantifies the mismatch between the synthetic data and the observed data is strictly not included in the Sifrian function. Sifrian combines augmented wavefield equations, augmented adjoint wavefields, and augmented gradient definitions, and is equal to zero at equilibrium. This invalidity extends to derivatives. The method can include generating an image of the measured medium using an updated model, wherein the image of the measured medium can be used to locate an object suitable for further detection.

[0023] In some embodiments, a method for waveform imaging is disclosed herein. A computing system receives observation data from a measured medium through a plurality of sensors. The computing system generates an image or model of the measured medium. The generation process requires an initial model as a preliminary guess at the properties of the medium. The generation further includes creating synthetic data and comparing it with the observed data, and determining whether the mismatch is within a matching threshold. The Lagrangian-like function referred to here as Sifrian is used to fully utilize the full second-order Hessian tensor to perform inversion and infer the wavefield properties of the measured medium. The embodiments discussed here use Sifrian inversion and inherent simplification to update the model, which can be used to generate a high-resolution image of the measured medium. Sifrian inversion solves the image resolution problem observed when using traditional methods to process data acquired for the measured propagation medium. The generation process further includes iteratively improving the model using the updates provided by the Sifrian inversion until the synthetic data matches the observed data within a preselected matching threshold.

[0024] The Sifrian inversion process is based on augmenting the wavefield equations, adjoint wavefields, and gradient definitions. Augmentation is performed by adding a first augmented variable, a second augmented variable, and a third augmented variable to each of the wavefield equations, adjoint wavefields, and gradient definitions and collecting the products into the same structure. The cost function that quantifies the mismatch between the synthetic data and the observed data is strictly omitted in the augmentation process. The change in the composite augmented structure, named the Sifrian functional, is the characteristic of the update sought. The generation further includes solving the Sifrian characterization, calculating the appropriate update steps, and synchronizing the model throughout the system starting from the initial model until the synthetic data matches the observed data within a predefined matching threshold. The final output of the computing system is an image or model of the measured medium, or any intermediate byproduct of the iterative process and the Sifrian inversion.

[0025] In some embodiments, a device for waveform imaging is disclosed herein. The device includes an interface and a data processing unit. The interface is configured to adapt / the interface is configured to accommodate the input of the initial model and store the observed data from the measured medium. The data processing unit is connected to the interface and carries the model update process and any underlying synchronization. The data processing unit is configured to generate synthetic data by passing an estimate of the energy source used to the initial model. The data processing unit is further configured to compare the calculated synthetic data with the collected and stored observed data. The data processing unit is further configured to determine that the observed data and the synthetic data are not within a matching threshold in order to continue the update sequence. The data processing unit is further configured to iteratively update the initial model and subsequent updated versions based on mismatch quantification using Sifrian functional characterization until the synthetic data matches the observed data. The iterative update is generated by the data processing unit by adding the first augmented variable, the second augmented variable, and the third augmented variable to each wave field equation to construct the Sifrian functional to augment the wave field equation, the adjoint wave field, and the gradient definition of the initial model. A cost function that quantifies the mismatch between the synthetic data and the observed data is strictly omitted in the Sifrian functional construction. The iterative update further includes a variational perturbation of Sifrian to generate a system that represents the sought-after update. The data processing unit is further configured to include a solver for the Sifrian representation and output and an improved model that reduces the mismatch between the synthetic data and the observed data. The data processing unit is further configured to iteratively run until a matching condition between the observed data and the synthetic data is met.

[0026] The Hessian or Hessian matrix is ​​a square matrix of the second-order partial derivatives of a scalar-valued function or scalar field. The Hessian describes the local curvature of a multivariate function.

[0027] The Hessian tensor is a rank-2 covariant symmetric tensor, or a second-order derivative.

[0028] Receiver geometry can refer to the positioning of one or more waveform sensors relative to the medium being measured and relative to one or more waveform sources. Receiver geometry can be two-dimensional or three-dimensional. Waveform sensors are designed to capture data related to waveform signals emitted from waveform sources. Waveform sources are used to transmit waveforms, or can correspond to natural phenomena, just like sources.

[0029] The waveform can be a mechanical wave, such as seismic waves, sound waves, acoustic waves, and elastic waves, or an electrical wave, such as electromagnetic waves.

[0030] Figure 1 An example receiver network 100 is shown according to an example embodiment. As shown, the receiver network 100 can be deployed below the water surface 110 to collect data on a measured medium 120 (e.g., the seafloor) including an object of interest 125 (e.g., a natural resource such as oil or gas). The receiver network 100 can include waveform sensors 101 to 104 and a waveform source 105. The positioning of the waveform sensors 101 to 104 and the optional location of the waveform source 105 can define the receiver geometry. A vessel 130 (e.g., a ship or buoy) can position the receiver network 100 at a desired location or move the network 100 to the surface of the water. The waveform source 105 can emit a waveform that penetrates and passes through the measured medium 120 before being detected by the waveform sensors 101 to 104 that generate observation data.

[0031] Figure 2 2 is a block diagram illustrating a computing environment 200 according to an example embodiment. The computing environment 200 may include a computing system 202 in communication with the receiver network 100. In some embodiments, the computing system 202 may communicate with the receiver network 100 via a network 205.

[0032] The network 205 may include any type of computer network arrangement for exchanging data. For example, the network 205 may be the Internet, a private data network, a virtual private network using a public network, and / or other suitable connections that enable components in the computing environment 200 to send and receive information between components of the computing environment 200.

[0033] The computing system 202 may be configured to process sensor data generated from the receiver network 100. The computing system 202 may include an image generation module 210. The image generation module 210 may be composed of one or more software modules. One or more software modules are a collection of codes or instructions stored on a medium (e.g., a memory of the computing system 202) that represent a series of machine instructions (e.g., program code) that implement one or more algorithm steps. Such machine instructions may be actual computer code that the processor of the back-end computing system 104 interprets to implement the instructions, or alternatively, may be a higher-level encoding of instructions that is interpreted to obtain the actual computer code. One or more software modules may also include one or more hardware components. One or more aspects of the example algorithm may be performed by the hardware component (e.g., circuit) itself, rather than as a result of the instructions.

[0034] The image generation module 210 can be configured to generate an image of the measured medium. For example, the image generation module 210 can generate an image of the measured medium using sensor data generated using the receiver network 100. To generate an image of the measured medium, the image generation module 210 can construct a model that is optimized to generate an image of the measured medium.

[0035] To build a model, the image generation module 210 can start with an initial model m0 for analyzing the observed data, and iteratively build or improve the initial model until the initial model reaches a threshold level of accuracy. The output model (e.g., output model 216) can be an optimized model capable of producing a high-resolution image of the measured medium.

[0036] The image generation module 210 may select or generate an initial model m0 to begin the optimization process. In some embodiments, the initial model may be based on an initial guess at the properties of the measured medium. In some embodiments, the initial model may be obtained through tomography, previous inversion work, and / or expert information about the material. If no information is available, the image generation module 210 may test multiple initial models. In some embodiments, the initial model may be represented by spatial coordinates (e.g., 2D or 3D). The data for the initial model may be represented as a time series of scalars or multiple components (i.e., vectors) and may be organized with reference to the locations of wavefield sources and wavefield sensors or receiver geometry.

[0037] The image generation module 210 can perform a forward modeling process. For example, the image generation module 210 can input the observation data d obs , can be provided as the input of the initial model m0. The initial model m0 can be based on the observed data d obs Generate modeling data d. Modeling data d can be used to obtain observation data d obsIn some embodiments, the modeling data d can be represented as the result of a forward modeling matrix operator F. The operator F may depend on the model m, the angular frequency ω. The operator F may be referred to as a forward propagation operator F(ω,m). The operator F may be represented as a matrix. In some embodiments, the forward modeling may be performed in the time domain and then transformed to the frequency domain using a Fourier transform.

[0038] In general, the modeled data generated by the initial model m0 may be different from the observed data d obs . This is because the initial model is not the real model, or the "real data". Depending on various factors, the operator F(m) can take different forms, such as but not limited to finite differences, finite element discretization methods or any other discretization method. The operator F(m) can also depend on the physical problem considered (acoustics, electromagnetics, seismology, etc.).

[0039] The image generation module 210 can combine the modeling data d with the observation data d obs The image generation module 210 can use these errors in the reverse modeling process. In the reverse modeling process, the image generation module 210 can generate a companion wavefield of the forward wavefield, that is, a known reverse wavefield.

[0040] In some embodiments, the image generation module 210 may further cross-correlate the forward wavefield and the reverse wavefield to generate gradients. The forward modeling step 525 and the reverse modeling step 540 produce forward and reverse operators F and F*, and forward and reverse sampling operators R and R*. The receiver geometry may be represented by RR*.

[0041] After collecting data from the sensors, the Sifrian system combines the wave field u with its companion λ and a derivative kernel This operation requires a device that can perform both forward and backward propagation in addition to the cascade.

[0042] As shown, the image generation module 210 may include an inversion module 212. The inversion module 212 may be configured to generate a Sifrian inversion for updating the initial model m0 and / or the subsequently updated model m. The Sifrian inversion process solves the image resolution problem observed when using a conventional scaling matrix to process data acquired for the measured propagation medium.

[0043] The inversion module 212 may include a preprocessing module 214. The preprocessing module 214 may be configured to preprocess the observation data in a manner compatible with the Sifrian inversion. For example, the preprocessing module 214 may include a gradient combination process, a wavefield expansion process, and an operator combination process.

[0044] The preprocessing module 214 may utilize an adjoint method. For example, the preprocessing module 214 may augment the gradient definition and the wavefield equations. In some embodiments, the augmentation may be performed by adding variables to each of the corresponding gradient definition, wavefield equations, and adjoint wavefields. For example, the preprocessing module 214 may augment the gradient definition within an auxiliary variable n and then combine by copying a data structure containing the gradient g and its conjugate g*. The augmented gradient definition may be expressed as:

[0045]

[0046] Where g is the gradient, ω is the frequency, is the first derivative of the reverse operator with respect to the model, λ is the adjoint, u* is the conjugate wavefield, and n is the augmented variable. The wavefield equation can be augmented by including the forward operator in the data structure and adding an additional variable γ. The augmented wavefield can be expressed as:

[0047] Wavefield augmented =∑ w <F(m)u-s,γ>

[0048] Where F is the forward operator, m is the model, u is the wavefield, s is the source, and γ is the augmented variable. The adjoint wavefield can be augmented with the reverse operator, the forward and reverse sampling operators, and the additional variable ζ. The augmented wavefield can be expressed as:

[0049]

[0050] Where ω is the frequency, is the first derivative of the cost function with respect to the wavefield, F* is the inverse operator, λ is the adjoint, and ζ is the augmentation variable. All these elements are injected into a data structure.

[0051] Once the data is preprocessed and the elements are injected into a single data structure, the image generation module 210 can use Sifrian inversion to invert the single data structure. In some embodiments, the inversion can be at one or several frequencies. For example, an augmented single frequency inversion process can be used because it combines efficiency with implicit regularization caused by preferential optimization of low frequencies. The forward operator, source, and cost function can be any choice and generally depend on the type of waves used for imaging and inversion.

[0052] In some embodiments, before inverting the data, the image generation module 210 may perform a full Newton directional operation. The Newton directional operation may represent a description of the balanced Hessian using augmented variables (γ, ζ, n) without performing the calculations necessary for the full Hessian.

[0053] The image generation module 210 may then update the initial model m0 with the results of the Sifrian inversion. The image generation module 210 may continue this iterative process until an optimized model m is generated. * .

[0054] Fig. 3 is a schematic diagram of data collection of a measured medium to generate a medium image from observation data. The left side 301 corresponds to the extraction of observation data related to the measured medium, and the right side 302 corresponds to the processing of the observation data to generate an image of the measured medium.

[0055] Figure 4 shows a schematic diagram of a conventional process for building a model for generating an image of a measured medium according to an example embodiment. Conventional image generation schemes use gradient-based inversion methods in the time domain or the frequency domain.

[0056] The observed data are used as targets to be matched with the output of the modeling step. After the gradient g is calculated, the initial model is iteratively updated and optionally scaled with a second-order tensor that approximates the inverse of the curvature matrix. When the error (misfit) is small, the iterative inversion stops and the resulting model is considered the final inversion result. Well-known methods in this category are gradient descent, truncated Gauss-Newton, BFGS, and Hessian Free + Conjugate Gradient (HF+CG) methods. Cross-correlation corresponds to multiplication with conjugate in the frequency domain. Forward modeling computes the response of the model excitation using a waveform source at the sensor location. The discretization of wave physics yields different matrix operators, depending on the chosen method, such as finite elements or finite differences, and different modeling parameters. Inverse modeling is the application of the adjoint operator of forward modeling and allows the back propagation of the misfit (error). An "educated" initial model may be necessary to start the iterative construction process, and often minor updates in the step selection process are implemented to avoid divergence problems.

[0057] As shown in Figure 4, without any approximation or truncation or special optimization steps to guess the appropriate update, the traditional method cannot fully exploit the benefits of the curvature information. The traditional method cannot utilize the exact second-order information. The traditional method must rely on approximation or truncation of the large-scale inversion. This is a limitation that leads to poor image quality and resolution.

[0058] One or more techniques described herein improve upon this conventional approach by accessing the total usage of the Hessian information. For example, the computing system 202 may pre-process the observation data in a manner that allows the use of a Sifrian inversion. The Sifrian inversion is the entire curvature inversion process (Hessian) without any truncation or approximation. Although the embodiments discussed herein may have a computational cost twice as high as conventional approaches due to the damping required to ensure a full rank inversion, this additional computational cost is compensated by higher results in terms of higher resolution, particularly for long-range / deep survey media. The term "long / deep" may vary from a few centimeters for ultrasonic imaging to a few meters / kilometers for subsurface terrain imaging.

[0059] For the measured medium, after being excited by the source, the wave propagation response is recorded by several sensors. This recorded data is called observation data here, d obs . The observation data are fed to the Sifrian image generation module, which performs several preprocessing steps to condition the data for Sifrian inversion. The preprocessing steps may include a data collection phase, a data augmentation phase, and a data combination phase. The data collection, data augmentation, and data combination phases make the data compatible with the Sifrian functional, which implements the full curvature inversion process (Hessian) without any truncation or approximation. The attenuation of the degenerate curvature is performed in the subspace derived for the augmented data.

[0060] FIG. 5 shows a schematic diagram of a process for building a model for generating an image of a measured medium according to an example embodiment.

[0061] For a given measured medium, at step 501, observation data may be collected by a sensor that detects wave propagation originating from a source. The observation data may be provided to the image generation module 210. As described above, and described in more detail below, the image generation module 210 may iteratively perform a series of steps to build a model that can produce a high-resolution image of the measured medium. In some embodiments, the series of steps may include a forward modeling step 525, a comparison step 530, an error determination step 535, a reverse modeling step 540, a cross-correlation step 545, and a Sifrian inversion step 550.

[0062] The image generation module 210 can be based on the observation data d obs Iteratively build the optimized model m * For example, the image generation module 210 may select an initial model m0 to start constructing a model m *An iterative process. The initial model m0 can be based on a preliminary guess at the properties of the measured medium or can be obtained through tomography, previous inversion work, or expert information on the materials. If no information is available, a number of constant initial models can be tested. The initial model can be expressed in terms of spatial coordinates (e.g., 2D or 3D). The data can be represented as a time series of scalars or multiple components (i.e., vectors) and organized with reference to the locations of wavefield sources and wavefield sensors or the receiver geometry.

[0063] Due to the collection process, the observed data d obs Random and coherent noise can be filtered out with minimal amplitude and phase changes. This filtering step is a standard procedure, depending on the type of data being acquired. Filtering can occur before the observation data is transmitted to the image generation module 210. If a synthetic reference is used, the filtering process may not be necessary.

[0064] In the forward modeling step 525, the image generation module 210 can use the initial model m0 to generate modeling data d. The modeling data d can be used to obtain the observation data d. obs The modeled data d can be expressed as the result of a forward modeling matrix operator F. The operator F may depend on the model m and the angular frequency ω, and is often referred to as the forward propagation operator F(ω,m), and in practice is often expressed as a matrix. Forward modeling can also be performed in the time domain and then converted to the frequency domain via a Fourier transform. The data d produced by the initial model m0 is usually different from the observed data d obs , since the initial model is not the real model, also called the "ground truth". The operator F(m) can take different forms according to the discretization method of finite differences, finite elements or any other discretization method. The operator F(m) can also depend on the physical problem considered (acoustics, electromagnetics, seismic, etc.). For further explanation, Maxwell's equations (Equation 1) and the acoustic wave equation (Equation 2) are given below as main examples of physical phenomena that can be modeled by the Sifrian inversion method according to the embodiments discussed here.

[0065]

[0066] Equation (1) shows Maxwell's equations for dissipative materials. E and H are the electric and magnetic fields, respectively, and M s and J s They act as magnetic source and power source respectively, (μ, ε, σ) are the properties of the propagation medium, i is a complex unit, and ω is the angular frequency.

[0067]

[0068] Equation (2) shows the acoustic wave equation with variable density p. The pressure wave field is P, K is the bulk modulus, and S is the source term.

[0069] Wave physics in the frequency domain is similar to equations (1) and (2), which for a large range of wave equations can be summarized as the wavefield response u to an excitation from a source s linked by a forward operator F. Equation (3) can represent a general model for the propagation of wave phenomena in the frequency domain:

[0070] F(x,ω,m)u(x,ω)=s(x,ω) (3)

[0071] Where x is the spatial coordinate, ω is the angular frequency, and m is the model.

[0072] Starting from the model m, forward propagation can generate a wave field u, which is used to recover the observation data d through the sampling operator R obs Equation (4) shows the receiver data sampling of the wavefield response:

[0073] Ru(x,ω)=d(ω) (4)

[0074] The inversion process creates a model m whose receiver data are consistent with the observation d obs For example, the cost function can be l 2 norm, and is used to quantify d and d obs The mismatch between is calculated and it is iteratively minimized using a gradient-based approach, as shown in the cost function of Equation (5):

[0075]

[0076] Starting from the initial model m0, several model updates are established according to the following general rule expressed in equation (6):

[0077]

[0078] The gradient is the local sensitivity of the cost function to model perturbations and is a major factor in model construction. It also needs to be rescaled by the matrix B. The scaling matrix can be the constant equation for first-order methods (i.e. gradient descent) or the inverse of the curvature, also known as the (Hessian), for second-order methods (Newton-type methods). The calculation of the gradient is well established via the adjoint state method, sometimes referred to as backpropagation. The forward equation (3) needs to be differentiated to establish a compact formula for the gradient that includes the backpropagated mismatch and cross-correlates them with the wavefield response u. The expression for the gradient using the Fréchet derivative is shown in equation (7):

[0079]

[0080] This amount is called the Fréchet derivative, and it is expensive to compute, but it can be circumvented by differentiating the forward model (Equation 3). Equation (8) shows the differentiation of the forward equation:

[0081]

[0082] Replacing the Fréchet derivative is crucial in simplifying the gradient computation to the cross-correlation between the wavefield u and the adjoint wavefield derived from the mismatch of the cost function. Equation (9) shows the gradient as the cross-correlation product:

[0083]

[0084] Besides the gradient, the scaling matrix is ​​the second factor that plays an important role in model building. Ideally, the scaling matrix should be the inverse of the Hessian in convex environments / convex settings. There are several methods in the literature to approximate the scaling matrix. However, all are based on truncation and approximation; the Hessian is usually degenerate (several zero eigenvalues) and concave in direction, corresponding to negative eigenvalues. The non-positive eigenvalues ​​of the Hessian ruin the whole process of cost minimization. Moreover, computing and storing the curvature (Hessian) is computationally expensive and infeasible outside of toy models. The scale of the Hessian is N 2 , where N is the number of inversion parameters. For medium-sized inversion tasks, N is usually around one million to tens of millions.

[0085] Traditional methods enforce the positivity of the scaling matrix B and avoid its direct computation (or storage) through essentially two types of methods. The first includes BFGS-type updates, which approximately approximate the scaling matrix in each iteration through a rank-two matrix update. The convergence and properties of BFGS-type methods are well studied and established. The second type of traditional methods, often called Hessian-Free, use an adjoint state method to characterize the product of the Hessian with a preselected update vector, and then add a search method, usually conjugate gradient, to find an update vector that approximately reconstructs the gradient after multiplication with the Hessian. Given that the Hessian is usually not positive (has negative eigenvalues), a variant of the Hessian-Free method is often used, to which the Gauss-Newton approximation applies, as well as the same damping (i.e., adding a small factor εI) to ensure strict positivity of the scaling matrix. The above two traditional methods for determining the scaling matrix avoid forming a curvature matrix and only store the result / update vector. The general workflow of the traditional methods is shown in Figure 4.

[0086] Referring back now to FIG. 5 , the image generation module 210 can utilize a preprocessing step to characterize second order directions, sometimes referred to as Newton directions, to enable the use of Sifrian inversion, which is different from and performs better than conventional inversion methods. The preprocessing step utilizes an adjoint method. One preprocessing step uses Lagrangian quantities for gradient calculation and extraction of adjoint systems. Equations (10A and 10B) show the gradient derived from the Lagrangian functional:

[0087]

[0088] The Lagrangian is the sum of the inner products between the cost function and the forward model (Equation 3) and the newly introduced variable λ. λ is called the adjoint wavefield and if it verifies Equation 10A (also called the adjoint equation), the calculation of the gradient g is significantly simplified and is given by Equation (10B).

[0089] The second-order update is established using a Lagrangian-like functional, called Sifrian, which is expressed by equation (11):

[0090]

[0091] Sifrian is a Lagrangian-like functional that is zero in equilibrium (i.e. the propagation equations are verified). Sifrian uses the forward and inverse equations and the gradient definition. Sifrian introduces three new adjoints (γ,ζ,n) to characterize the expression of the Newtonian directions Hn=g. These adjoints are augmented variables attached to the wavefield equations, the adjoint wavefield equations and the gradient definition. The HessianH is neither formed nor explicitly expressed, only its effect is exploited to recover the Newtonian directions, which in turn yields the gradient g. If the forward equations, the inverse equations and the gradient definition are verified, then Sifrian is in equilibrium, it is equal to zero.

[0092] If the forward equation, the inverse equation, and the gradient definition are verified, then the Sifrian is zero. As with the Lagrangian, differentiation leads to a block matrix structure, preferably solved for a unique frequency, and produces n as a second-order update. The block structure showing the Sifrian system is detailed in equation (12) below:

[0093]

[0094] The modeled data d is provided to an error determination step 535 so that the mismatch errors between the modeled data d and the observed data can be determined. These errors can be provided to an inverse modeling step 540, which produces a companion to the forward wavefield, called the inverse wavefield. At step 545, the forward wavefield and the inverse wavefield can be cross-correlated to produce a gradient g. The forward modeling step 525 and the inverse modeling step 540 can produce forward and inverse operators F and F*, and forward and inverse sampling operators R and R*. The receiver geometry can be represented by RR*. After collecting data from the sensor, the Sifrian system combines the wavefield u with its companion λ and a derivative kernel This operation requires a device that can perform forward and backward propagation in addition to cascading.

[0095] Step 555 can implement data combination using equation (13):

[0096]

[0097] The combination of wave operators completes the expanded wave field.

[0098] Step 555 can implement operator combination using equation (14):

[0099]

[0100] The Sifrian system can be recombined with combined data and combined operators. In some embodiments, the Sifrian system may be suppressed. The resolution of the Sifrian system produces new types of updates related to model construction. The efficiency of this method stems from the augmentation and combination steps, which require appropriate allocation of computational structures to perform the Sifrian inversion and the simplifications therein. The inversion can be performed at one or several frequencies, but the augmented single-frequency inversion process is preferred because it combines efficiency with implicit regularization caused by preferential optimization of low frequencies. The forward operator, source, and cost function can be of any choice and generally depend on the type of waves used for imaging and inversion.

[0101] The forward wavefield, reverse wavefield, forward operator, reverse operator, forward sampling operator, reverse sampling operator and gradient can be used in the Sifrian inversion step 550, such as in pre-processing operations (i.e., steps 555 and 560) to condition the data for Sifrian inversion. The Sifrian inversion step 550 can include an expansion step 555, a combination step 560, a full Newton direction step 570 and an update step 575.

[0102] In step 555, the gradient definitions and wavefield equations may be augmented. Augmentation may be performed by adding variables to each of the corresponding gradient definitions, wavefield equations, and adjoint wavefields. For example, the gradient definition g may be augmented in auxiliary variables n and then by replicating the gradient g and its conjugate g. * The expanded gradient is defined in the following equation (15):

[0103]

[0104] Where g is the gradient, ω is the frequency, is the first-order derivative of the inverse operator with respect to the model, λ is the adjoint, u * is a conjugate wave field, and n is an augmented variable. The wave field equation can be augmented by including a forward operator in the data structure and adding an additional variable γ. The augmented wave field is represented by the following equation (16):

[0105] Wavefield augmented =∑ w <F(m)u-s,γ> (16)

[0106] Where F is the forward operator, m is the model, u is the wavefield, s is the source, and γ is the augmented variable. The adjoint wavefield can be augmented with the reverse operator, the forward and reverse sampling operators, and the additional variable ζ. The augmented wavefield can be expressed in the following equation (17):

[0107]

[0108] Where ω is the frequency, is the first-order derivative of the cost function with respect to the wavefield, F * is the inverse operator, λ is the adjoint, and ζ is the augmented variable. All of these elements are injected into a data structure. Equations 3 to 5 are combined in step 460 to produce the Sifrian function represented in equation (11) above.

[0109] The Newton direction step 570 may represent the use of the augmented variables (γ, ζ, n) to balance the description of the Hessian without performing the calculations necessary for the full Hessian. At step 575, the image generation module 210 may update the initial model m0, or any iteration of the initial model, with the results of the Sifrian inversion. The image generation module 210 may continue to update the model through an iterative process derived from the equilibrium conditions of the Sifrian (i.e., its zero values ​​and derivatives). Once the image generation module 210 determines that the model has converged (e.g., meets the desired level of accuracy), the final updated model m0 is * Can be used to generate an image of the measured medium.

[0110] Figure 6 6 is a flow chart illustrating a method 600 of generating an image of a measured medium according to an example embodiment. The method 600 may begin at step 602 .

[0111] At step 602, computing system 202 may collect or receive observation data of the measured medium. In some embodiments, computing system 202 may collect or receive observation data from receiver network 100. In some embodiments, the observation data may be obtained from a plurality of sensors detecting signals originating from a source.

[0112] At step 604, the computing system 202 may select an initial model. In some embodiments, the initial model may be based on an initial guess at the properties of the measured medium. In some embodiments, the initial model may be obtained through tomography, previous inversion work, and / or expert information about the material. If no information is available, the image generation module 210 may test multiple initial models. In some embodiments, the initial model may be represented by spatial coordinates (e.g., 2D or 3D). The data for the initial model may be represented as a time series of scalars or multiple components (i.e., vectors) and may be organized with reference to the locations of wavefield sources and wavefield sensors or receiver geometry.

[0113] The computing system 202 can derive synthetic data from the model at step 606. For example, the image generation module 210 can generate synthetic data using the observed data and the initial model.

[0114] At step 608, computing system 202 may determine whether the model has reached a threshold level of accuracy. If, at step 608, computing system 202 determines that the model has reached a threshold level of accuracy, method 600 may proceed to step 612 (discussed below). However, if computing system 202 determines that the model has not reached a threshold level of accuracy, method 600 may proceed to step 610.

[0115] At step 610, computing system 202 may update the model. For example, computing system 202 may update the model with the goal of generating a model that can generate synthetic data that better matches the observed data. To update the model, computing system 202 may utilize a Sifrian function, such as the function described above in conjunction with Equation 11. The Sifrian function may augment the wavefield equations, adjoint wavefields, and gradient definitions by adding corresponding first, second, and third augmented variables to each of the wavefield equations, adjoint wavefields, and gradient definitions. A cost function that quantifies the mismatch between the synthetic data and the observed data is omitted.

[0116] This updating process may continue iteratively until computing system 202 generates a model that is capable of generating synthetic data that matches observed data within a predefined mismatch threshold.

[0117] Once a model is generated that matches the observed data within a predefined mismatch threshold, the computing system 202 can generate an image of the measured medium at step 612. The image of the measured medium can be used to locate objects suitable for further exploration.

[0118] It should be understood that Figure 6 The steps shown in are merely illustrative, and additional steps may be added, the order of steps may be rearranged, and some steps may be simplified or omitted.

[0119] Figure 7 to Fig. 9 Synthetic examples are shown demonstrating superior image generation of acoustic waveforms from Sifrian inversion compared to conventional image generation methods. Conventional methods include Gradient Descent (first order), BFGS (second order), and HessianFree + Conjugate Gradient (second order). Sifrian inversion is tested for acoustic frequency domain FWI. Three well-known geophysical benchmark models (such as Marmousi, Sigsbee, and Inverse Mask models) are used to compare Sifrian inversion results with the three conventional methods. Gradient Descent is a basic and relatively old inversion method that simply updates the velocity model with the gradient of the objective function. Limited memory, L-BFGS, is another well-known and effective second-order inversion method that iteratively includes curvature information through several rank-one updates (two per iteration). This method is built on several theoretical proofs. Hessian Free (+ Conjugate Gradient): This method utilizes a matrix-vector product algorithm to calculate the product of the curvature and a preselected update; the preselected update is varied by a conjugate gradient (CG). From low to high, all methods are implemented with the same frequency band. The same step size selection method is applied. No further fine-tuning is applied. Fig. 9 Each of shows a ground truth image, an initial model image, and images created by a gradient descent model, an L-BFGS model, a Hessian-Free+CG model, and a Sifrian inversion model. The initial model for all four inversions is the same.

[0120] Figure 7 shows the Marmousi model, a well-known geophysical seismic velocity model widely used to test new inversion methods. The large amount of lateral variation at small scales makes the synthetic model difficult to invert. The inversion frequencies range from 4 Hz to 32 Hz, with a maximum of 30 iterations at each frequency. The image resolution of the Sifrian inversion is clearer than that of the other three models.

[0121] Figure 8 shows the Sigsbee model, a large 2D model with a large number of salt bodies (shown by black dots). Salt is highly reflective to seismic waves and is difficult to depict. The inversion frequencies ranged from 2 Hz to 32 Hz, with 30 iterations at each frequency. Note how the Sifrian inversion recovers the salt bodies much better than the other three models.

[0122] Fig. 9 A reverse fault model is shown, corresponding to structures often encountered in geological structures. The recovery of the low-velocity layer (about 1.7 km deep) is a challenging task. The inversion frequency ranges from 4 Hz to 62 Hz, with 30 iterations for each frequency. Note that only the Sifrian inversion is able to recover the deep low-velocity layer.

[0123] Fig.10 1 is a block diagram of a special-purpose computer system 1000 according to one embodiment. The methods and processes described herein may similarly be implemented by tangible, non-transitory computer-readable storage media and / or computer program products that instruct a computer system to perform the actions of the methods and processes described herein. Each such computer program product may include a set of instructions (e.g., code) contained on a computer-readable medium that instructs a processor of a computer system to perform corresponding operations. These instructions may be configured to run sequentially, or in parallel (e.g., under different processing threads), or in combination.

[0124] The special purpose computer system 1000 includes a computer 1002, a display 1004 coupled to the computer 1002, one or more additional user output devices 1006 (optional) coupled to the computer 1002, one or more user input devices 1008 (e.g., keyboard, mouse, trackball, touch screen) coupled to the computer 1002, an optional communication interface 1010 coupled to the computer 1002, and a computer program product including a tangible computer readable storage medium 1012 in or accessible to the computer 1002. The instructions stored on the computer readable storage medium 1012 can direct the system 1000 to perform the methods and processes described herein. The computer 1002 may include one or more processors 1014 that communicate with a number of peripheral devices via a bus subsystem 1016. These peripherals may include user output devices 1006, user input devices 1008, communications interface 1010, and storage subsystems such as random access memory (RAM) 1018 and non-volatile storage drives 1020 (e.g., magnetic disk drives, optical drives, solid-state drives), which are forms of tangible computer-readable memory.

[0125] The computer readable medium 1012 may be loaded into the random access memory 1018, stored in the non-volatile storage drive 1020, or accessed by one or more components of the computer 1002. Each processor 1014 may include a microprocessor, such as a processor from or Advanced Micro Devices, To support the computer-readable medium 1012, the computer 1002 runs an operating system that handles communications between the computer-readable medium 1012 and the above components and supports communications between the above components of the computer-readable medium 1012. Exemplary operating systems include those from Microsoft Corporation. etc., from Sun Microsystems LINUX, UNIX, etc. In many embodiments, as described herein, a computer program product may be a device (e.g., a hard drive including a housing, a read / write head, etc., a computer disk including a housing, a memory card including a connector, a housing, etc.) that includes a computer-readable medium (e.g., a disk, a memory chip, etc.). In other embodiments, a computer program product may include an instruction set or a code module itself and be contained on a computer-readable medium.

[0126] User input devices 1008 include all possible types of devices and mechanisms for inputting information to computer system 1002. These may include keyboards, keypads, mice, scanners, digital drawing tablets, touch screens integrated into displays, audio input devices such as voice recognition systems, microphones, and other types of input devices. In various embodiments, user input devices 1008 are typically embodied as computer mice, trackballs, trackpads, joysticks, wireless remote controls, drawing tablets, and voice command systems. User input devices 1008 typically allow a user to select objects, icons, text, etc. that appear on display 1004 through commands such as clicking buttons. User output devices 1006 include all possible types of devices and mechanisms for outputting information from computer 1002. These may include displays (e.g., display 1004), printers, non-visual displays such as audio output devices, and the like.

[0127] The communication interface 1010 provides an interface to other communication networks and devices, and may act as an interface to receive and send data from and to other systems, WANs, and / or the Internet via a wired or wireless communication network 1022. Embodiments of the communication interface 1010 typically include an Ethernet card, a modem (telephone, satellite, cable, ISDN), a (asynchronous) digital subscriber line (DSL) unit, interface, USB interface, wireless network adapter, etc. For example, the communication interface 1010 can be coupled to a computer network, In other embodiments, the communication interface 1010 may be physically integrated on a motherboard of the computer 1002 and / or may be a software program or the like.

[0128] RAM 1018 and non-volatile storage drive 1020 are examples of tangible computer-readable media configured to store data, such as computer program product embodiments of the present invention, including executable computer code, human-readable code, etc. Other types of tangible computer-readable media include floppy disks, removable hard disks, optical storage media such as CD-ROMs, DVDs, bar codes, and semiconductor memories such as flash memory, read-only memory (ROM), battery-powered volatile memory, network storage devices, etc. As described above, RAM 1018 and non-volatile storage drive 1020 can be configured to store basic programming and data structures that provide the functionality of various embodiments of the present invention.

[0129] The software instruction sets that provide the functionality of the embodiments discussed herein may be stored in the computer-readable medium 1012, the RAM 1018, and / or the non-volatile storage drive 1020. These instruction sets or codes may be executed by the processor 1014. The computer-readable medium 1012, the RAM 1018, and / or the non-volatile storage drive 1020 may also provide a repository to store data and data structures used according to the embodiments discussed herein. The RAM 1018 and the non-volatile storage drive 1020 may include multiple memories, including a main random access memory (RAM) that stores instructions and data during program execution and a read-only memory (ROM) that stores fixed instructions. The RAM 1018 and the non-volatile storage drive 1020 may include a file storage subsystem that provides permanent (non-volatile) storage of program and / or data files. The RAM 1018 and the non-volatile storage drive 1020 may also include a removable storage system, such as a removable flash memory.

[0130] The bus subsystem 1016 provides a mechanism to allow the various components and subsystems of the computer 1002 to communicate with each other as intended. Although the bus subsystem 1016 is shown schematically as a single bus, alternative embodiments of the bus subsystem may utilize multiple buses or communication paths within the computer 1002.

[0131] For firmware and / or software implementations, the methods can be implemented with modules (e.g., programs, functions, etc.) that perform the functions described herein. Any machine-readable medium that tangibly contains instructions can be used to implement the methods described herein. For example, software code can be stored in a memory. The memory can be implemented inside a processor or outside a processor. As used herein, the term "memory" refers to any type of long-term, short-term, volatile, non-volatile, or other storage medium, and is not limited to any particular type of memory or amount of memory, or the type of medium that stores the memory.

[0132] In addition, as disclosed herein, the term "storage medium" may represent one or more memories for storing data, including read-only memory (ROM), random access memory (RAM), magnetic RAM, magnetic core memory, magnetic disk storage media, optical storage media, flash memory devices, and / or other machine-readable media for storing information. The term "machine-readable medium" includes, but is not limited to, portable or fixed storage devices, optical storage devices, wireless channels, and / or various other storage media capable of storing or carrying instructions and / or data.

[0133] While many changes and modifications to the present invention will no doubt become apparent to those of ordinary skill in the art after reading the foregoing description, it is to be understood that the particular embodiments shown and described by way of example are not intended to be considered limiting in any way.

[0134] In addition, regarding Figures 1 to 10The processes described in one or more of the above, as well as any other aspects of the present disclosure, can each be implemented by software, but can also be implemented by hardware, firmware, or any combination of software, hardware, and firmware. The instructions for performing these processes can also be embodied as machine or computer readable codes recorded on a machine or computer readable medium. In some embodiments, the computer readable medium can be a non-transitory computer readable medium. Examples of such non-transitory computer readable media include, but are not limited to, read-only memory, random access memory, flash memory, CD-ROM, DVD, tape, removable memory card, and optical data storage device. In other embodiments, the computer readable medium can be a temporary computer readable medium. In such an embodiment, the temporary computer readable medium can be distributed on a network-coupled computer system so that the computer readable code is stored and executed in a distributed manner. For example, such a temporary computer readable medium can be transmitted from one electronic device to another electronic device using any suitable communication protocol. Such a temporary computer readable medium can contain computer readable code, instructions, data structures, program modules, or other data in a modulated data signal, such as a carrier wave or other transmission mechanism, and can include any information transfer medium. The modulated data signal can be a signal that sets or changes one or more of its characteristics in a manner that encodes information in the signal.

[0135] It should be understood that any or each module of any one or more of any system, device, or server may be provided as a software construct, a firmware construct, one or more hardware components, or a combination thereof, and may be described in the general context of computer-executable instructions, such as program modules that may be executed by one or more computers or other devices. In general, a program module may include one or more routines, programs, objects, components, and / or data structures that may perform one or more specific tasks or may implement one or more specific abstract data types. It should also be understood that the number, configuration, functionality, and interconnection of any one or more of the modules of any system device or server are merely illustrative, and the number, configuration, functionality, and interconnection of existing modules may be modified or omitted, additional modules may be added, and the interconnection of specific modules may be changed.

[0136] Although systems, methods, and computer-readable media for enabling an electronic user device to effectively control a media application at an electronic media device have been described, it should be understood that many changes may be made therein without departing from the spirit and scope of the present disclosure. Insubstantial changes to the claimed subject matter now known or later devised from the perspective of one of ordinary skill in the art are expressly deemed to be equivalent to being within the scope of the claims. Therefore, obvious substitutions now or later known to one of ordinary skill in the art are defined to be within the scope of the defined elements.

[0137] Accordingly, those skilled in the art will appreciate that the present invention can be practiced by embodiments other than those described, which are presented for purposes of illustration and not limitation.

Claims

1. A method for waveform imaging, comprising: The computing system receives observation data from the measured medium via a plurality of sensors; The computing system generates a model or image of the measured medium, wherein the generating includes: obtaining the model; generating synthetic data using the model; comparing the synthesized data with the observed data; determining that the observed data and the synthesized data are not within a matching threshold; Based on the determination, iteratively updating the model using Sifrian inversion until the synthetic data matches the observed data within the matching threshold, the iterative updating comprising: augmenting the wavefield equations, the adjoint wavefields, and the gradient definitions of the model by adding first, second, and third augmented variables to each of the wavefield equations, adjoint wavefields, and gradient definitions, wherein a cost function quantifying a mismatch between the synthetic data and the observed data is strictly omitted; combining the expanded wavefield equation, the expanded adjoint wavefield, and the expanded gradient definition using the first expanded variable, the second expanded variable, and the third expanded variable to create a Sifrian functional; and performing a variational perturbation of the update and characterizing the update; and updating the model based on output from the Sifrian functional; and The computing system generates a final model or a final image of the measured medium.

2. The method according to claim 1, wherein: The first augmentation variable, the second augmentation variable, and the third augmentation variable used to augment each of the wavefield equation, the adjoint wavefield, and the gradient definition enable a Sifrian function when equal to zero.

3. The method according to claim 1, wherein: The Sifrian functional is a function of: Forward operator model, F; Inverse operator model, F * ; Wavefield response, source, With TRON, gradient, Angular frequency, Reverse operator, Conjugate wavefield response, The first extended variable, the second expanded variable, and The third extended variable, Therein, the Sifrian functional produces an effective representation of the second-order updates.

4. The method according to claim 3, wherein: The first extended variable, the second extended variable and the third extended variable are used to characterize Newton directions without forming or expressing Hessian.

5. The method according to claim 1, wherein: The Sifrian functionals and characterizations are configured as: Expand operator; inhibiting the expansion operator; and The suppression augmentation operator is simplified to efficiently generate second-order updates for model building.

6. The method according to claim 1, wherein: The Sifrian functional utilizes the Hessian without explicitly computing the Hessian or the inverse or approximation of the Hessian.

7. The method according to claim 1, wherein: The measured medium is a subterranean formation, and wherein the image of the measured medium is an image of the subterranean formation.

8. The method according to claim 1, wherein: The measured medium is a biomedical medium or a tissue, and wherein the image of the measured medium is an image of the biomedical medium or the tissue.

9. The method according to claim 1, wherein: The measured medium is a metal or an organic structure, and wherein the image of the measured medium is an image of the metal or organic structure.

10. A device for waveform imaging and model building, the device comprising: An interface configured to accommodate an initial model and observation data of a measured medium; as well as a data processing unit connected to the interface and configured to: Generate synthetic data; comparing the synthesized data with the observed data; determining that the observed data and the synthesized data are not within a matching threshold; Based on the determination, iteratively updating the initial model using Sifrian inversion until the synthetic data matches the observed data within the matching threshold, the iterative updating comprising: augmenting the wavefield equations, the adjoint wavefields, and the gradient definitions of the initial model by adding first, second, and third augmented variables to each of the wavefield equations, adjoint wavefields, and gradient definitions, wherein a cost function quantifying a mismatch between the synthetic data and the observed data is omitted; and combining the expanded wavefield equation, the expanded adjoint wavefield, and the expanded gradient definition using the first expanded variable, the second expanded variable, and the third expanded variable to create a Sifrian functional; and Perform variational characterization, suppression, and interpretation of second-order updates; and outputting a model based on iteratively updating the initial model until the synthetic data matches the observed data within the matching threshold; and An image of the measured medium or the model is generated.

11. The device according to claim 10, wherein: The first augmented variable, the second augmented variable, and the third augmented variable enable the Sifrian functional to characterize a second-order update when it is equal to zero at equilibrium.

12. The device according to claim 10, wherein: The Sifrian functional is a function of: Forward operator model, F; Inverse operator model, F * ; Wavefield response, source, With TRON, gradient, Angular frequency, Reverse operator, Conjugate wavefield response, The first extended variable, the second expanded variable, and The third extended variable, Therein, the Sifrian functional produces a second-order update.

13. The device according to claim 12, wherein: The first extended variable, the second extended variable and the third extended variable are used to characterize Newton directions without forming or expressing Hessian.

14. The apparatus according to claim 10, wherein: The Sifrian functional exploits the Hessian without explicitly computing the Hessian or its inverse.

15. The apparatus according to claim 10, wherein: The measured medium is a subterranean formation, and wherein the image of the measured medium is an image of the subterranean formation.

16. The apparatus according to claim 10, wherein: The measured medium is a biomedical medium or a tissue, and wherein the image of the measured medium is an image of the biomedical medium or the tissue.

17. The apparatus according to claim 10, wherein: The measured medium is a metal or an organic structure, and wherein the image of the measured medium is an image of the metal or organic structure.

18. A method for waveform imaging and model building, comprising: Collecting observation data of the measured medium from multiple sensors; obtaining an initial model and synthetic data derived from the initial model; Iteratively updating the initial model until the synthetic data matches the observed data within a matching threshold, wherein the initial model includes a wavefield equation, an adjoint wavefield, and a gradient definition, wherein iteratively updating the initial model includes using Sifrian inversion, wherein the Sifrian inversion: augmenting the wavefield equations, the companion wavefields, and the gradient definitions by adding respective first, second, and third augmented variables to each of the wavefield equations, the companion wavefields, and the gradient definitions, wherein a cost function quantifying a mismatch between the synthetic data and the observed data is omitted; and combining the expanded wavefield equation, the expanded adjoint wavefield and the expanded gradient definition; and performing a variational characterization of a second-order update; and An image of the measured medium is generated using the updated model, wherein the image of the measured medium can be used to locate objects suitable for further exploration.

19. The method of claim 18, wherein the Sifrian inversion utilizes the Hessian without explicitly computing the Hessian or its inverse.

20. The method according to claim 18, wherein: The first augmented variable, the second augmented variable, and the third augmented variable used to augment each of the wavefield equation, the adjoint wavefield, and the gradient definition enable the Sifrian inversion to skillfully characterize second order updates.

21. A method for waveform imaging, comprising: The computing system receives observation data from the measured medium via a plurality of sensors; Obtaining, by the computing system, an available model of the measured medium; generating, by the computing system, synthetic data using the available model; comparing, by the computing system, the synthesized data with the observed data to determine whether the observed data and the synthesized data are within a matching threshold; determining, by the computing system, that the observed data and the synthesized data are not within the matching threshold; based on the determination, iteratively updating the available model using Sifrian inversion until the synthetic data matches the observed data within the matching threshold; in response to the synthetic data matching the observed data within the matching threshold, updating, by the computing system, the usable model; as well as The updated usable model is used by the computing system to generate a final model or a final image of the measured medium.

22. The method according to claim 21, wherein: Iteratively updating the available model using the Sifrian inversion until the synthetic data matches the observed data within the matching threshold comprises: A Sifrian functional is generated based on one or more of a wavefield equation, an adjoint wavefield, and a gradient definition of the available model.

23. The method according to claim 22, wherein: Generating the Sifrian functional based on one or more of the wavefield equation of the available model, the adjoint wavefield, and the gradient definition comprises: augmenting the wavefield equation by adding a plurality of augmented variables to the wavefield equation; and The Sifrian functional is generated based on the extended wave field equation.

24. The method according to claim 22, wherein: Generating the Sifrian functional based on the one or more of the wavefield equation of the available model, the adjoint wavefield, and the gradient definition comprises: augmenting the companion wavefield by adding a plurality of augmentation variables to the companion wavefield; and The Sifrian functional is generated based on the adjoint wavefield.

25. The method according to claim 22, wherein: Generating the Sifrian functional based on the one or more of the wavefield equation of the available model, the adjoint wavefield, and the gradient definition comprises: augmenting the gradient definition by adding a plurality of augmented variables to the gradient definition; and The Sifrian functional is generated based on the gradient definition.

26. The method of claim 22, wherein: A cost function that quantifies the mismatch between the synthetic data and the observed data is omitted from the Sifrian functional.

27. The method of claim 22, wherein: When in equilibrium, the Sifrian functional is zero.

28. A system for waveform imaging and model building, comprising: processor; as well as A memory having programming instructions stored thereon, which, when executed by the processor, cause the system to perform operations, including: Receiving observation data from the measured medium via a plurality of sensors; Obtaining a usable model of the measured medium; generating synthetic data using the available model; comparing the synthesized data to the observed data to determine whether the observed data and the synthesized data are within a matching threshold; determining that the observed data and the synthesized data are not within a matching threshold; based on the determination, iteratively updating the available model using Sifrian inversion until the synthetic data matches the observed data within the matching threshold; In response to the synthetic data matching the observed data within the matching threshold, updating the available model; and The updated available model is used to generate a final model or a final image of the measured medium.

29. The system of claim 28, wherein: Iteratively updating the available model using the Sifrian inversion until the synthetic data matches the observed data within the matching threshold, comprising: A Sifrian functional is generated based on one or more of a wavefield equation, an adjoint wavefield, and a gradient definition of the available model.

30. The system of claim 29, wherein: Generating the Sifrian functional based on the one or more of the wavefield equation of the available model, the adjoint wavefield, and the gradient definition comprises: augmenting the wavefield equation by adding a plurality of augmented variables to the wavefield equation; and The Sifrian functional is generated based on the extended wave field equation.

31. The system of claim 29, wherein: Generating the Sifrian functional based on the one or more of the wavefield equation of the available model, the adjoint wavefield, and the gradient definition comprises: augmenting the companion wavefield by adding a plurality of augmentation variables to the companion wavefield; and The Sifrian functional is generated based on the adjoint wavefield.

32. The system of claim 29, wherein: Generating the Sifrian functional based on the one or more of the wavefield equation of the available model, the adjoint wavefield, and the gradient definition comprises: augmenting the gradient definition by adding a plurality of augmented variables to the gradient definition; and The Sifrian functional is generated based on the gradient definition.

33. The system of claim 29, wherein: The cost function that quantifies the mismatch between the synthetic data and the observed data is omitted from the Sifrian functional.

34. The system of claim 29, wherein: When in equilibrium, the Sifrian functional is zero.

35. A non-transitory computer readable medium comprising one or more sequences of instructions that, when executed by one or more processors, cause a computing system to perform operations comprising: The computing system receives observation data from the measured medium via a plurality of sensors; Obtaining, by the computing system, an available model of the measured medium; generating, by the computing system, synthetic data using the available model; comparing, by the computing system, the synthesized data with the observed data to determine whether the observed data and the synthesized data are within a matching threshold; determining, by the computing system, that the observed data and the synthesized data are not within the matching threshold; based on the determination, iteratively updating the available model using Sifrian inversion until the synthetic data matches the observed data within the matching threshold; in response to the synthetic data matching the observed data within the matching threshold, updating, by the computing system, the usable model; as well as The updated usable model is used by the computing system to generate a final model or a final image of the measured medium.

36. The non-transitory computer readable medium of claim 35, wherein: Iteratively updating the available model using the Sifrian inversion until the synthetic data matches the observed data within the matching threshold comprises: A Sifrian functional is generated based on one or more of a wavefield equation, an adjoint wavefield, and a gradient definition of the available model.

37. The non-transitory computer readable medium of claim 36, wherein: Generating the Sifrian functional based on the one or more of the wavefield equations of the available model, the adjoint wavefield, and the gradient definition comprises: augmenting the wavefield equation by adding a plurality of augmented variables to the wavefield equation; and The Sifrian functional is generated based on the extended wave field equation.

38. The non-transitory computer readable medium of claim 36, wherein: Generating the Sifrian functional based on the one or more of the wavefield equations of the available model, the adjoint wavefield, and the gradient definition comprises: augmenting the companion wavefield by adding a plurality of augmentation variables to the companion wavefield; and The Sifrian functional is generated based on the adjoint wavefield.

39. The non-transitory computer readable medium of claim 36, wherein: Generating the Sifrian functional based on the one or more of the wavefield equations of the available model, the adjoint wavefield, and the gradient definition comprises: augmenting the gradient definition by adding a plurality of augmented variables to the gradient definition; and The Sifrian functional is generated based on the gradient definition.

40. The non-transitory computer readable medium of claim 36, wherein: The cost function that quantifies the mismatch between the synthetic data and the observed data is omitted from the Sifrian functional.

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