CN114782276B - Resistivity imaging dislocation correction method based on adaptive gradient projection - Google Patents
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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 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.
Description
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 wobbling 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 approach 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 realize image dislocation correction and simultaneously ensure that the image loss is the lowest possible, 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 serial number of the electrode to obtain pre-corrected data X = [ X ] 1 ,x 2 ,...,x n ]Wherein x is i The method is a column vector with the size of m multiplied by 1 obtained after the data X column is divided into blocks, wherein i =1, \8230;, 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:
wherein P is a transformation matrix (·) T Is a matrix transpose operator, Y is a corrected image in the subspace corresponding to the transform matrix P,
is a horizontal second-order gradient matrix, in which the first and last columns are zero vectors, | · caly | calving F The Frobenius norm is obtained, R is a difference matrix, and the expression is as follows:
the constraint condition two:
the constraint condition three:
constructing an objective function l:
wherein, λ is a regularization factor of a gradient domain target function, σ is a regularization factor of a least square term, and Tr (-) is a matrix tracing operation;
the resulting transformation matrix formula:
wherein,
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 * And obtaining a corrected image Y under the corresponding subspace:
Y＝(P * ) T X
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 search method, and returning to the step two until the boundary is reached.
Further, the specific process of the step four of discriminating 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 min λ,σ |AGD|+|ECE|.
wherein the edge correction error is as follows:
wherein, c ij For the discrimination coefficient, the expression is as follows:
wherein x is i,j 、x i,j+1 Respectively 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:
wherein,
in the horizontal partial derivative, in>
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 micro-displacement, and utilizes a least square term to ensure that the loss and distortion of an image are minimum; 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.
Drawings
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 is directed to a new subspace in which the image is updated under a plurality of constraints and can be determined by calculating a corresponding transformation matrix, which is expressed as:
Y＝P T X (1)
wherein，
For a displaced original image, a>
For changing the matrix, is>
For the corrected image corresponding subspace found by the transformation matrix P (·) T The 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:
wherein,
is a horizontal second-order gradient matrix, in which the first and last columns are zero vectors, | · caly | calving F The Frobenius norm is obtained, R is a difference matrix, and the expression is as follows:
furthermore, in order to ensure that the loss of the transformed image is minimal, the present embodiment introduces a least square term between the images before and after transformation, and the expression is as follows:
meanwhile, for controlling the multi-solution of the method, the invention sets a decompression item, and the expression is as follows:
to sum up all constraints, the objective function l is constructed by using a regularization method as follows:
wherein, λ is a regularization factor of the gradient domain target function, and σ is a regularization factor of the least square term. Each item is added with a 1/2 coefficient to facilitate subsequent operation.
Next, an objective function l is optimized, and by regularizing the above constraints, l can be written as an equivalent expression as follows:
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:
wherein,
is an identity matrix.
And (3) continuously finishing to obtain:
calculated P * Can pass throughThe generalized inverse is calculated as follows:
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:
wherein, c ij For the discrimination coefficient, the expression is as follows:
wherein x is i,j 、x i,j+1 Respectively 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 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:
wherein,
in the horizontal partial derivative, in>
Is the vertical partial derivative. Since the corrected image has a loss compared to the original image, the index is negative. The larger the numerical value, the closer to zero, the smaller the loss amount of image information.
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 min λ,σ |AGD|+|ECE| (14)
in summary, the steps of the adaptive gradient projection correction method of the present embodiment are shown in fig. 2, which are as follows:
step one, data preprocessing, aligning each electrode image according to the odd-even serial number of the electrode to obtain pre-corrected data X = [ X ] 1 ,x 2 ,...,x n ]Wherein x is i The column vector with size m × 1 is obtained after dividing the data X columns into blocks, where i =1, \ 8230;, 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 projective transformation matrix are subjected to Y = (P) according to a formula * ) T And X is calculated to obtain a corrected image Y under the corresponding subspace.
And step four, optimizing the parameters, calculating corresponding indexes of the corrected images under the parameters according to the formulas (11) to (14) and the sum of absolute values, if the range of the optimized boundary is not reached, updating the regularization factor according to the set step length by using a grid search 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, so that the hypothesis is proved to be correct.
The optimal parameters obtained by this embodiment are σ =0.001 and λ =0.001, the corresponding corrected image is shown in fig. 4, and fig. 4 lists the corrected images corresponding to the two filters under the respective optimal parameters. Comparison shows that the adaptive gradient domain projection correction method provided by the embodiment effectively keeps balance between dislocation correction and imaging quality, and can better realize dislocation correction with low information loss compared with the traditional filter method.
In summary, the adaptive gradient domain projection correction method provided by the invention is different from the traditional filter and complex image stitching methods, the offset correction is realized by adopting the quadratic derivative minimization of the gradient domain, the correction target is realized in a self-adaptive mode, no complex calculation process is needed, the error is accurately corrected, the time and resource cost is 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 images of each electrode according to odd-even serial numbers of the electrodes to obtain pre-corrected data X = [ X [ ] 1 ,x 2 ,...,x n ]Wherein x is i The column vector with the size of m multiplied by 1 is obtained after the data X column is divided into blocks, wherein i =1, \ 8230;, 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 * ，
Constraint one:
wherein P is a transformation matrix (·) T For the matrix transpose operator, Y is the corrected image in the subspace corresponding to the transform matrix P,
is a horizontal second-order gradient matrix, in which the first and last columns are zero vectors, | · survival F The Frobenius norm is obtained, R is a difference matrix, and the expression is as follows:
and the constraint condition two is as follows:
constraint condition three:
constructing an objective function l:
wherein, λ is a regularization factor of a gradient domain target function, σ is a regularization factor of a least square term, and Tr (-) is a matrix tracing operation;
the resulting transformation matrix formula:
wherein,
is an identity matrix;
step three, imaging correction, namely, the pre-corrected data X obtained in the step one and the projection transformation obtained in the step twoMatrix P * And obtaining a corrected image Y under the corresponding subspace:
Y＝(P * ) T X
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 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 min λ,σ |AGD|+|ECE|.
wherein the edge correction error is as follows:
wherein, c ij For the discrimination coefficient, the expression is as follows:
wherein x is i,j 、x i,j+1 Respectively 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:
wherein,
is a horizontal partial derivative, is based on>
Is the vertical partial derivative. />
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