A Multi-parameter Analytical Forward Modeling Method for Spatial Domain Gravity of a Rotating Rectangular Prism

Abstract

This invention is applicable to the fields of gravimetry and gravity exploration, providing a spatial domain multi-parameter analytical forward modeling method for gravity using a rotating rectangular prism. The method includes the following steps: establishing an observation coordinate system and a local coordinate system of the source; providing parameters for the rectangular prism model and the observation system; defining the attitude of the rotating prism using Euler angles, determining the rotation matrix, and completing the coordinate transformation; calculating the analytical solutions for the three-directional vectors of the gravity field and the analytical solutions for each component of the gravity gradient tensor in the spatial domain of the source local coordinate system; and deriving the three-directional vector data of the gravity field in the observation coordinate system, as well as the data for each component of the gravity gradient tensor in the observation coordinate system. This invention is of great significance for constructing high-precision, full-space multi-parameter gravity field models. It offers high accuracy and fast computation, and can be flexibly applied to gravity field numerical simulation, qualitative analysis of gravity data, and gravity field modeling, effectively serving the teaching and research fields of gravimetry and gravity exploration.

Description

Technical Field

[0001] This invention belongs to the field of gravimetry and gravity exploration, and particularly relates to a spatial domain gravity multi-parameter analytical forward modeling method for a rotating rectangular prism. Background Technology

[0002] Gravity forward modeling, under the condition of knowing the shape, attitude, and physical properties of a geological body, theoretically calculates the magnitude of the field generated by it at the observation point. It is used to study the spatial distribution characteristics and variation patterns of density field source anomalies. In a rectangular coordinate system, the gravity vector (V...) x V y V z Let V be the first derivative of the gravitational potential (V) generated by the source at the observation point in the x, y, and z directions, and let the gravity gradient tensor be the second derivative of the gravitational potential in the three directions. The gravitational field outside the source is conservative and irrotational; therefore, the gravity gradient tensor is not diagonally symmetric, and the sum of its diagonal components is zero. Five independent components, obtained through actual measurement or theoretical simulation, provide the complete gravity gradient tensor data. Forward modeling of the gravity vector and gravity gradient tensor is a crucial step in the processing and interpretation of gravity data and a prerequisite for inversion problems. For a long time, numerous theoretical methods for forward modeling of the spatial domain of the gravity vector and gravity gradient field have been studied, mainly categorized into analytical methods and numerical simulation methods. The former offers higher accuracy, but the analytical formulas for complex shapes are complex and difficult to derive; the latter is more applicable to arbitrary shapes, but its computational accuracy is lower.

[0003] Since the successive proposals of forward modeling equations for the gravity field and gravity gradient tensor of upright rectangular prisms in Cartesian coordinates, and based on the superposition property of gravity fields, using rectangular prisms as discrete basic units, an effective method has been provided for simulating field sources of complex shapes and for gravity / gravity gradient terrain correction. Forward modeling methods for the gravity field of upright prisms with arbitrary polygonal top and bottom surfaces, and forward modeling methods for the gravity field and gradient tensor of upright prisms with non-uniform density parameters (which vary linearly and non-linearly with depth), have become the main methods and computational tools developed to date. These methods can all be used to calculate the gravity vector and gravity gradient tensor field of upright prisms (each edge of which is parallel to the three coordinate axes of the observation coordinate system). However, the calculation of the gravity vector and gradient tensor response of rectangular prisms with arbitrary orientations in Cartesian coordinates has not yet been considered or described in detail. In theory, the forward modeling of a rectangular prism in any orientation can be calculated using analytical formulas for any polyhedron or numerical simulations. However, the former is more complex and the latter is an approximate solution with some error, both of which limit flexibility. Specific simulation results are also available in relevant published literature and reports.

[0004] In gravity field modeling and human-computer interaction modeling interpretation, the constructed rectangular prism units are usually not strictly parallel to the observation coordinate system; and in the actual data acquisition process, due to the unknown nature of underground targets, it is difficult to ensure that the observation network is strictly parallel to the major and minor axes of the blocky geological body's field source.

[0005] Previous research and analysis have shown that existing methods and technologies primarily focus on forward modeling of gravity vectors and gravity gradient tensors for upright prism field sources. These methods strictly require each edge of the model to be parallel to the three coordinate axes of the observation coordinate system, thereby calculating the response at the observation point using analytical formulas for the prism's gravity and gravity gradient. However, existing analytical formulas for rectangular prisms are not applicable to simulating the gravity vector and gravity gradient tensor responses of rectangular prisms in arbitrary orientations within a Cartesian coordinate system, and therefore cannot flexibly meet the needs for rapid modeling and interpretation of real-world gravity fields. Summary of the Invention

[0006] The purpose of this invention is to provide a spatial domain gravity multi-parameter analytical forward modeling method for rotating rectangular prisms. This method aims to address the problem that existing methods and techniques primarily focus on forward modeling the gravity vector and gravity gradient tensor of upright prism field sources, strictly requiring each edge of the model to be parallel to the three coordinate axes of the observation coordinate system. This allows for the calculation of the response at the observation point using analytical formulas for the prism's gravity and gravity gradient. However, existing analytical calculation formulas for rectangular prisms are not applicable to simulating the gravity vector and gravity gradient tensor responses of rectangular prisms in arbitrary orientations within a Cartesian coordinate system, thus failing to flexibly meet the needs for rapid modeling and interpretation of actual gravity fields.

[0007] The present invention is implemented as follows: a spatial domain gravity multi-parameter analytical forward modeling method for a rotating rectangular prism, the method comprising the following steps:

[0008] Step 1: Establish the observation coordinate system and the local coordinate system of the field source;

[0009] Step 2: Given the parameters of the rectangular prism model and the observation system parameters;

[0010] Step 3: Define the orientation of the prism of revolution using Euler angles, determine the rotation matrix, and complete the coordinate transformation;

[0011] Step 4: Calculate the analytical solutions of the three-directional vectors of the gravity field and the analytical solutions of each component of the gravity gradient tensor in the local coordinate system space domain of the field source;

[0012] Step 5: Based on the solution in Step 4, deduce the three-directional vector data of the gravity field in the observation coordinate system, as well as the data of each component of the gravity gradient tensor in the observation coordinate system.

[0013] As a further embodiment of the present invention, the establishment of the observation coordinate system and the local coordinate system of the field source in step one specifically includes:

[0014] S1. Define both the observation coordinate system O-XYZ and the local coordinate system O'-X'Y'Z' of the prism field source as rectangular coordinate systems;

[0015] S2. Define the X, Y, and Z coordinate axes in the observation coordinate system as azimuth coordinate systems indicating the north, east, and vertical downward directions, respectively, with the origin at O.

[0016] S3. Define the X, Y, and Z coordinate axes in the field source coordinate system as the directions of the three mutually perpendicular edges of the rectangular prism, and the origin O' is a vertex of the prism.

[0017] As a further embodiment of the present invention, the given rectangular prism model parameters and observation system parameters mentioned in step two specifically include:

[0018] The given rectangular prism model parameters and observation system parameters are defined as follows: model length is defined as a, width as b, height as c, model density as ρ, and the coordinates of the prism center Ct are defined as (x...). c y c , z c );

[0019] The observation system parameters include the observation area x, y, and the direction range x. min x max y max y min And the observation interval dx, dy.

[0020] As a further embodiment of the present invention, step three, which involves defining the orientation of the rotating prism using Euler angles, determining the rotation matrix, and completing the coordinate transformation, specifically includes:

[0021] S1. Determine the rotation matrix R by defining the orientation of any rectangular prism placed in space using Euler angles. The combination formula is as follows:

[0022]

[0023] Wherein, the Euler angles (α, β, γ) represent the rotation of the observation coordinate system about the z-axis by α degrees, about the x-axis by β degrees, and about the y-axis by γ degrees, respectively, R z R x R y This indicates that three rotations form three independent direction cosine angles;

[0024] S2. By translating the origin of the coordinate system and rotating the coordinate axes, the observed coordinate system is made to coincide with the local coordinate system of the field source. The coordinate transformation is then performed using the rotation matrix R to obtain the coordinates (x', y', z') of the calculation point in the local coordinate system of the field source. The calculation formula is as follows:

[0025]

[0026] As a further embodiment of the present invention, the calculation of the analytical solutions of the three-directional vectors of the gravitational field and the analytical solutions of each component of the gravitational gradient tensor in step four specifically includes:

[0027] The gravity vector (V) at each calculation point in the local coordinate system is calculated using the analytical forward modeling expression of the gravity vector and gravity gradient tensor space domain of a vertical prism. x ', V y ', V z ') and gravity gradient tensor (V xx ', V xy ', V xz ', V yy ', V yz ', V zz The calculation formula is:

[0028]

[0029] Where G is the gravitational constant, r = (x 2 +y 2 +z 2 ) 1 / 2 ;

[0030] x1, x2, y1, y2, z1, and z2 are the coordinates of the corner points of the cuboid after coordinate transformation using formula (2).

[0031] As a further embodiment of the present invention, the calculation of the three-directional vector data of the gravity field in the observation coordinate system and the data of each component of the gravity gradient tensor in the observation coordinate system in step five specifically includes:

[0032] Based on the vector transformation formula (4) and the tensor transformation formula (5), the components of gravity are transformed to the original observation coordinate system, thereby obtaining the three-directional gravity vector (V) of the rotating rectangular prism. x V y V z ) and each component of the gravity gradient tensor (V xx V xy V xz V yy V yz V zz ).

[0033]

[0034]

[0035] The spatial domain gravity multi-parameter analytical forward modeling method for a rotating rectangular prism provided in this invention has the following beneficial effects:

[0036] This invention is of great significance for the construction of high-precision full-space multi-parameter gravity field models. The model test results show that it is suitable for the fusion of gravity data with arbitrary parameters.

[0037] Considering the complexity of actual geological bodies, the method proposed in this invention can effectively calculate the gravity vector and gravity gradient tensor of rectangular prisms with arbitrary spatial orientations. It provides certain methodological and technical support and contributions to the numerical simulation, qualitative analysis and gravity field modeling of actual gravity vector fields and gradient tensor fields.

[0038] This invention takes into account the complexity of actual geological targets and the requirements of gravity processing and interpretation methods, and proposes an analytical forward modeling method for the spatial domain of gravity vector field and gradient tensor field of a rotating rectangular prism. It has high accuracy and fast calculation, and can be flexibly used for gravity field numerical simulation, qualitative analysis of gravity data and gravity field modeling, and effectively serves the teaching and research fields of gravimetry and gravity exploration. Attached Figure Description

[0039] Figure 1 A flowchart of a spatial domain gravity multi-parameter analytical forward modeling method for a rotating rectangular prism provided in an embodiment of the present invention;

[0040] Figure 2 This is a schematic diagram of the observation coordinate system and the local coordinate system of the field source;

[0041] Figure 3 A schematic diagram of Euler angles for coordinate system rotation;

[0042] Figure 4 A schematic diagram of a rectangular prism model rotated 45° around the z-axis;

[0043] Figure 5 A schematic diagram of the observed gravity field vector results for a rectangular prism model;

[0044] Figure 6 This is a schematic diagram of the observed gravity gradient tensor results for a rectangular prism model.

[0045] Figure 7 This is a schematic diagram of a prism of revolution.

[0046] Figure 8 A schematic diagram of the observed gravity field vector results for a rectangular prism model;

[0047] Figure 9This is a schematic diagram of the observed gravity gradient tensor results for a rectangular prism model. Detailed Implementation

[0048] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0049] The specific implementation of the present invention will be described in detail below with reference to specific embodiments.

[0050] like Figure 1 As shown, in an embodiment of the present invention,

[0051] A method for analytical forward modeling of spatial domain gravity of a rotating rectangular prism using multiple parameters, comprising the following steps:

[0052] Step 1: Establish the observation coordinate system and the local coordinate system of the field source;

[0053] Step 2: Given the parameters of the rectangular prism model and the observation system parameters;

[0054] Step 3: Define the orientation of the prism of revolution using Euler angles, determine the rotation matrix, and complete the coordinate transformation;

[0055] Step 4: Calculate the analytical solutions of the three-directional vectors of the gravity field and the analytical solutions of each component of the gravity gradient tensor in the local coordinate system space domain of the field source;

[0056] Step 5: Based on the solution in Step 4, deduce the three-directional vector data of the gravity field in the observation coordinate system, as well as the data of each component of the gravity gradient tensor in the observation coordinate system.

[0057] like Figure 2 As shown, in this embodiment of the invention, the establishment of the observation coordinate system and the local coordinate system of the field source in step one specifically includes:

[0058] S1. Define both the observation coordinate system O-XYZ and the local coordinate system O'-X'Y'Z' of the prism field source as rectangular coordinate systems;

[0059] S2. Define the X, Y, and Z coordinate axes in the observation coordinate system as azimuth coordinate systems indicating the north, east, and vertical downward directions, respectively, with the origin at O.

[0060] S3. Define the X, Y, and Z coordinate axes in the field source coordinate system as the directions of the three mutually perpendicular edges of the rectangular prism, and the origin O' is a vertex of the prism.

[0061] like Figure 3As shown in this embodiment of the invention, the given rectangular prism model parameters and observation system parameters mentioned in step two specifically include:

[0062] The given rectangular prism model parameters and observation system parameters are defined as follows: model length is defined as a, width as b, height as c, model density as ρ, and the coordinates of the prism center Ct are defined as (x...). c y c , z c );

[0063] The observation system parameters include the observation area x, y, and the direction range x. min x max y max y min and the observation intervals dx and dy;

[0064] like Figure 3 As shown, in this embodiment of the invention, step three, which involves defining the orientation of the rotating prism using Euler angles, determining the rotation matrix, and completing the coordinate transformation, specifically includes:

[0065] S1. Determine the rotation matrix R by defining the orientation of any rectangular prism placed in space using Euler angles. The combination formula is as follows:

[0066]

[0067] Wherein, the Euler angles (α, β, γ) represent the rotation of the observation coordinate system about the z-axis by α degrees, about the x-axis by β degrees, and about the y-axis by γ degrees, respectively, R z R x R y This indicates that three rotations form three independent direction cosine angles;

[0068] S2. By translating the origin of the coordinate system and rotating the coordinate axes, the observed coordinate system is made to coincide with the local coordinate system of the field source. The coordinate transformation is then performed using the rotation matrix R to obtain the coordinates (x', y', z') of the calculation point in the local coordinate system of the field source. The calculation formula is as follows:

[0069]

[0070] In this embodiment of the invention, step four, which involves calculating the analytical solutions of the three-directional vectors of the gravitational field and the analytical solutions of each component of the gravitational gradient tensor, specifically includes:

[0071] The gravity vector (V) at each calculation point in the local coordinate system is calculated using the analytical forward modeling expression of the gravity vector and gravity gradient tensor space domain of a vertical prism. x ', V y ', V z ') and gravity gradient tensor (V xx ', Vxy ', V xz ', V yy ', V yz ', V zz The calculation formula is:

[0072]

[0073] G is the gravitational constant, r = (x 2 +y 2 +z 2 ) 1 / 2 ;

[0074] x1, x2, y1, y2, z1, and z2 are the coordinates of the corner points of the cuboid after coordinate transformation using formula (2).

[0075] 6. The spatial domain gravity multi-parameter analytical forward modeling method for a rotating rectangular prism according to claim 1, characterized in that, the calculation of the three-directional vector data of the gravity field in the observation coordinate system and the data of each component of the gravity gradient tensor in the observation coordinate system in step five specifically includes:

[0076] Based on the vector transformation formula (4) and the tensor transformation formula (5), the components of gravity are transformed to the original observation coordinate system, thereby obtaining the three-directional gravity vector (V) of the rotating rectangular prism. x V y V z ) and each component of the gravity gradient tensor (V xx V xy V xz V yy V yz V zz ).

[0077]

[0078]

[0079] Example 1:

[0080] Figure 4 A rectangular prism model is given, rotating only 45° around the Z-axis, with center coordinates (10km, 12km, 0.75km), length 8km, width 4km, height 0.5km, density 1g / cm³, and observation area ranging from 0 to 20km in both x and y directions, and observation interval dx = dy = 0.1km. The three-directional vectors of the gravity field are calculated using this method in the spatial domain forward modeling. Figure 5 ) and each component of the gravity gradient tensor ( Figure 6 There are no singularities, and the calculation speed is relatively fast, with the entire process taking only 2 seconds on a regular computer.

[0081] Example 2:

[0082] Figure 7 A rectangular prism model is given, rotated sequentially around the Z-axis by 45°, X-axis by 30°, and Y-axis by 10°. It has a density of 1 g / cm³, a center coordinate of (10 km, 12 km, 2.5 km), a length of 8 km, a width of 4 km, and a height of 1 km. The observation area ranges from 0 to 20 km in both the x and y directions, and the observation interval is dx = dy = 0.1 km. The three-directional vectors of the gravitational field are calculated using this method in the spatial domain forward modeling. Figure 8 ) and each component of the gravity gradient tensor ( Figure 9 Similarly, none of them have singularities, and the calculation speed is relatively fast, with the entire process taking only 2 seconds on a regular computer.

[0083] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A spatial domain analytical forward modeling method for a rotating rectangular prism, characterized in that, The spatial domain analytical forward modeling method for the rotated rectangular prism includes the following steps: Step 1: Establish the observation coordinate system and the local coordinate system of the field source; Step 2: Given the parameters of the rectangular prism model and the observation system parameters; Step 3: Define the orientation of the prism of revolution using Euler angles, determine the rotation matrix, and complete the coordinate transformation; Step 4: Calculate the analytical solutions of the three-directional vectors of the gravity field and the analytical solutions of each component of the gravity gradient tensor in the local coordinate system space domain of the field source; Step 5: Based on the solution in Step 4, deduce the three-directional vector data of the gravity field in the observation coordinate system, as well as the data of each component of the gravity gradient tensor in the observation coordinate system. Specifically, the establishment of the observation coordinate system and the local coordinate system of the field source in step one includes: S1. Define both the observation coordinate system O-XYZ and the local coordinate system O'-X'Y'Z' of the prism field source as rectangular coordinate systems; S2. Define the X, Y, and Z coordinate axes in the observation coordinate system as azimuth coordinate systems indicating the north, east, and vertical downward directions, respectively, with the origin at O. S3. Define the X, Y, and Z coordinate axes in the field source coordinate system as the directions of the three mutually perpendicular edges of the rectangular prism, and the origin O' is a vertex of the prism. Step three, which involves defining the orientation of the prism of revolution using Euler angles, determining the rotation matrix, and completing the coordinate transformation, specifically includes: S1. Use Euler angles to describe the attitude of the rotating rectangular prism, establish the transformation relationship from the local coordinate system of the field source to the observation coordinate system, and complete the coordinate transformation of the calculation points in the local coordinate system of the field source. S2. Based on the actual posture of the rotating prism, the observation coordinate system is made to coincide with the local coordinate system of the field source by translating the origin of the observation coordinate system and rotating the coordinate axes, thus establishing the rotation matrix R and completing the coordinate transformation. S3. Define the orientation of any rectangular prism placed in space using Euler angles, and determine the rotation matrix R. The combination formula is as follows: （1）； Wherein, the Euler angles (α, β, γ) represent the rotation of the observation coordinate system about the z-axis by α degrees, about the x-axis by β degrees, and about the y-axis by γ degrees, respectively, R z R x R y This indicates that three rotations form three independent direction cosine angles; S4. Use a rotation matrix to perform coordinate transformation to obtain the coordinates (x', y', z') of the calculation point in the local coordinate system of the field source. The calculation formula is as follows: （2）； Step four, which involves calculating the analytical solutions of the three-directional vectors of the gravitational field and the analytical solutions of each component of the gravitational gradient tensor, specifically includes: The gravity vector (V) at each calculation point in the local coordinate system is calculated using the analytical forward modeling expression of the gravity vector and gravity gradient tensor space domain of a vertical prism. x ', V y ', V z ') and gravity gradient tensor (V xx ', V xy ', V xz ', V yy ', V yz ', V zz The calculation formula is: （3）； Where G is the gravitational constant, and r = (x² + y² + z²) 1 / 2 ; x1, x2, y1, y2, z1, and z2 are the coordinates of the corner points of the cuboid after coordinate transformation by formula (2).

2. The spatial domain analytical forward modeling method for a rotating rectangular prism according to claim 1, characterized in that, The specific parameters of the given rectangular prism model and the observation system parameters mentioned in step two include: The given rectangular prism model parameters and observation system parameters are defined as follows: model length is defined as a, width as b, height as c, model density as ρ, and the coordinates of the prism center Ct are defined as (x...). c y c , z c ); The observation system parameters include the observation area x, y, and the direction range x. min x max y max y min and observation interval d x d y .

3. The spatial domain analytical forward modeling method for a rotating rectangular prism according to claim 1, characterized in that, Step five, which involves calculating the three-directional vector data of the gravity field in the observation coordinate system and the data of each component of the gravity gradient tensor in the observation coordinate system, specifically includes: Based on the vector transformation formula (4) and the tensor transformation formula (5), the components of gravity are transformed to the original observation coordinate system, thereby obtaining the three-directional gravity vector (V) of the rotating rectangular prism. x V y V z ) and each component of the gravity gradient tensor (V xx V xy V xz V yy V yz V zz ); （4）； （5）。

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