CN107491411B - Gravity anomaly inversion method based on N-order polynomial density function - Google Patents

Gravity anomaly inversion method based on N-order polynomial density function Download PDF

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CN107491411B
CN107491411B CN201710488596.8A CN201710488596A CN107491411B CN 107491411 B CN107491411 B CN 107491411B CN 201710488596 A CN201710488596 A CN 201710488596A CN 107491411 B CN107491411 B CN 107491411B
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张建中
刘洁
江丽
马飞
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Ocean University of China
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Abstract

The invention relates to a gravity anomaly inversion method based on an N-order polynomial density function, and belongs to the technical field of geophysical exploration. Which comprises the following steps: firstly, preprocessing actual measurement gravity anomaly to obtain gravity anomaly generated by a research target body; second, unit division, i.e. dividing the research area into vertically side-by-side NrEach rectangular unit has the same longitudinal length and is greater than or equal to the depth of the research target area; thirdly, setting the maximum order of the density polynomial function and constructing a kernel function matrix of gravity inversion; fourthly, constructing an inversion target function, establishing a corresponding gravity inversion linear equation set, solving an equation, and making the calculated residual error of the theoretical gravity anomaly and the actually measured gravity anomaly within an allowable range to obtain a polynomial coefficient of each corresponding rectangular unit density function; and fifthly, calculating the underground density distribution of the whole research area by using the inverted polynomial coefficient. The invention realizes the inversion of the gravity anomaly of the research area.

Description

Gravity anomaly inversion method based on N-order polynomial density function
Technical Field
The invention relates to a gravity anomaly inversion method based on an N-order polynomial density function, and belongs to the technical field of geophysical exploration.
Background
At present, gravity inversion is mainly divided into two categories, one is interface inversion, the method generally assumes that the density distribution of the stratum is known, and determines the depth of a certain stratum interface by observing the fitting between gravity anomaly and theoretically calculating the gravity anomaly, and the method is widely applied to solving the base depth of a basin; the other type is inversion of physical properties, the underground is subjected to gridding subdivision and is divided into a plurality of rectangular units, and the density value of each rectangular unit is a constant. During inversion, the density values of all units are continuously modified to enable the residual error between the observed gravity anomaly and the calculated gravity anomaly to be within an allowable range, and the underground density distribution is continuously obtained in an iterative mode.
In recent years, many scholars have focused on the problem of gravity anomaly calculation of a variable density body. Considering the situation that the density changes along with the depth, the underground density distribution is expressed as different functional relations such as an exponential function, a parabolic function, a hyperbolic function and the like related to the depth, a related forward formula is deduced, the basin foundation depth is researched by using a variable density function, and some research results are obtained. However, the following problems still remain: (1) the variation relation of the density along with the depth is various, and the linear or exponential, parabolic, hyperbolic and other functional relations are not enough to describe the underground real density distribution characteristics; (2) due to the diversification of the density function form, a certain density-depth function is difficult to select in inversion, and inversion gravity kernel functions cannot be unified; (3) at present, most inversion considers an inversion interface and inversion density distribution separately, namely a specific density function is assumed when the interface is inverted, and actually, underground real density distribution is different from the assumed density function, which undoubtedly increases inversion errors; (4) the existing physical property inversion method, whether a regular grid, such as a rectangular grid, or an irregular grid, such as a triangular grid, is divided in the horizontal direction and the vertical direction, if the number of grid units is too small, the inversion is not fine enough, and if the number of grid units is more, the multi-resolution problem is more serious.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a novel gravity anomaly inversion method based on an N-order polynomial density function. On the basis of establishing a target body gravity anomaly analytic expression of an N-order polynomial density function, determining the density distribution of the underground substances by inverting the coefficient of the polynomial density function.
The invention is realized by adopting the following technical scheme: the gravity anomaly inversion method based on the N-order polynomial density function comprises the following steps:
(1) preprocessing the actually measured gravity anomaly to obtain the gravity anomaly generated by the research target body;
(2) dividing a unit: divide the study area into N vertically side by siderEach rectangular unit has the same longitudinal length and is greater than or equal to the depth of the research target area;
(3) setting the maximum order of the density polynomial function, and constructing a gravity kernel function matrix: dividing the region of interest into NrVertical side by side rectangular units, the density of each vertical rectangular unit representingAs a function of the depth z:
Figure GDA0002465278180000021
wherein N iszIs the maximum order of z, ajRepresenting the coefficients of the terms in the polynomial. Then, one upright rectangular cell is at the measurement point (x)o,zo) The resulting gravity anomaly may be calculated by:
Figure GDA0002465278180000022
then the gravity anomaly of all the upright rectangular units at one measuring point is as follows:
Figure GDA0002465278180000023
the above formula can be abbreviated as
Figure GDA0002465278180000024
Wherein the content of the first and second substances,
Figure GDA0002465278180000025
if the number of the abnormal gravity measurement points is NmThe coefficients of the gravity anomaly and the polynomial density function can be expressed in the form of a matrix as follows:
Figure GDA0002465278180000026
wherein the content of the first and second substances,
Figure GDA0002465278180000027
is a gravity kernel function;
Figure GDA0002465278180000028
polynomial coefficients to be solved;
Figure GDA0002465278180000029
the measured gravity abnormal value is obtained;
(4) constructing an inversion target function, establishing a corresponding gravity inversion linear equation set, solving an equation, and making a calculated theoretical gravity anomaly and an actually measured gravity anomaly residual error within an allowable range to obtain a polynomial coefficient of each corresponding rectangular unit density function;
the construction form of the objective function is as follows:
Figure GDA0002465278180000031
wherein the content of the first and second substances,
the first item is a data residual item; the second term is prior model constraint; the third term is the lateral smoothing constraint; the fourth term is a longitudinal smoothing constraint; lambda [ alpha ]1,λ2,λ3The weight coefficients of the items are respectively;
p is a diagonal block matrix and,
Figure GDA0002465278180000032
Figure GDA0002465278180000037
the block matrixes are respectively, and can be unified into the following forms:
Figure GDA0002465278180000033
h＝1,2,…Nr(ii) a d is the number of depth points in each rectangular cell to which the constraint is applied, z1,z2,…,zdDepth values, σ, of the respective constraint points0For the prior density difference model, when the trend of the density model is unknown, the sigma is taken0＝0；
H is a density difference first order transverse difference matrix,
Figure GDA0002465278180000034
v is a density difference longitudinal second order difference matrix,
Figure GDA0002465278180000035
Figure GDA0002465278180000038
the block matrixes are respectively, and can be unified into the following forms:
Figure GDA0002465278180000036
(5) the subsurface density distribution is calculated for the entire study area using the inverted polynomial coefficients. Through the processing of the specific steps, the inversion of the gravity anomaly in the research area is realized.
The principle of the invention is as follows: inversion of subsurface density distributions using gravity anomalies is one of the main targets of gravity exploration. The density dependence on depth is expressed as a polynomial function of arbitrary order. The higher the maximum order of the polynomial function, the more complex the subsurface density profile can be described. Given polynomial coefficient, the gravity anomaly generated correspondingly can be calculated through forward calculation, so that the calculated theoretical gravity anomaly and the actually measured gravity anomaly are minimum under certain constraint, and the underground density distribution can be obtained. The core of the invention is that: (1) a polynomial function representation of subsurface material density; (2) constructing a kernel function matrix of gravity inversion by taking coefficients of a polynomial density function as unknowns; (3) applying constraint conditions and prior information; (4) and establishing a gravity inversion linear equation set and solving polynomial coefficients of the density function.
The invention has the beneficial effects that: by adopting the gravity anomaly inversion method based on the N-order polynomial density function, the density is expressed by the polynomial function and is more consistent with the actual situation than the constant density hypothesis, the gravity anomaly calculation precision is higher, and the inversion result is more consistent with the geological rule; (2) the underground is divided into vertical and parallel rectangular units, only polynomial coefficients of each rectangular unit need to be inverted, the number of the rectangular units is less than that of conventional grid subdivision units, the solution unknowns are small, the inversion multi-solution is reduced, and the inversion calculation efficiency is also improved; (3) the constructed objective function is flexible, and can conveniently apply constraint and prior information obtained by geological survey, well logging and other geophysical means.
Drawings
FIG. 1 is a block flow diagram of the present invention.
FIG. 2(a) is a diagram of a basin model using a polynomial of order 2 to represent density as a function of depth.
FIG. 2(b) is a graph of negative gravity anomaly values and inversion errors generated by the basin.
FIG. 2(c) is a density profile inverted by a conventional method.
FIG. 2(d) shows the process (N) according to the inventionz9) inversion of the resulting subsurface density profile.
FIG. 3(a) is a diagram of a rectangular and parallelogram local density anomaly model.
Fig. 3(b) is a diagram showing a gravity anomaly value and an inversion error generated by the local density anomaly model.
FIG. 3(c) is a density profile inverted by a conventional method.
FIG. 3(d) is a density profile inverted by the method of the present invention.
Detailed Description
In order to make the purpose and technical solution of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings.
The detailed technical operation diagram of the invention is shown in figure 1. The key points of the main technology of the invention are four as follows: (1) a polynomial function representation of subsurface material density; (2) constructing a kernel function matrix of gravity inversion by taking coefficients of a polynomial density function as unknowns; (3) applying constraint conditions and prior information; (4) and establishing a gravity inversion linear equation set and solving polynomial coefficients of the density function.
The first embodiment is as follows:
as shown in FIG. 2(a), a basin model with density varying with depth is obtained, and the density function is expressed by a polynomial of order 2, Δ σ ═ 400+274.8z-47.3z2Wherein z is km and Delta sigma is kg/m3The superficial density difference is large and can reach-0.4 g/cm at the earth surface3. The horizontal extension of the basin is 5.4km, the maximum base depth of the basin is 2.3 km. FIG. 2(b) shows the basin in solid linesThe resulting negative gravity anomaly. According to the process of the invention, the gravity inversion is carried out: the research area is divided into 60 vertical rectangular units side by side, and the density distribution of each rectangular unit is represented by 9-order polynomial which changes along with the depth, namely NzA gravity kernel function matrix is constructed, 9. Take sigmamin＝-0.6g/cm3,σmax＝0.6g/cm3，σ 00, d 300, weight λ1＝10-4,λ2＝10-3,λ3＝10-3And summarizing the calculated polynomial coefficients of each rectangular unit in a table I.
Table one embodiment a table of density function polynomial coefficient calculation results
Figure GDA0002465278180000051
Figure GDA0002465278180000061
FIG. 2(c) is a density distribution inverted by a conventional method assuming a linear change in density, and FIG. 2(d) is a subsurface density distribution obtained by the inventive method by inversion with a 9 th order polynomial. The black line frame in the graph represents the real position of the density abnormal body, and the comparison shows that the inversion result of the invention is better matched with the real density, the trend of large shallow density difference and small deep density difference is presented, the basin foundation is matched with the whole contour of the inversion density distribution, and the average absolute error of the inversion density distribution and the real density distribution is calculated to be 0.042g/cm3. The dotted line in fig. 2(b) shows the theoretical gravity anomaly of the density distribution calculation inverted in fig. 2(d), it can be seen that the observed gravity anomaly is better matched with the calculated gravity anomaly, the dashed line in fig. 2(b) shows the difference between the observed gravity anomaly and the theoretical calculated gravity anomaly, and the residual error between the observed gravity anomaly and the theoretical calculated gravity anomaly is very small and is below 0.4 mGal.
Example two:
as shown in FIG. 3(a), the model is a local density anomaly model, in which the rectangular volume is a positive density volume and the density value is 0.9g/cm3The parallelogram is a negative density bodyA value of-0.9 g/cm3The depth of the centers of both the abnormal bodies is 1.5km, the negative density abnormal body inclines to the positive density abnormal body side with the increase of the depth, and the solid line in fig. 3(b) shows the gravity abnormal value generated by the local abnormal body. According to the process of the invention, the gravity inversion is carried out: the research area is divided into 60 vertical rectangular units side by side, and the density distribution of each rectangular unit is represented by 9-order polynomial which changes along with the depth, namely NzA gravity kernel function matrix is constructed, 9. Take sigmamin＝-1g/cm3,σmax＝1g/cm3，σ 00, d 300, weight λ1＝10-2,λ2＝10-4,λ3＝10-4Depth weighting β ═ 2, z00.5. The calculated polynomial coefficients of each rectangular unit are summarized in table two.
Table two: example table of the results of calculating coefficients of two-density function polynomial
Figure GDA0002465278180000071
Figure GDA0002465278180000081
Fig. 3(c) and fig. 3(d) are comparisons of inversion results obtained by conventional smooth constraint inversion and the inversion method of the present invention, and a black line frame in the diagram represents a real position of a density anomaly, and it can be found that density distribution generated by conventional smooth constraint inversion is smoother, although a density difference extremum is displayed in the center of the anomaly, an inversion value is greatly different from an actual value, and a boundary of a local anomaly is not clearly depicted; the method can effectively depict the real density distribution and the boundary position of the local abnormal body by inverting the coefficient of the high-order polynomial. The dotted line in fig. 3(b) shows the theoretical gravity anomaly calculated from the density distribution inverted in fig. 3(d), and the dashed line in fig. 3(b) shows the difference between the observed gravity anomaly and the theoretical gravity anomaly, and the residual error between the observed gravity anomaly and the theoretical gravity anomaly is small and less than 0.04 mGal.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, but rather the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.

Claims (6)

1. A gravity anomaly inversion method based on an N-order polynomial density function is characterized by comprising the following steps:
(1) preprocessing the actually measured gravity anomaly to obtain the gravity anomaly generated by the research target body;
(2) dividing a unit: divide the study area into N vertically side by siderA plurality of rectangular units;
(3) setting the maximum order of a density polynomial function, and constructing a gravity kernel function matrix;
(4) constructing an inversion target function, establishing a corresponding gravity inversion linear equation set, solving an equation, and making a calculated theoretical gravity anomaly and an actually measured gravity anomaly residual error within an allowable range to obtain a polynomial coefficient of each corresponding rectangular unit density function;
(5) calculating the subsurface density distribution of the whole research area by utilizing the inverted polynomial coefficients; through the processing of the specific steps, the inversion of the gravity anomaly in the research area is realized.
2. The method for inverting gravity anomaly based on N-order polynomial density function according to claim 1, wherein the unit division method in step (2) is that the longitudinal length of each rectangular unit is the same and is greater than or equal to the depth of the research target area.
3. The N-order polynomial density function-based gravity anomaly inversion method according to claim 1, wherein the specific construction method of the kernel function matrix for gravity inversion in the step (3) is as follows:
dividing the region of interest into NrVertical side by side rectangular elements, the density of each upright rectangular element being expressed as a function of the depth z:
Figure FDA0002465278170000011
wherein N iszIs the maximum order of z, ajCoefficients representing terms in the polynomial; then, one upright rectangular cell is at the measurement point (x)o,zo) The resulting gravity anomaly may be calculated by:
Figure FDA0002465278170000012
then the gravity anomaly of all the upright rectangular units at one measuring point is as follows:
Figure FDA0002465278170000013
the above formula can be abbreviated as:
Figure FDA0002465278170000014
wherein the content of the first and second substances,
Figure FDA0002465278170000015
if the number of the abnormal gravity measurement points is NmThe coefficients of the gravity anomaly and the polynomial density function can be expressed in the form of a matrix as follows:
Figure FDA0002465278170000021
wherein the content of the first and second substances,
Figure FDA0002465278170000022
is a gravity kernel function;
Figure FDA0002465278170000023
polynomial coefficients to be solved;
Figure FDA0002465278170000024
the measured gravity abnormal value is obtained.
4. The method for inverting gravity anomaly based on N-order polynomial density function according to claim 1, wherein the target function in the step (4) is constructed in the following specific form:
Figure FDA0002465278170000025
wherein, the first item is a data residual item; the second term is prior model constraint; the third term is the lateral smoothing constraint; the fourth term is a longitudinal smoothing constraint; lambda [ alpha ]1,λ2,λ3The weight coefficients of the items are respectively; wherein P is a diagonal block matrix, H is a density difference transverse first-order difference matrix, and V is a density difference longitudinal second-order difference matrix; sigma0For the prior density difference model, when the trend of the density model is unknown, the sigma is taken0＝0。
5. The method for inverting gravity anomaly based on N-th order polynomial density function according to claim 4, wherein the diagonal matrix P in the step (4) is specifically expressed as:
Figure FDA0002465278170000026
Q1,Q2,…,
Figure FDA0002465278170000027
the block matrixes are respectively, and can be unified into the following forms:
Figure FDA0002465278170000028
h＝1,2,…Nr(ii) a d is the number of depth points in each rectangular cell to which the constraint is applied, z1,z2,…,zdThe depth values of the constraint points are respectively.
6. The order-N polynomial based of claim 4The gravity anomaly inversion method of the formula density function is characterized in that the density difference transverse first-order difference matrix H in the step (4) is specifically expressed as:
Figure FDA0002465278170000029
the density difference longitudinal second order difference matrix V is specifically expressed as:
Figure FDA0002465278170000031
W1,W2,…,
Figure FDA0002465278170000032
the block matrixes are respectively, and can be unified into the following forms:
Figure FDA0002465278170000033
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