CN114782276A - Resistivity imaging dislocation correction method based on adaptive gradient projection - Google Patents

Resistivity imaging dislocation correction method based on adaptive gradient projection Download PDF

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CN114782276A
CN114782276A CN202210464197.9A CN202210464197A CN114782276A CN 114782276 A CN114782276 A CN 114782276A CN 202210464197 A CN202210464197 A CN 202210464197A CN 114782276 A CN114782276 A CN 114782276A
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李廷军
田雨桐
杨海宁
赵泽轩
杨旭
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a self-adaptive gradient projection correction method for resistivity array imaging logging, which is based on horizontal term second-order gradient promotion caused by dislocation, and adopts a gradient domain self-adaptive method to correct micro-displacement and ensures the minimum loss and distortion of an image by using a least square term; meanwhile, the invention adopts simple linear transformation to realize dislocation correction, so that the dislocation of the transformed image is effectively inhibited, the minimum loss of image information can be ensured, and the balance between dislocation correction and image information loss is ensured. The method is quick and effective, not only improves the resolution and the imaging quality of the resistivity array imaging, but also efficiently solves the dislocation problem of the resistivity array imaging with the lowest computation complexity.

Description

Resistivity imaging dislocation correction method based on adaptive gradient projection
Technical Field
The invention belongs to the technical field of resistivity imaging, and relates to a method for correcting imaging of logging equipment.
Background
Resistivity imaging technology has been widely used in the fields of complex oil and gas resource exploration, biomedicine, agricultural detection, geological exploration and the like. Resistivity logging equipment is instrument equipment which realizes underground formation imaging by applying resistivity imaging technology. Resistivity array imaging is one of the most promising developments in resistivity logging, which utilizes a plurality of button electrodes to detect the medium current around the well and give an image of the resistivity distribution around the well, providing a high resolution resistivity image in an intuitive manner.
The resistivity logging equipment firstly sends out an excitation signal from the transmitting electrode and then measures the loop current of the button electrode. Differences in the formation composition, structure and resistivity of the borehole may result in changes in the loop current, which may be used to infer the resistivity of the borehole wall. However, due to irregular rotation and oscillation of the downhole resistivity array tool, time-varying misalignments of the horizontal positions of the electrodes occur, reducing the resolution of the imaging.
Currently, there are few methods to improve the resolution of borehole resistivity images while reducing image loss. There are three methods available:
one method is to use conventional filters for image processing, such as averaging filters, gaussian filters, convolution filters, etc. Although these methods have been developed over decades and can almost completely eliminate the misalignment phenomenon, practical simulation results show that the filter can achieve the misalignment elimination while causing serious image distortion and information loss, which is fatal to imaging. That is, the conventional filter directly processes an image globally without considering its inherent characteristics, and although the misalignment problem is corrected, the processed image is more blurred and distorted.
The second method is to calculate the offset of the adjacent pole images and add or subtract the average value of the images of each pole so as to realize the image alignment between the adjacent poles. Although this method does not cause image information loss, it does not consider the irregular condition of the rotation and swing of the downhole instrument, resulting in poor local area calibration.
And the third method is to realize dislocation correction by using resistivity imaging inversion and an image alignment splicing algorithm in the field of computer vision. The methods can ensure that the image loss is as minimum as possible while realizing the image dislocation correction, however, the methods usually need a large amount of auxiliary data for model fitting, and are time-consuming and labor-consuming.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides a self-adaptive gradient projection correction method for resistivity array imaging logging.
The specific technical scheme of the invention is as follows: a self-adaptive gradient projection correction method for resistivity array imaging logging comprises the following steps:
step one, data preprocessing, aligning each electrode image according to the odd-even number of the electrode, obtaining the data after precorrection X ═ X1,x2,...,xn]Wherein x isiThe column vector with the size of m × 1 obtained after the data X column is partitioned, wherein i is 1, …, n;
step two, projection transformation matrix calculation, namely substituting the data X obtained in the step one and the estimated regularization factor into a transformation matrix formula for calculation to obtain a projection transformation matrix P meeting the constraint condition*，
The constraint condition one is as follows:
Figure BDA0003623018780000021
wherein P is a transformation matrix (·)TIs a matrix transpose operator, Y is a corrected image in the subspace corresponding to the transform matrix P,
Figure BDA0003623018780000022
is a horizontal second-order gradient matrix, in which the first and last columns are zero vectors, | · caly | calvingFIs Frobenius norm, R is a difference matrix, and the expression is as follows:
Figure BDA0003623018780000023
the second constraint condition is as follows:
Figure BDA0003623018780000024
the constraint condition three:
Figure BDA0003623018780000025
constructing an objective function l:
Figure BDA0003623018780000026
wherein, lambda is a regularization factor of a gradient domain target function, sigma is a regularization factor of a least square term, and Tr (-) is a matrix tracing operation;
the resulting transformation matrix equation:
Figure BDA0003623018780000031
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003623018780000032
is an identity matrix;
step three, imaging correction, namely, pre-corrected data X obtained in step one and projective transformation matrix P obtained in step two*And obtaining a corrected image Y under the corresponding subspace:
Y＝(P*)TX
and step four, optimizing the parameters, judging the corrected image Y generated in the step three, if the optimized boundary range is not reached, updating the regularization factor according to the set step length by using a grid searching method, and returning to the step two until the boundary is reached.
Further, the specific process of the step four is as follows:
determining the optimal parameters of the corresponding dislocation images by searching the minimum value of the sum of the edge correction error absolute value ECE and the average gradient difference index AGD absolute value and the corresponding regularization factor, wherein the expression is as follows:
arg minλ,σ|AGD|+|ECE|.
wherein the edge correction error is as follows:
Figure BDA0003623018780000033
wherein, cijFor the discrimination coefficient, the expression is as follows:
Figure BDA0003623018780000034
wherein x isi,j、xi,j+1Respectively, a certain element of the data X and the right element thereof, and max and min are respectively the maximum element and the minimum element in the data X;
the average gradient difference index is as follows:
Figure BDA0003623018780000035
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003623018780000036
in the form of the partial derivative in the horizontal direction,
Figure BDA0003623018780000037
is the vertical partial derivative.
The invention has the beneficial effects that: based on horizontal term second-order gradient promotion caused by dislocation, the method adopts a gradient domain self-adaptive method to correct the micro-displacement, and utilizes a least square term to ensure the minimum loss and distortion of the image; meanwhile, the invention adopts simple linear transformation to realize dislocation correction, so that the dislocation of the transformed image is effectively inhibited, the minimum loss of image information can be ensured, and the balance between the dislocation correction and the loss of the image information is ensured. The method is quick and effective, not only improves the resolution and the imaging quality of the resistivity array imaging, but also efficiently solves the dislocation problem of the resistivity array imaging with the lowest possible calculation complexity.
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FIG. 1 is a schematic diagram of the structure and operation of a resistivity array imaging system in normal and irregular rotational wobble states, according to an embodiment of the present invention.
FIG. 2 is a schematic diagram of an adaptive gradient projection correction method according to an embodiment of the invention.
FIG. 3 is a graph of simulation data comparison and horizontal second order gradient comparison for an embodiment of the present invention.
Fig. 4 is a comparison diagram of an image processed by three methods under the corresponding optimal parameters according to the embodiment of the present invention.
Detailed Description
The embodiments of the present invention will be further described with reference to the accompanying drawings.
The resistivity array imaging system is constructed and operated according to the principle shown in fig. 1(a), and the horizontal position of the electrode generates time-varying dislocation due to irregular rotation and swing of the downhole resistivity array instrument, so that the resolution of a shaft image is reduced, as shown in fig. 1 (b). Some potential methods are either difficult to balance between the effect of the correction and the loss of image information or are not optimized enough in terms of correction efficiency and cost. The invention provides a self-adaptive gradient projection correction method, which mainly utilizes minimum constraint and least square constraint of a gradient domain to calculate a group of linear projections so as to realize self-adaptive projection correction aiming at a dislocation image, thereby not only realizing the balance between dislocation correction and information loss, but also improving the correction efficiency and reducing the calculation cost.
The designed adaptive gradient projection correction method aims at a new subspace, in which the image is updated under a plurality of constraint conditions, and can be determined by calculating a corresponding transformation matrix, which is expressed as:
Y＝PTX (1)
wherein the content of the first and second substances,
Figure BDA0003623018780000041
in order for the original image to be misaligned,
Figure BDA0003623018780000042
in order to transform the matrix, the matrix is,
Figure BDA0003623018780000043
for the corrected image corresponding subspace, derived from the transformation matrix P, (·)TThe operator is matrix transpose.
The invention designs an objective function for determining the transformation matrix P on the assumption that the shift of the electrode imaging causes the enhancement of the image gradient domain. Considering that the image is discrete, gradient domain adaptation can be achieved by minimizing the horizontal axis differential only, so the objective function can be written as:
Figure BDA0003623018780000044
wherein the content of the first and second substances,
Figure BDA0003623018780000045
is a horizontal second-order gradient matrix, in which the first and last columns are zero vectors, | · survivalFIs Frobenius norm, R is a difference matrix, and the expression is as follows:
Figure BDA0003623018780000051
in addition, in order to ensure that the loss of the transformed image is minimum, the embodiment introduces a least square term between the images before and after transformation, and the expression is as follows:
Figure BDA0003623018780000052
meanwhile, for controlling the multi-solution of the method, the invention sets a decompression item, and the expression is as follows:
Figure BDA0003623018780000053
to sum up all constraints, the objective function l is constructed by using a regularization method as follows:
Figure BDA0003623018780000054
wherein, λ is a regularization factor of the gradient domain objective function, and σ is a regularization factor of the least square term. Each term has 1/2 coefficients added to facilitate subsequent operations.
Next, an objective function l is optimized, and by regularizing the above constraints, l can be written as an equivalent expression as follows:
Figure BDA0003623018780000055
wherein Tr (-) is a matrix tracing operation.
Obviously, the objective function is an unconstrained problem, and the value of P can be directly determined by solving the partial derivative of P for l to make it zero.
The partial derivative of l with respect to P is expressed as follows:
Figure BDA0003623018780000056
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003623018780000057
is an identity matrix.
And (3) continuously finishing to obtain:
Figure BDA0003623018780000058
calculated P*Can be calculated by the generalized inverse as follows:
Figure BDA0003623018780000061
for the two regularization factors λ and σ involved in the estimation method, two non-reference image evaluation indexes are designed in this embodiment.
For the evaluation of the misalignment correction, the present embodiment considers the case of numerical continuity between adjacent poles, and designs the edge correction error as follows:
Figure BDA0003623018780000062
wherein, cijFor the discrimination coefficient, the expression is as follows:
Figure BDA0003623018780000063
wherein x isi,j、xi,j+1Respectively, a certain element of the data X and the right element thereof, and max and min are respectively the maximum element and the minimum element in the data X;
the smaller the index value is, the better the inter-pole image continuity is, and the better the error correction is.
However, considering that image information loss inevitably occurs when misalignment correction is simply implemented, the present embodiment designs an average gradient difference index as follows for evaluation of image quality:
Figure BDA0003623018780000064
wherein the content of the first and second substances,
Figure BDA0003623018780000065
is the partial derivative of the horizontal direction,
Figure BDA0003623018780000066
is the vertical partial derivative. The indicator is negative because the corrected image is lost compared to the original image. The larger the value, the closer to zero, indicating a smaller amount of image information loss.
The two indexes have a game relation and respectively correspond to two items of constraints designed by the method, so that the minimum value of the sum of the absolute values of the two indexes is found, the corresponding regularization factor is determined, and the optimal parameter of the corresponding dislocation image can be determined, wherein the expression of the regularization factor is as follows:
arg minλ,σ|AGD|+|ECE| (14)
in summary, the steps of the method for adaptive gradient projection correction of this embodiment are shown in fig. 2, which are specifically as follows:
step one, data preprocessing, aligning each electrode image according to the odd-even number of the electrode, obtaining the data after precorrection X ═ X1,x2,...,xn]Wherein x isiThe column vector with the size of m × 1 obtained after column partitioning of the data X is shown, where i is 1, …, n.
Step two, projective transformation matrix calculation, substituting the given data set and the estimated regularization factor into a formula (10) for calculation to obtain a projective transformation matrix P meeting the constraint condition*。
Step three, imaging correction, namely, manually corrected original imaging data and a projection transformation matrix are corrected according to a formula Y (P)*)TAnd X is calculated to obtain a corrected image Y under the corresponding subspace.
And step four, optimizing the parameters, calculating corresponding indexes and the sum of absolute values of the corrected images under the parameters according to the formulas (11) to (14), if the optimal boundary range is not reached, updating the regularization factor according to a set step length by using a grid searching method, and returning to the step two until the boundary is reached.
Fig. 3(a) is a corresponding ideal dislocation-free image, and fig. 3(b) is a simulated dislocation image set according to actual conditions, wherein three dislocations are set, and the dislocation condition at each position is different, as shown by a square frame in the figure. The group of data adopts 90 receiving button electrodes, and 180 sampling points are set. According to the hypothesis, the horizontal second-order gradients of the two groups of images are calculated to respectively obtain (c) and (d), and obvious gradient promotion is found at the corresponding dislocation positions to prove that the hypothesis is correct.
The optimal parameters obtained in this embodiment are σ ═ 0.001 and λ ═ 0.001, the corresponding corrected images are shown in fig. 4, and fig. 4 lists the corrected images corresponding to the two filters under the respective optimal parameters. The comparison shows that the adaptive gradient domain projection correction method provided by the embodiment effectively keeps the balance between the dislocation correction and the imaging quality, and can better realize the dislocation correction with low information loss than the traditional filter method.
In conclusion, the adaptive gradient domain projection correction method provided by the invention is different from the traditional filter and complex image splicing method, the offset correction is realized by adopting the quadratic derivative minimization of the gradient domain, the correction target is realized in an adaptive mode, no complex calculation process is needed, the error is accurately corrected, the time and resource cost are reduced to the minimum, and the method is easier to realize on a hardware platform.

Claims (2)

1. A self-adaptive gradient projection correction method for resistivity array imaging logging comprises the following steps:
step one, data preprocessing, aligning each electrode image according to the odd-even number of the electrode, obtaining the data after precorrection X ═ X1,x2,...,xn]Wherein x isiA column vector with the size of m × 1 obtained after partitioning the data X columns, where i is 1, …, n;
step two, projection transformation matrix calculation, namely substituting the data X obtained in the step one and the estimated regularization factor into a transformation matrix formula for calculation to obtain a projection transformation matrix P meeting the constraint condition*，
The constraint condition one is as follows:
Figure FDA0003623018770000011
wherein P is a transformation matrix (·)TFor the matrix transpose operator, Y is the corrected image in the subspace corresponding to the transform matrix P,
Figure FDA0003623018770000012
is a horizontal second-order gradient matrix, in which the first and last columns are zero vectors, | · survivalFThe Frobenius norm is obtained, R is a difference matrix, and the expression is as follows:
Figure FDA0003623018770000013
the second constraint condition is as follows:
Figure FDA0003623018770000014
the constraint condition three:
Figure FDA0003623018770000015
constructing an objective function l:
Figure FDA0003623018770000016
wherein, lambda is a regularization factor of a gradient domain target function, sigma is a regularization factor of a least square term, and Tr (-) is a matrix tracing operation;
the resulting transformation matrix formula:
Figure FDA0003623018770000021
wherein the content of the first and second substances,
Figure FDA0003623018770000022
is a unit matrix;
step three, imaging correction, namely, pre-corrected data X obtained in step one and a projective transformation matrix P obtained in step two*To obtain the corresponding sub-spaceCorrection image Y at intervals:
Y＝(P*)TX
and step four, optimizing the parameters, judging the corrected image Y generated in the step three, if the optimized boundary range is not reached, updating the regularization factor according to the set step length by using a grid searching method, and returning to the step two until the boundary is reached.
2. The method of claim 1, wherein the fourth step of determining comprises the following steps:
determining the optimal parameters of the corresponding dislocation images by searching the minimum value of the sum of the edge correction error absolute value ECE and the average gradient difference index AGD absolute value and the corresponding regularization factor, wherein the expression is as follows:
arg minλ,σ|AGD|+|ECE|.
wherein the edge correction error is as follows:
Figure FDA0003623018770000023
wherein, cijFor the discrimination coefficient, the expression is as follows:
Figure FDA0003623018770000024
wherein x isi,j、xi,j+1Respectively a certain element of the data X and a right element thereof, and max and min are respectively the maximum element and the minimum element in the data X;
the average gradient difference index is as follows:
Figure FDA0003623018770000025
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003623018770000026
in the form of the partial derivative in the horizontal direction,
Figure FDA0003623018770000027
is the vertical partial derivative.
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