CN110794469B - Gravity inversion method based on minimum geological feature unit constraint - Google Patents

Gravity inversion method based on minimum geological feature unit constraint Download PDF

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CN110794469B
CN110794469B CN201910163596.XA CN201910163596A CN110794469B CN 110794469 B CN110794469 B CN 110794469B CN 201910163596 A CN201910163596 A CN 201910163596A CN 110794469 B CN110794469 B CN 110794469B
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gravity
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CN110794469A (en
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宋明水
王金铎
于会臻
王千军
陈学国
毕丽飞
李守济
王有涛
相鹏
王圣柱
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China Petroleum and Chemical Corp
Exploration and Development Research Institute of Sinopec Shengli Oilfield Co
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Exploration and Development Research Institute of Sinopec Shengli Oilfield Co
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Abstract

The invention provides a gravity inversion method based on minimum geological feature unit constraint, which comprises the following steps: step 1, establishing a minimum geological feature unit model set; step 2, establishing a gravity inversion target function based on minimum geological feature unit model constraint; step 3, establishing a density initial model; step 4, solving a gravity forward kernel function; step 5, calculating a minimum geological feature unit weight coefficient; step 6, recovering the model based on the minimum geological feature unit and the weight coefficient; and 7, outputting the final inversion model. According to the gravity inversion method based on the minimum geological feature unit constraint, geological knowledge of a research target area is subjected to a structural feature extraction mode to obtain a minimum geological feature unit model set, and the minimum geological feature unit model set is applied to a gravity inversion process to ensure that an inversion result comes from the combination of the minimum geological feature unit models, so that the existing model constraint method is perfected, and the resolution and reliability of an interpretation result are improved.

Description

Gravity inversion method based on minimum geological feature unit constraint
Technical Field
The invention relates to the field of exploration geophysics, in particular to a gravity inversion method based on minimum geological feature unit constraint.
Background
Gravity exploration, as a classic geophysical exploration method, has been widely applied to deep geological exploration and exploration of resources such as mineral products, water, petroleum, geothermal heat and the like. In recent years, with the improvement of the precision of the gravity instrument, the application range of the gravity instrument is not limited to the early regional exploration, but infiltrates into each link of resource exploration.
Gravity inversion is an important technology for effectively recovering underground density distribution and determining exploration targets, and scholars at home and abroad have obtained a great deal of research achievements and the application effect is gradually improved. In many research directions, how to apply more effective model constraints is one of the key issues.
The gravity model constraint term can reduce the solution space by limiting the model density distribution so as to increase the reliability of the inversion result. Model constraints are typically applied in two categories: in areas with abundant known data, inversion results can be controlled by adopting a mode of well constraint expansion and physical property value range limitation; for the regions with less known data and low exploration degree, the adopted scheme comprises the technical means of applying smooth constraint of the model to obtain a constructed continuous model, applying focusing constraint to improve the inversion resolution of the isolated field source body and the like. Although the two measures can reduce the multi-solution of the gravity inversion to a certain extent, the former depends too much on the accuracy and the richness of known data and can only ensure the accuracy of inversion results near well logging data, and the latter is too wide and cannot analyze geological data and interpretation results in a research area according to local conditions, so that the geological structure characteristics of the research area are lost, the resolution and the reliability of the inversion results are difficult to ensure, and the application level of the gravity exploration technology is reduced. Therefore, for high-precision gravity inversion, further research needs to be carried out by using a model constraint method with strong pertinence and flexible application. Therefore, a new gravity inversion method based on minimum geological feature unit constraint is invented, and the technical problems are solved.
Disclosure of Invention
The invention aims to provide a gravity inversion method based on minimum geological feature unit constraint, which improves the resolution and reliability of an explanation result by improving the existing model constraint method.
The object of the invention can be achieved by the following technical measures: the gravity inversion method based on the minimum geological feature unit constraint comprises the following steps: step 1, establishing a minimum geological feature unit model set; step 2, establishing a gravity inversion target function based on minimum geological feature unit model constraint; step 3, establishing a density initial model; step 4, solving a gravity forward kernel function; step 5, calculating a minimum geological feature unit weight coefficient; step 6, recovering the model based on the minimum geological feature unit and the weight coefficient; and 7, outputting the final inversion model.
The object of the invention can also be achieved by the following technical measures:
in step 1, a geological interpretation model is used as basic sample data, and a minimum geological feature unit model set capable of representing original sample data is solved in a manual or automatic construction mode.
In step 1, defining the minimum geological feature unit set as a geological feature set in a local space range, namely, a geological model in a work area is formed by the weighted combination of the sets; the method for establishing the minimum geological feature unit model set comprises the following two modes: the method comprises the following steps of firstly, manually drawing a geological unit according to known knowledge, wherein the geological unit comprises cuboids, spheres, arcs and Gaussian sources with different sizes, and the range of the geological unit does not exceed the range of an area to be inverted; secondly, an automatic construction mode is adopted, and the geological model is combined into a sample data set in a sliding window mode; then, a solving method based on sparse constraint is utilized, a dictionary learning method is used, and a minimum geological feature unit model set is obtained, wherein an optimization objective function of the minimum geological feature unit model set is as follows:
Figure RE-GDA0002330503840000021
wherein the content of the first and second substances,
Figure RE-GDA0002330503840000022
for a set of geological model samples, existing geological model sections from the area of study
Figure RE-GDA0002330503840000023
Acquiring a sliding window;
Figure RE-GDA0002330503840000024
is the minimum geological feature unit to be learned;
Figure RE-GDA0002330503840000025
is the corresponding sparse weight; n is a radical ofx、NzWidth and height of the original geological model respectively; n isx、nzRespectively the width and the height of the window, l is the number of geological model samples, and k is the number of minimum geological feature units; λ is the regularization parameter.
Solving the formula (1) by adopting an ADMM (Alternating Direction Method of Multipliers) algorithm to obtain gamma and D corresponding to gamma which simultaneously meet the sparse condition, and converting different columns of data in the D matrix into nx×nzThe size is given to the central position of the model grid to be inverted, and the construction of the minimum geological feature unit set is completed at the moment; in order to characterize geologic bodies of different scales, a mode of searching a geologic model S by a multi-scale window is adopted to obtain a minimum geologic feature unit set D with more complete characterization capability.
In step 2, the principle of constructing the inversion objective function is to ensure that the gravity fitting residual is minimum, and pursue to use minimum geological feature units as possible to represent two conditions of the inversion model, so as to ensure that structural features in the geological prior constraint are added into the inversion process, and further reduce the inversion solution space.
In step 2, the gravity forward calculation formula is:
b＝AM
(2)
wherein:
Figure RE-GDA0002330503840000031
the gravity observation data is obtained;
Figure RE-GDA0002330503840000032
a gravity forward kernel function;
Figure RE-GDA0002330503840000033
the density model vector to be solved; wherein q is the number of observed data, and p is the number of models;
assuming that the constructed minimum geological feature unit set can represent a density model to be solved, namely, the minimum geological feature unit and the corresponding weight coefficient vector are subjected to convolution and summation to obtain a geological model:
Figure RE-GDA0002330503840000034
wherein, represents a two-dimensional convolution,
Figure RE-GDA0002330503840000035
a sparse weighting coefficient set corresponding to the minimum geological feature unit set D to be solved;
equation (3) is expressed in the form of a matrix as follows:
Figure RE-GDA0002330503840000036
wherein, CD＝[CD1,CD2,…,CDk]；
Figure RE-GDA0002330503840000037
Constructing a produced convolution operator matrix for the unit representing the kth minimum geological feature;
Figure RE-GDA0002330503840000041
represents a k-th sparse weighting coefficient vector;
the forward formula of gravity (2) is then re-expressed as follows:
Figure RE-GDA0002330503840000042
the gravity inversion objective function is as follows:
Figure RE-GDA0002330503840000043
wherein C isbA covariance matrix which is observation data b;
is provided with
Figure RE-GDA0002330503840000044
Figure RE-GDA0002330503840000045
At this point, the original gravity inversion problem becomes a sparse weight coefficient under the constraint of the minimum set of geologic feature cells
Figure RE-GDA0002330503840000046
And solving the problem.
In step 5, the minimum geologic feature unit weight coefficient meeting the sparse condition is solved to ensure that the inversion model is represented by the minimum geologic feature unit.
In step 5, in the formula (7)
Figure RE-GDA0002330503840000047
The terms are not microminiature, and the solution is difficult; in order to obtain the optimal sparse solution, the formula (7) is transformed and solved by adopting a optimization-Minimization framework, and a more easy optimization function is utilized to gradually approximate
Figure RE-GDA0002330503840000048
The sparse solution formula is derived as follows:
Figure RE-GDA0002330503840000049
wherein the content of the first and second substances,
Figure RE-GDA0002330503840000051
in the formula
Figure RE-GDA0002330503840000052
Is a pair of
Figure RE-GDA0002330503840000053
Is used to determine the threshold value of the threshold value function,
Figure RE-GDA0002330503840000054
is composed of
Figure RE-GDA0002330503840000055
First derivative of, i.e. assurance
Figure RE-GDA0002330503840000056
Setting the partial result to be less than a certain threshold value and setting the partial result to be 0; since a large number of 0 values appear in the process of solving the sparse weight coefficient M, the formula (9) shows
Figure RE-GDA0002330503840000057
At the denominator, singular values will appear; for this purpose, the inversion term in the formula is used
Figure RE-GDA0002330503840000058
Replacing the steps as follows:
Figure RE-GDA0002330503840000059
at this time, the sparse solution can be obtained by combining the formulas (8) and (10) in an iterative loop manner
Figure RE-GDA00023305038400000510
In step 3, the model is respectively processed along x and z coordinate axes under a Cartesian coordinate systemSpace division into Nx、NzA plurality of rectangular grid cells; the distance is delta x and delta z, and the initial models adopt uniform half-space models.
In step 4, the gravity anomaly forward modeling formula is calculated for any polygonal prism.
In step 6, the obtained minimum geologic feature cell weight coefficient is convolved with the minimum geologic feature cell model to obtain an actual inverted density model.
In step 7, forward calculation is carried out on the recovered model, fitting is carried out on the model and the actually measured gravity data, whether calculation is terminated or not is judged according to inversion termination conditions such as the size of a fitting residual error and the maximum iteration number, if the conditions are met, the density model is output as a final inversion result, and if not, the step 1 is returned.
According to the gravity inversion method based on the minimum geological feature unit constraint, geological knowledge of a research target area is subjected to structural feature extraction to obtain a minimum geological feature unit model set, and the minimum geological feature unit model set is applied to a gravity inversion process to ensure that an inversion result comes from the combination of the minimum geological feature unit models, so that the existing model constraint method is perfected, and the resolution and reliability of an interpretation result are improved. Compared with the prior art, the invention has the main innovation points that:
(1) the model constraint comes from local structural features, while the traditional gravity inversion model constraint is too wide, and the overall distribution features of the model are controlled, and if the smooth constraint and the focusing constraint are adopted, the solution space is difficult to effectively reduce; or hard constraints such as density value range constraints and logging constraints are directly adopted, point-to-point constraints are achieved, and the expansion and popularization capacity is low. This is one of the important differences between the present invention and conventional gravity inversion.
(2) The original gravity inversion problem is converted into a minimum geological feature unit weight coefficient under sparse constraint to be solved, the minimum geological feature unit and convolution summation of the minimum geological feature unit and the minimum geological feature unit is used for obtaining an inversion result, the minimum geological feature unit is pursued to recover a density model, the traditional gravity inversion method is used for directly solving the density model, although the solution is simple, key features capable of reflecting the distribution of the density model are difficult to extract, and the interpretability of the inversion result is poor. This is yet another important difference between the present invention and conventional gravity inversion.
According to the method, geological knowledge of a research target area is effectively quantitatively fused in a gravity inversion process, a final density inversion model is obtained by solving an optimal minimum geological feature unit model and a sparse weight coefficient corresponding to the optimal minimum geological feature unit model, and compared with a conventional gravity inversion method, the method emphasizes matching of an inversion result and structural features of a prior geological model, and effectively improves resolution and reliability of the inversion result.
Drawings
FIG. 1 is a flow diagram of one embodiment of a minimum geological feature cell constraint-based gravity inversion method of the present invention;
FIG. 2 is a diagram illustrating a density model and a forward modeling result of gravity anomaly according to an embodiment of the present invention;
FIG. 3 is a diagram of conventional gravity inversion results in an embodiment of the present invention;
FIG. 4 is a sample set of known geological models in accordance with an embodiment of the present invention;
FIG. 5 is a diagram of a set of four minimum geologic feature unit models computed in accordance with an embodiment of the present invention;
FIG. 6 is a graph of sparse weight coefficients inverted based on the minimum set of geologic feature unit models of FIG. 5(a) in an embodiment of the present invention;
FIG. 7 is a sparse weight coefficient inverted based on the minimum set of geologic feature unit models of FIG. 5(b) in an embodiment of the present invention;
FIG. 8 is a sparse weight coefficient inverted based on the minimum set of geologic feature unit models of FIG. 5(c) in an embodiment of the present invention;
FIG. 9 is a sparse weight coefficient obtained by inversion based on the minimum set of geologic feature unit models of FIG. 5(d) in an embodiment of the present invention;
FIG. 10 is a graph of a density model obtained for inversion in an embodiment of the present invention.
Detailed Description
In order to make the aforementioned and other objects, features and advantages of the present invention comprehensible, preferred embodiments accompanied with figures are described in detail below.
As shown in fig. 1, an embodiment of the gravity inversion method based on minimum geological feature unit constraint according to the present invention is a flowchart, and the specific steps of the embodiment are as follows:
(1) and establishing a minimum geological feature unit model set. Taking a geological interpretation model as basic sample data, and solving a minimum geological feature unit model set capable of representing original sample data in a manual or automatic construction mode; and defining the minimum geological feature unit set as a geological feature set in a local space range, namely, a geological model in the work area can be formed by the weighted combination of the sets. The method for establishing the minimum geological feature unit model set comprises the following two modes: firstly, a manual construction mode. Geological units can be manually drawn according to known knowledge, such as cuboids, spheres, arcs, Gaussian sources and the like with different sizes, and the range of the general geological units does not exceed the range of an area to be inverted; the second is an automatic construction mode. Combining the geological models into a sample data set in a sliding window mode, and then obtaining a minimum geological feature unit model set by a dictionary learning method by utilizing a solving method based on sparse constraint, wherein an optimization objective function of the minimum geological feature unit model set is as follows:
Figure RE-GDA0002330503840000071
wherein the content of the first and second substances,
Figure RE-GDA0002330503840000072
for a set of geological model samples, existing geological model sections from the area of study
Figure RE-GDA0002330503840000073
Acquiring a sliding window;
Figure RE-GDA0002330503840000074
for minimum geology to be learnedA feature unit;
Figure RE-GDA0002330503840000075
is the corresponding sparse weight. N is a radical ofx、NzWidth and height of the original geological model respectively; n isx、nzRespectively the width and the height of the window (namely the size of the minimum geological feature unit model), l is the number of geological model samples, and k is the number of minimum geological feature units; λ is the regularization parameter.
Solving the formula (1) by adopting an ADMM (Alternating Direction Method of Multipliers) algorithm summarized by Stephen Boyd (2011), obtaining gamma satisfying sparse conditions and D corresponding to gamma, and converting different columns of data in the D matrix into nx×nzSize and assigned to the model mesh to be inverted (size N)x×Nz) The minimum set of geologic feature cells is now complete. In order to characterize geologic bodies of different scales, a mode of searching a geologic model S by a multi-scale window is adopted to obtain a minimum geologic feature unit set D with more complete characterization capability.
(2) And establishing a gravity inversion target function based on minimum geological feature unit model constraints. The principle of constructing the inversion target function is to ensure that the gravity fitting residual is minimum, and pursue two conditions of representing an inversion model by using minimum geological feature units as few as possible so as to ensure that structural features in a geological prior constraint are added into an inversion process and further reduce an inversion solution space; the gravity forward calculation formula is as follows:
b＝AM
(2)
wherein:
Figure RE-GDA0002330503840000081
the gravity observation data is obtained;
Figure RE-GDA0002330503840000082
a gravity forward kernel function;
Figure RE-GDA0002330503840000083
is the density model vector to be solved. Wherein q is the number of observed data, and p is the number of models.
Assuming that the constructed minimum geological feature unit set can represent a density model to be solved, namely, the minimum geological feature unit and the corresponding weight coefficient vector are subjected to convolution and summation to obtain a geological model:
Figure RE-GDA0002330503840000084
wherein, represents a two-dimensional convolution,
Figure RE-GDA0002330503840000085
and obtaining a sparse weighting coefficient set corresponding to the minimum geological feature unit set D to be solved.
Equation (3) can be expressed using the following matrix form:
Figure RE-GDA0002330503840000086
wherein, CD＝[CD1,CD2,…,CDk]。
Figure RE-GDA0002330503840000087
And constructing a produced convolution operator matrix for representing the kth minimum geological feature unit.
Figure RE-GDA0002330503840000091
Representing the k-th sparse weighting coefficient vector.
The forward formula of gravity (2) can be re-expressed as follows:
Figure RE-GDA0002330503840000092
the gravity inversion objective function is as follows:
Figure RE-GDA0002330503840000093
wherein C isbIs a covariance matrix of the observation data b.
Is provided with
Figure RE-GDA0002330503840000094
Figure RE-GDA0002330503840000095
At this point, the original gravity inversion problem becomes a sparse weight coefficient under the constraint of the minimum set of geologic feature cells
Figure RE-GDA0002330503840000096
And solving the problem.
(3) Establishing a density initial model; dividing the model space into N along x and z coordinate axes under Cartesian coordinate systemx、NzA rectangular grid cell. The distance is delta x and delta z, and the initial models adopt uniform half-space models.
(4) Solving a gravity forward kernel function; the gravity anomaly forward equation uses the calculations proposed by Singh (2002) for any polygonal prism.
(5) And calculating the minimum geological feature unit weight coefficient. Solving the weight coefficient of the minimum geological feature unit meeting the sparse condition to ensure that the minimum geological feature unit is used for representing the inversion model; in formula (7)
Figure RE-GDA0002330503840000097
The terms are not trivial and the solution is difficult. In order to obtain the optimal sparse solution, the formula (7) is transformed and solved by adopting a optimization-Minimization framework, and a more easy optimization function is utilized to gradually approximate
Figure RE-GDA0002330503840000101
The sparse solution formula is derived as follows.
Figure RE-GDA0002330503840000102
Wherein the content of the first and second substances,
Figure RE-GDA0002330503840000103
in the formula
Figure RE-GDA0002330503840000104
Is a pair of
Figure RE-GDA0002330503840000105
Is used to determine the threshold value of the threshold value function,
Figure RE-GDA0002330503840000106
is composed of
Figure RE-GDA0002330503840000107
First derivative of, i.e. assurance
Figure RE-GDA0002330503840000108
The medium result is set to 0 when less than a certain threshold. Since a large number of 0 values appear in the process of solving the sparse weight coefficient M, the formula (9) shows
Figure RE-GDA0002330503840000109
At the denominator, singular values will appear. For this purpose, the inversion term in the formula is used
Figure RE-GDA00023305038400001010
Replacing the steps as follows:
Figure RE-GDA00023305038400001011
at this time, the sparse solution can be obtained by combining the formulas (8) and (10) in an iterative loop manner
Figure RE-GDA00023305038400001012
(6) And recovering the model based on the minimum geological feature unit and the weight coefficient. Convolving the obtained minimum geological feature unit weight coefficient with the minimum geological feature unit model to obtain an actually inverted density model; in one embodiment, the minimum geologic feature unit weight coefficient obtained by solving is used
Figure RE-GDA00023305038400001013
Convolve with the minimum geological feature unit model set D and sum to obtain the density model, which can be referred to as formula (4).
(7) And finally outputting the inversion model. And (3) forward calculation is carried out on the recovered model, fitting is carried out on the model and the actually measured gravity data, whether the calculation is terminated or not is judged according to other inversion termination conditions such as the fitting residual error size, the maximum iteration times and the like, if the conditions are met, the density model is output as a final inversion result, and if the conditions are not met, the step (1) is returned. The gravity inversion method based on the geological feature unit constraint can be better integrated with geological prior knowledge, and the reliability of gravity inversion is effectively improved.
The present invention is further illustrated by the following specific examples.
The experimental analysis fully considers the problem that the gravity inversion calculation precision is influenced by observation data noise, background density interference and the like, and meanwhile, in order to verify the robustness of the method to grid subdivision errors, the inversion grid moves 5m and 3m in the x direction and the y direction compared with the forward grid.
Fig. 2 is a diagram of a density model and a forward result of gravity anomaly according to the present invention. As shown in the figure, a two-dimensional density model combination is designed to verify the inversion effect, the subdivision x-direction range of the underground half-space is 50-2950 m, the spacing is 100m, and the number of grids is 29; the range of the z direction is 0-290 m, the distance is 10m, and the number of grids is 29. The gravity observation point range is 100-2900 m, the distance is 100m, the number of observation points is 29, and the height of the observation points is 0 m. The total number of the cuboid density models is 3, and the residual density of the block at the upper left corner is 0.4g/cm3And the residual density of the block at the upper right corner is 0.4g/cm3The lower residual density is-0.2 g/cm3. To ensure the practicability of the method, the mean value is set to be 0.01 Gauss density disturbancePerforming low-pass filtering on the density model, and taking the density model as background density; meanwhile, random noise with the maximum amplitude of 2% of the original gravity anomaly is added in the gravity anomaly forward modeling result.
Fig. 3 is a conventional gravity inversion result, and it can be seen that the resolution of the inversion result is low, and the density value is also greatly different from that of the real model.
FIG. 4 is a schematic diagram of a sample set of known geological models. And selecting models with different block distributions to construct a geological model sample set. Two of which are shown.
FIG. 5 is a minimum set of earth feature unit models computed by the present invention. The grid size of the sliding window in the x direction and the z direction is 11 multiplied by 7, and the distance is the same as the model distance. For convenience of illustration, fig. 5(a) -5 (d) respectively show 4 minimum geologic feature units, and it can be seen that the obtained minimum geologic feature unit model set extracts different block structural features in the known geologic model.
FIG. 6 is a graph of the results of the gravity inversion of the present invention. Fig. 6 to 9 are sparse weight coefficients corresponding to the minimum geological feature unit model sets of fig. 5(a) to 5(d) obtained by inversion, respectively; FIG. 10 shows the inversion results of the density model of the present invention. From fig. 6-9, it can be seen that the sparse weight coefficient has a larger amplitude only in the result corresponding to the 2 nd and 4 th block minimum geological models, so that the key features of the geological structure are effectively utilized, and the underground density model is effectively recovered by performing convolution summation on the key features and the minimum geological feature unit set. Compared with the conventional technology, the density inversion result (figure 10) not only keeps the structural characteristics of the known geological model and improves the inversion resolution, but also is very close to the spatial distribution and value range of the real geological model, thereby proving the accuracy and effectiveness of the method.
The above-mentioned embodiments are further illustrative of the objects, technical solutions and effects of the present invention, and it should be understood that the above-mentioned embodiments are only illustrative of the present invention and are not intended to limit the scope of the present invention, and any modifications, equivalent substitutions, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (7)

1. The gravity inversion method based on the minimum geological feature unit constraint is characterized by comprising the following steps of:
step 1, establishing a minimum geological feature unit model set;
step 2, establishing a gravity inversion target function based on minimum geological feature unit model constraint;
step 3, establishing a density initial model;
step 4, solving a gravity forward kernel function;
step 5, calculating a minimum geological feature unit weight coefficient;
step 6, recovering the model based on the minimum geological feature unit and the weight coefficient;
step 7, outputting a final inversion model;
in the step 1, a geological interpretation model is used as basic sample data, and a minimum geological feature unit model set capable of representing original sample data is solved in a manual or automatic construction mode;
defining the minimum geological feature unit set as a geological feature set in a local space range, namely, a geological model in a work area is formed by the weighted combination of the sets; the method for establishing the minimum geological feature unit model set is a manual construction method, geological units are manually drawn according to known knowledge and comprise cuboids, spheres and arcs with different sizes, and the range of the geological units does not exceed the range of an area to be inverted; or an automatic construction mode, combining the geological model into a sample data set in a sliding window mode; then, a solving method based on sparse constraint is utilized, a dictionary learning method is used, and a minimum geological feature unit model set is obtained, wherein an optimization objective function of the minimum geological feature unit model set is as follows:
Figure FDA0003046441800000011
wherein the content of the first and second substances,
Figure FDA0003046441800000021
for a set of geological model samples, existing geological model sections from the area of study
Figure FDA0003046441800000022
Acquiring a sliding window;
Figure FDA0003046441800000023
is the minimum geological feature unit to be learned;
Figure FDA0003046441800000024
is the corresponding sparse weight coefficient; n is a radical ofx、NzWidth and height of the original geological model respectively; n isx、nzRespectively the width and the height of the window, l is the number of geological model samples, and k is the number of minimum geological feature units; λ is a regularization parameter;
solving the formula (1) by adopting an alternating direction multiplier algorithm to obtain gamma and D corresponding to gamma which simultaneously meet the sparse condition, and then converting different rows of data in a D matrix into nx×nzThe size is given to the central position of the model grid to be inverted, and the construction of the minimum geological feature unit set is completed at the moment; in order to characterize geologic bodies of different scales, a mode of searching a geologic model sample set S by a multi-scale window is adopted to obtain a minimum geologic feature unit set D with more complete characterization capability.
2. The minimum geological feature cell constraint-based gravity inversion method according to claim 1, wherein in step 2, the gravity forward calculation formula is:
b＝AM
(2)
wherein:
Figure FDA0003046441800000025
the gravity observation data is obtained;
Figure FDA0003046441800000026
a gravity forward kernel function;
Figure FDA0003046441800000027
the density model vector to be solved; wherein q is the number of observed data, and p is the number of models;
assuming that the constructed minimum geological feature unit set can represent a density model to be solved, namely, the minimum geological feature unit and the corresponding weight coefficient vector are subjected to convolution and summation to obtain a geological model:
Figure FDA0003046441800000028
wherein, represents a two-dimensional convolution, mL1A sparse weighting coefficient set corresponding to the minimum geological feature unit set D to be solved;
equation (3) is expressed in the form of a matrix as follows:
Figure FDA0003046441800000031
wherein, CD＝[CD1,CD2,…,CDk]；
Figure FDA0003046441800000032
Constructing a produced convolution operator matrix for the unit representing the kth minimum geological feature;
Figure FDA0003046441800000033
represents a k-th sparse weighting coefficient vector;
the forward formula of gravity (2) is then re-expressed as follows:
Figure FDA0003046441800000034
the gravity inversion objective function is as follows:
Figure FDA0003046441800000035
wherein C isbA covariance matrix which is observation data b;
is provided with
Figure FDA0003046441800000036
Figure FDA0003046441800000037
At this point, the original gravity inversion problem becomes a sparse weight coefficient under the constraint of the minimum set of geologic feature cells
Figure FDA0003046441800000038
And solving the problem.
3. The minimum geological feature cell constraint-based gravity inversion method according to claim 2, wherein in step 5, in equation (7)
Figure FDA0003046441800000041
The terms are not microminiature, and the solution is difficult; to obtain the optimal sparse solution, the equation (7) is transformed and solved by using an optimization minimization framework, and a more easy optimization function is used for successive approximation
Figure FDA0003046441800000042
The sparse solution formula is derived as follows:
Figure FDA0003046441800000043
wherein the content of the first and second substances,
Figure FDA0003046441800000044
in the formula
Figure FDA0003046441800000045
Is a pair of
Figure FDA0003046441800000046
Is used to determine the threshold value of the threshold value function,
Figure FDA0003046441800000047
is composed of
Figure FDA0003046441800000048
First derivative of, i.e. assurance
Figure FDA0003046441800000049
Setting the partial result to be less than a certain threshold value and setting the partial result to be 0; since a large number of 0 values appear in the process of solving the sparse weight coefficient M, the formula (9) shows
Figure FDA00030464418000000410
At the denominator, singular values will appear; for this purpose, the inversion term in the formula is used
Figure FDA00030464418000000411
Replacing the steps as follows:
Figure FDA00030464418000000412
in this case, I is an identity matrix, and the sparse solution can be obtained by combining equations (8) and (10) in an iterative loop manner
Figure FDA00030464418000000413
4. The method according to claim 1The gravity inversion method of minimum geological feature unit constraint is characterized in that in step 3, a model space is divided into N along x and z coordinate axes under a Cartesian coordinate systemx、NzA plurality of rectangular grid cells; the distance is delta x and delta z, and the initial model adopts a uniform half-space model.
5. The minimum geological feature cell constraint-based gravity inversion method according to claim 1, wherein in step 4, the gravity anomaly forward equation is calculated by using any polygonal prism.
6. The minimum geologic feature cell constraint-based gravity inversion method of claim 1, wherein in step 6, the obtained minimum geologic feature cell weight coefficients are convolved with the minimum geologic feature cell model to obtain the actual inverted density model.
7. The gravity inversion method based on the minimum geological feature unit constraint according to claim 1, characterized in that in step 7, forward calculation is performed on the restored model, fitting is performed on the restored model and actual measurement gravity data, whether the calculation is terminated or not is judged according to inversion termination conditions such as the size of a fitting residual error and the maximum iteration number, if the conditions are met, the density model is output as a final inversion result, and if the conditions are not met, the step 1 is returned.
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