Cross-correlation function in the interpretation of potential fields anomalies of strike-slip nature

. In the study, a comparison of correlation characteristics that can be obtained when analyzing areal geophysical data in COSCAD 3D, GIS INTEGRO software, as well as in custom software with a novel approach, has been conducted. It is proposed to utilize the cross-correlation function for mapping strike-slip deformations while establishing shift parameters. A synthetic model field has been constructed, and calculations have been performed in various software packages, demonstrating the advantages of calculating the cross-correlation function within a sliding window.


Introduction
Strike-slip disturbances are actively studied in various regions due to the resolution of numerous practical tasks.For instance, strike-slips are of interest in the exploration and prospecting of hydrocarbon deposits, as they give rise to pull-apart basins, typically rhomboid depressions ranging in size from small basins to hundreds of kilometers [1,2].The structure of ore bodies of noble metals deposits is often controlled by strike-slips, as the resulting faults and fractures create void space for ore deposition in a well-permeable fluid zone [3,4].Additionally, strike-slips are undoubtedly of interest from the standpoint of general geology, as they provide insights into the refinement of the structure and evolution of geological territories [5,6].

Strike-slip structures in geophysical data
The identification of fault disruptions is a standard task for potential field geophysical methods.Faults, both dips and upthrows, which represent vertical displacements, are effectively distinguished by their gravitational and magnetic fields as gradient zones or zones with a change in field characteristics [7,8].Lineaments for the identification of such zones can be accurately traced both manually and automatically based on the fields themselves or their transformations [9][10][11].
The identification of strike-slip disruptions is one of the most challenging tasks for potential field methods, as strike-slips manifest as a complex chain of anomalies with varying signs within the disruption zone, along with a shift in the axes of anomalies between displaced blocks [12].Typically, seismic exploration plays a significant role in studying such structures, where "flower structures" are recognized in cross-sections.
The goal of this study is to propose a transformation that could assist in identifying strikeslip zones and calculating displacement along fault disruptions.In this work, we focus exclusively on one aspect of the strike-slip, namely the displacement, without considering the formation of anomalies associated with strike-slip structures, such as basins, thrust folds, and others.

Correlation functions in geophysical data interpretation
Correlation characteristics are widely employed in the analysis of random and deterministic signals in geosciences [13][14][15].For instance, calculating linear and rank correlation coefficients between two fields within a moving window allows for the assessment of the strength of the relationship between the two fields and the tracking of its variation across the studied area.The primary functions used in the analysis of potential fields are autocorrelation functions (ACF) and cross-correlation functions (CCF).
The autocorrelation function   () characterizes the dependence of the relationship between the function (signal)   and its argument-shifted copy  − on the magnitude of the shift .
Calculations of autocorrelation functions (ACFs) can be used for the following purposes: Evaluating material quality: Inexplicable (geologically unexplained) variations in the ACF, easily detectable through visual analysis, can serve as an indication of defects.
Qualitative field analysis: By examining the shape of the ACF, conclusions can be drawn about the field components, anomaly sizes, their correlation properties, and energy relationships.
Selection of filtering type and parameters [16].Calculating correlation intervals using the two-dimensional ACF allows for the determination of correlation intervals (both magnitude and direction) within a moving window.The obtained characteristics can be used for: Assessing the correlation properties of the field.Zoning fields.
Estimating the position and depth of major gravity and magnetic surfaces [16].
The cross-correlation function () is defined between a pair of functions U and V using the formula: where: () is the cross-correlation function.
represents the i-th sample of signal U.   represents the i-th sample of signal V.  is the shift magnitude.Typically, signals U and V are derived from data obtained from two survey profiles, and the infinite summation limits are replaced by finite ones corresponding to the data arrays under investigation.
Cross-correlation functions (CCF) are typically used for: -assessing the correlation characteristics of the signal, particularly when there is little variation in the signal's shape from one profile to another and when the noise is uncorrelated.

Numerical experiments on synthesized data
It is advisable to initially compare the results of geophysical data interpretation using simple models, which are synthesized for an "ideal" case with no noise.In this scenario, the well-defined problem formulation allows for the identification of the most suitable interpretation method, such as the optimal field transformation.

Model field description
Let's consider a model of the disposition of an anomalous object (characterized by properties such as density and magnetization) within a homogeneous medium (see Figure 1).During its development, this object was fractured by two strike-slip faults.The field is expressed in arbitrary units, with the assumption that it corresponds to the units of measurement for gravity or magnetic anomalies in geophysics.A simple field G0 has been generated in which synthetic linear positive anomalies of meridional extension are presented with a displacement in the y-coordinate at 9 and 15 km.The displacement, simulating a fault, has a strike-slip character -making it a straightforward model for interpretation.The displacement is implemented in a westward direction (leftlateral strike-slip) by 4 km at y = 9 km and 5 km at y = 15 km.To better correspond to realworld field data, random noise with an amplitude of 1 arbitrary unit has been added.

Correlation calculations in COSCAD 3D software
The computer technology for statistical and spectral-correlation analysis of geodata, known as "COSCAD 3D," is designed for processing and interpreting geological-geophysical information organized in one-dimensional, two-dimensional, and three-dimensional regular grids using probabilistic-statistical methods.Within the "Correlation Characteristics" module, various submodules are integrated, allowing for the assessment of correlation characteristics of geofields.These include onedimensional and two-dimensional autocorrelation functions, one-dimensional and twodimensional cross-correlation functions, three-dimensional autocorrelation functions, and more [17,18].
"Autocorrelation Function" is used to compute the autocorrelation function for each profile in the original grid.
"Cross-Correlation Function between Profiles" calculates the cross-correlation function between neighboring layers' profiles in the original grid.
"Cross-Correlation Function between Fields" is used to calculate the cross-correlation function between two different features from two grids.
"Two-Dimensional Autocorrelation Function" is designed for evaluating the field's correlation properties over an area by computing the two-dimensional autocorrelation function.
"Two-Dimensional Cross-Correlation Function" calculates the two-dimensional crosscorrelation function between two features of the grid.It assesses the correlation relationships between the two features over an area and is used for building two-dimensional filters.
"Three-Dimensional Autocorrelation Function" is used to compute the three-dimensional autocorrelation function for a grid feature.This function allows for a detailed study of the field's correlation characteristics in space and is used to calculate the weighting coefficients for three-dimensional filters [17].
Let's test the operation of the CCF function between profiles for the model field (Figure 1, on the left).The calculation of this function is performed along profiles extending in the strike-slip direction.The calculation window's width is equal to an entire profile, i.e., the number of stations, and the maximum displacement is specified independently (in this case, it is set to 10).The final number of profiles is y-1 (the initial number of profiles minus one), as the calculation is carried out between two adjacent profiles.
As a result of the calculation, a per-profile representation of the one-dimensional CCF is obtained in the form of a map (Figure 2).The resulting map displays the magnitude of displacement between anomalous objects relative to each other, which is 4 km for both shifts (the actual displacements are 4 km and 5 km).It can also be assumed that the strike-slip is left-lateral, as the anomalous objects are to the left of the indicated shifts when there is no displacement.However, the obtained maps do not allow for the precise determination of the specific locations of the shifts within the examined area, making the interpretation process challenging.

Calculations in GIS INTEGRO software
The geoinformation system INTEGRO [16] includes a module called "GEOPHYSICS" with a section of functions labeled "Correlation-Spectral Analysis."This section combines procedures for calculating correlation and spectral functions for two-dimensional areal data [19][20].Two groups of procedures are highlighted: the first treats two-dimensional data as a set of one-dimensional (profile) data and calculates one-dimensional functions, while the second calculates two-dimensional correlation and spectral functions [16].
Here are the procedures in this section: "One-Dimensional Autocorrelation Functions" -This procedure allows for the assessment of the field's correlation properties along profiles.
"One-Dimensional Cross-Correlation Functions" -This procedure enables the evaluation of cross-correlation properties between profiles of the processed field, both for the entire profile and within a specified network.
"One-Dimensional Spectra" -This procedure allows for the assessment of the spectral properties of the field, both for the entire profile and within a specified network.It is possible to calculate standard amplitude, energy, and phase spectra of Fourier, as well as the power spectrum using the maximum entropy method.
"Two-Dimensional Autocorrelation Functions" -This procedure facilitates the evaluation of the correlation properties of the two-dimensional field, both for the entire field and locally within a specified network.
"Calculation of Correlation Intervals Using Two-Dimensional ACF" -This procedure allows for the estimation of the magnitude and direction of correlation intervals within a specified window of the two-dimensional ACF.
For your specific task, a normalized CCF is considered, calculated as the dependence of the sample correlation coefficient between the shared parts of displaced profiles on the magnitude of displacement.A positive displacement corresponds to the direction of extension in a clockwise manner from the line of equal stations.
In general, the CCF is calculated based on a network within the original field's network, where the stations of the new network coincide with those of the original network.The profiles run in between the profiles of the original field.The window from which data is selected for CCF calculation includes fragments of two neighboring profiles and is symmetrically placed around a node (calculation point).The most well-known case is the classical special case, where one CCF is calculated for each pair of adjacent profiles, and the window encompasses both profiles entirely.As a result of this operation, a new network with a single property is created.The number of objects in this network is equal to the number of nodes in the resulting network (when the "Classical" calculation option is chosen, the number of objects is equal to the number of profiles in the original network minus one).Each object in this network corresponds to one cross-correlation function.It stores a number (a normalization factor) and a sequence of values of the normalized CCF: CCF (negative maximum shift by stations), ..., CCF (-1), CCF (0), CCF (+1), ..., CCF (positive maximum shift by stations).
For the original field shown in Figure 1 (on the right), calculations were performed in both "Classical" and "Network" modes.In the "Classical" mode, a series of values is obtained, where, sequentially, one CCF is calculated for each pair of adjacent profiles (1,2); (2,3); ... of the original field.A graph depicting the calculated CCF values between two neighboring profiles is also generated and displayed alongside the map accordingly (Figure 3).The expected result was that within the anomaly regions, the CCF values should be high, while outside these regions, they should be randomly low.However, the graph did not show the expected result.For two profiles, the CCF value reached one (which is not reliable), and the other values fluctuated between -0.19 and 0.37.It is not possible to provide a correct explanation for the obtained result.
In the "Network" mode, values were calculated for different CCF shifts for all original grid cells.The results are displayed in the form of value tables when selecting a specific cell and involve individual analysis of each point without data export.Let's consider a cell in the area circled in red (cell #241) in Figure 4.For this cell, the CCF function was calculated with a window length of 19.The function was calculated between profiles 9 and 10, where the shift of anomalous objects is observed.
In the resulting graph, there is a peak in the CCF at a shift of -5, while the actual displacement in magnitude is 4.However, the discrepancy may be due to the non-uniformity of the positive anomaly.
The obtained results (Figure 3, Figure 4) allow for the examination of crosscorrelation functions between profiles when exporting the calculation results and visualizing them in the form of graphs.However, these results do not strongly support a spatial analysis of the data, making it challenging to create a map of a complex parameter or identify strike-slip disruptions in this context.

New approach
The proposed approach involves calculating the cross-correlation function (CCF) or correlation coefficient [21] in a sliding window between parallel profiles, limited to the sizes of the study area in specified directions.Various azimuths are considered, with the selection of the most reliable shift direction.In this case, the prior choice for the profile expansion direction is meridional.
The window size is set before the calculation begins and is determined by the scale of the studied objects and the parameters of the input data (grids).The dimensions of the sliding window are specified in terms of the number of cells.For each calculation point, two calculation parameters are recorded: CCFmax and τ, at which the maximum is observed.
Next, the sliding window is moved one unit to the right or upwards.This process calculates the CCF for each point in the original field.The algorithm produces two output grids.The first grid characterizes the displacement magnitude along the fault, and the second grid represents the maximum value of the CCF.The latter grid can be used to identify shift zones and filter the obtained anomalies.
Figure 5 shows the results of the calculation using the proposed algorithm.On the left is the map of the CCF maximum, and on the right is the result of the displacement calculation.The displacement (Figure 6, right) is determined in the range from -5 to 7 km.Two pronounced "negative" anomalies follow along the shifts that needed to be identified.There is also a chaotic pattern in the area where no shift is expected.Since the chaotic distribution of displacements is a manifestation of noise unrelated to the shift, such areas of the displacement map can be screened using the map of the maximum CCF value (Figure 5, left).
This way, the regions with well-defined anomalies become clearly visible, and two distinct shifts in the region of profiles at 9 and 15 km are highlighted.Positive displacement values correspond to right-lateral shifts, while negative values correspond to left-lateral shifts.Therefore, this pair of maps allows for the identification of the position of the shift, with the main result being the displacement (right), and the supplementary map for assessing reliability on the left in Figure 5.

Discussion
The model field allowed for the determination of the most suitable parameter for mapping strike-slip disturbances -the displacement at which the maximum crosscorrelation function (CCF) is achieved between shifting parallel profiles according to the proposed "WindowXCorr" algorithm.However, for practical application, there appear to be several challenges.The most significant challenges in the application of the algorithm likely relate to the possibility of shifts occurring in arbitrary directions, with the axes of extension of the main anomalies occurring in arbitrary directions as well (in the current example, the specified directions are orthogonal).It is probable that the identification of real shifts will also be complicated by the formation of derivatives from structural shifts, such as ridges, basins, and fault systems.Nevertheless, the work appears promising and of practical value.

Conclusions
The study compared the results of three software products oriented towards utilizing correlation characteristics of areal data: COSCAD 3D, GIS INTEGRO, and "WindowXCorr."The proposed algorithm represents a new approach that allows for the computation and visualization of fundamentally new information in the form of a map characterizing the displacement associated with strike-slip tectonic deformations.This algorithm has been chosen as the optimal one and will be further used for testing with real geophysical data.

Fig. 1 .
Fig. 1.The model field in arbitrary units -representing the idea of the model G0 (on the left); the map with the addition of random noise to the field G (on the right).The lines depict the simulated displacements of anomalies.

Fig. 2 .
Fig. 2. The model field (on the left) and the cross-correlation function (CCF) between profiles obtained using COSCAD 3D (on the right).

Fig. 3 .
Fig. 3.The result of the calculation in GIS INTEGRO in "Classical" modetable and graph of CCF value along lines

Fig. 4 .
Fig. 4. The result of the calculation in GIS INTEGRO in "Network" modetable and graph of CCF value at marked point

Fig. 5 .
Fig. 5.The result of the calculation in the author's program "WindowXCorr" for identifying shifts: a map of the value of the maximum CCF (left) and the shift at which the maximum is reached (right)