A sidereal day filter correction method for multipath effect in landslide deformation monitoring

Abstract

The invention discloses a multipath effect sidereal day filter correction method suitable for landslide deformation monitoring, comprising: S1, collecting GNSS monitoring data; S2, preprocessing the GNSS monitoring data; S3, identifying landslide critical points: identifying the number and position of critical points of the landslide deformation stage of the monitoring point according to the coordinate residual sequence set, and determining the stage of the landslide deformation by the number of critical points; S4, determining the selected coordinate residual sequence and the coordinate residual sequence to be corrected according to the number and position of the landslide deformation critical points; S5, calculating the multipath correlation delay and constructing a multipath repetition model; S6, using the multipath repetition model to correct the coordinate residual sequence to be corrected in S4, and obtaining the corrected coordinate residual sequence. The method of the invention has low complexity, is simple and fast, retains the landslide deformation information of the monitoring point, and effectively reduces the influence of the multipath effect on the monitoring accuracy.

Description

Technical Field

[0001] The invention relates to the technical field of multipath effect correction, and in particular to a multipath effect sidereal day filtering correction method suitable for landslide deformation monitoring. Background Art

[0002] The Global Navigation Satellite System (GNSS) has the advantages of all-weather, high-precision, real-time and automation, and has been widely used in transportation, geological disaster monitoring, agriculture and other fields. The advantages of GNSS are more prominent in landslide deformation monitoring. Through high-precision positioning technologies such as RTK, GNSS can achieve millimeter-level displacement monitoring and adapt to landslide monitoring under complex terrain and environmental conditions. The real-time data transmission and automated monitoring functions of GNSS help to quickly obtain deformation information in the landslide area, and can continuously monitor the dynamic changes in the landslide area, providing key support for landslide early warning and disaster management.

[0003] When GNSS is used for landslide deformation monitoring, the multipath effect caused by the special terrain in the landslide area and surrounding areas has an impact on positioning accuracy that cannot be ignored. There are three commonly used multipath effect correction methods:

[0004] (1) Reducing the multipath effect based on hardware design, such as using a choke antenna, but this has problems such as limited receiver size and increased cost.

[0005] (2) Multipath effects are reduced based on received signal processing strategies, such as multipath delay locked loop and extended Kalman filter. However, these strategies have some problems, such as poor application effects in some scenarios and residual multipath errors cannot be ignored.

[0006] (3) Use data post-processing methods to reduce the multipath effect, such as the commonly used sidereal day filtering method and signal-to-noise ratio method, among which the sidereal day filtering method is the most widely used.

[0007] The traditional correction method of multipath effect sidereal day filtering usually uses the coordinate residual sequence of the monitoring point one day before to construct a multipath repetition model, such as by using wavelet transform, local mean decomposition and other methods to extract low-frequency signals to construct a multipath repetition model. This method is usually only applicable to multipath effect correction in the case of no displacement deformation. However, in GNSS landslide deformation monitoring, the displacement deformation of the landslide (usually divided into three stages: initial stage, uniform deformation and accelerated deformation) cannot be ignored. At this time, the multipath error and landslide displacement deformation are mainly concentrated in the low-frequency part, which is difficult to further distinguish through low-pass filtering. Therefore, if this traditional method is used for multipath correction, important displacement information of the landslide may be lost, resulting in the failure of the corrected monitoring sequence to reflect the displacement trend of the landslide body itself.

[0008] Therefore, a sidereal day filtering correction method that considers the multipath effect of the landslide deformation stage is urgently needed. Summary of the invention

[0009] In view of the problems existing in the prior art, the present invention provides a multi-path effect stellar day filtering correction method suitable for landslide deformation monitoring, which solves the problem that when traditional stellar day filtering is used to extract a multi-path repetitive model in landslide deformation monitoring, the deformation displacement of the monitoring point is not considered to correct the multi-path effect, which affects the subsequent monitoring accuracy.

[0010] To this end, the present invention adopts the following technical solutions:

[0011] 1. A multipath effect sidereal day filtering correction method suitable for landslide deformation monitoring, comprising the following steps:

[0012] S1, GNSS monitoring data collection:

[0013] Deploy GNSS equipment at a monitoring point in the landslide area to obtain GNSS monitoring data of the monitoring point for multiple consecutive days;

[0014] S2, preprocessing of GNSS monitoring data:

[0015] Eliminate gross errors in GNSS monitoring data, interpolate missing data in GNSS monitoring data, and obtain a set of coordinate residual sequences;

[0016] S3, identification of landslide critical point:

[0017] According to the coordinate residual sequence set, the number and position of the critical points of the landslide deformation stage of the monitoring point are identified, and the stage of the landslide deformation is determined by the number of critical points. The landslide deformation stage includes the initial deformation stage, the uniform deformation stage and the accelerated deformation stage;

[0018] The parameters of the ischange function are set, including the linear change detection method and the maximum number of change points, where the maximum number of change points is 2; the ischange function determines the number and position of critical points of each coordinate residual sequence in the coordinate residual sequence set X according to the sudden change of the slope and the intercept;

[0019] S4, determine the selected coordinate residual sequence and the coordinate residual sequence to be corrected:

[0020] According to the number and position of the critical points of landslide deformation, when the landslide deformation enters a new stage, if the new stage is a uniform deformation stage, the coordinate residual sequence of the initial stage is used to calculate the multipath correlation delay and construct a multipath repetition model, and is used to correct the coordinate residual sequence of the uniform deformation stage; if the new stage is an accelerated deformation stage, the coordinate residual sequence of the uniform deformation stage is used to calculate the multipath correlation delay and construct a multipath repetition model, and is used to correct the coordinate residual sequence of the accelerated deformation stage;

[0021] S5, calculating the multipath related delay and constructing a multipath repetition model, including the following steps:

[0022] S51, calculate the correlation coefficient R based on the selected coordinate residual sequence xy (m), and use the correlation coefficient R xy (m) Estimate the multipath correlation delay T(m) corresponding to the coordinate residual sequence of adjacent days. The calculation formula is as follows:

[0023]

[0024] T(m)=mN

[0025] Among them, R xy 9m) is the correlation coefficient, x n and n is the selected coordinate residual sequence and the coordinate residual sequence to be corrected, E is the expected operator, * is the conjugate complex number, N is the length of the coordinate residual sequence, m = 1, 2, ..., 2N-1, if R = R xy (m) is the maximum value, then the offset T(m) corresponding to m is taken as the multipath correlation delay;

[0026] S52, using a local mean decomposition algorithm to perform noise reduction processing on the selected coordinate residual sequence to construct a multi-path repetition model;

[0027] Among them, the method of constructing a multi-path repetition model is:

[0028]

[0029] in, For the multi-path repetitive model, PF i (t) is the decomposed PF component, u k (t) is the residual component after decomposition, k is the number of components obtained after performing local mean decomposition on the coordinate residual sequence, and j is the dividing point between the high-frequency PF noise component and the low-frequency PF multipath component, which is calculated by the following formula:

[0030]

[0031] Where CMSE() represents the mean square error function, Respectively represent PF l PF l+1 The signal after starting reconstruction, N is the length of the coordinate residual sequence, l = 1, 2, ..., k, j is the minimum point of the above formula (5), that is, the dividing point;

[0032] S6, using a multipath repetition model to correct the coordinate residual sequence to be corrected in S4, to obtain a corrected coordinate residual sequence.

[0033] The method for eliminating gross errors described in S2 is:

[0034] The triple interquartile range method is used to calculate the outlier limit of GNSS monitoring data, and the data exceeding the outlier limit in the GNSS monitoring data are removed as gross errors. The outlier limit calculation formula is:

[0035] [Q1-3×IQR,Q3+3×IQR]

[0036] Where Q1 and Q3 are the first quartile and the third quartile of the GNSS monitoring data, respectively, IQR is the interquartile range, and IQR = Q3-Q1.

[0037] The interpolation process in S2 adopts the spline interpolation method.

[0038] In S2, a coordinate residual sequence is data within 24 consecutive hours.

[0039] In S3, when the following equation holds true, it means that the coordinate residual sequence X contains critical points:

[0040] C(S1)+C(S2)+ε<C(X)

[0041] Among them, C is the cost function, ε is the set positive threshold, and S1 and S2 are two sequences of coordinate residual sequences.

[0042] In S4, if the number of critical points of landslide deformation is 1, the coordinate residual sequence of the initial stage before the critical point is used to construct a multipath repetition model and calculate the multipath correlation delay, and is used to correct the coordinate residual sequence of the uniform deformation stage; if the number of critical points of landslide deformation is 2, the coordinate residual sequence of the initial stage before the first critical point is used to construct a multipath repetition model and calculate the multipath correlation delay, and is used to correct the coordinate residual sequence of the uniform deformation stage, and the coordinate residual sequence of the uniform deformation stage before the second critical point is used to construct a multipath repetition model and calculate the multipath correlation delay, and is used to correct the coordinate residual sequence of the accelerated deformation stage.

[0043] In S51, the 30 seconds before and after the multipath nominal repetition period of 236 seconds is used as the search range, the correlation coefficients of adjacent coordinate residual sequences are traversed and calculated, and the maximum value of the correlation coefficient is used as the advance amount. The traversal interval is 1s; if the coordinate residual sequence of three days or more is used to calculate the advance amount, the search range is adjusted to: 236 seconds × interval time ± 30 seconds, interval time = total number of coordinate residual sequences - 1.

[0044] In S6, when the correction process is performed, the translation step length of the coordinate residual sequence to be corrected is the maximum value of the correlation coefficient calculated in S4.

[0045] The multipath effect sidereal day filtering correction method suitable for landslide deformation monitoring of the present invention pre-processes the original monitoring data, including gross error detection and repair, then identifies the critical point of the landslide deformation stage, determines the current deformation stage of the monitoring point, then calculates the multipath repetition period and uses the local mean decomposition method to decompose the GNSS coordinate residual sequence to eliminate high-frequency random noise and construct a multipath repetitive signal, and finally corrects the subsequent GNSS monitoring data to achieve multipath effect correction while taking into account the deformation displacement of the monitoring point, thereby improving the accuracy of deformation monitoring.

[0046] The present invention has the following beneficial effects:

[0047] 1. The present invention can use continuous GNSS deformation monitoring data to extract a multipath repetitive model, and perform multipath effect correction processing on subsequent GNSS coordinate residual sequences, thereby effectively reducing the impact of the multipath effect on the GNSS monitoring positioning accuracy;

[0048] 2. Compared with the traditional sidereal day filtering method, the present invention also takes into account the deformation stage of the landslide. On this basis, the GNSS monitoring data of the appropriate stage is selected to construct a multi-path repetition model, which will not lose the important displacement information of the landslide contained in the monitoring data and make the multi-path correction result more accurate.

[0049] 3. Compared with the traditional method of reducing the multipath effect by hardware design, the present invention reduces the multipath effect from the algorithm level, which is simple to operate and has lower cost. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 A flowchart of the correction method of the present invention;

[0051] Figure 2 The figure is an effect diagram of multi-path correction of a landslide monitoring point using the method of the present invention. DETAILED DESCRIPTION

[0052] The technical solution of the present invention is described in detail below in conjunction with the accompanying drawings and embodiments.

[0053] Figure 1 FIG. 1 is a flow chart of a multipath effect sidereal day filtering correction method for landslide deformation monitoring according to the present invention. Figure 1 As shown, the method comprises the following steps:

[0054] S1, GNSS monitoring data collection:

[0055] A GNSS device is deployed at a monitoring point in the landslide area to obtain GNSS monitoring data of the monitoring point for multiple consecutive days.

[0056] S2, preprocessing of GNSS monitoring data:

[0057] The gross errors of the GNSS monitoring data are eliminated, and the missing data in the GNSS monitoring data are interpolated to obtain a set of coordinate residual sequences, wherein the set of coordinate residual sequences includes multiple coordinate residual sequences, and one coordinate residual sequence is data within 24 consecutive hours.

[0058] The method for eliminating gross errors is: using the triple interquartile range method to calculate the outlier limit of the GNSS monitoring data, and eliminating the data in the GNSS monitoring data that exceeds the outlier limit as gross errors. The outlier limit calculation formula is:

[0059] [Q1-3×IQR,Q3+3×IQR]

[0060] Where Q1 and Q3 are the first quartile and the third quartile of the GNSS monitoring data, respectively, IQR is the interquartile range, and IQR = Q3-Q1.

[0061] The interpolation process adopts a spline interpolation method.

[0062] S3, identification of landslide critical point:

[0063] The number and position of critical points of the landslide deformation stage of the monitoring point are identified according to the set of coordinate residual sequences. The stage of landslide deformation is determined by the number of critical points. The landslide deformation stage includes the initial deformation stage, uniform deformation stage and accelerated deformation stage.

[0064] The number and position of critical points of each coordinate residual sequence in the set are identified by the ischange function (matlab), and the parameters of the ischange function are set (linear change detection method, maximum number of change points is 2). The ischange function determines the number and position of critical points of landslide deformation in the coordinate residual sequence set X according to the sudden change of slope and intercept.

[0065] When the following equation holds true, it means that the coordinate residual sequence X contains critical points:

[0066] C(S1)+C(S2)+ε<C(X)

[0067] Among them, C is the cost function, ε is the set positive threshold (to prevent over-segmentation), and S1 and S2 are two sequences of coordinate residual sequences.

[0068] S4, determining the selected coordinate residual sequence and the coordinate residual sequence to be corrected.

[0069] According to the number and position of the critical points of landslide deformation, when the landslide deformation enters a new stage, if the new stage is a uniform deformation stage, the coordinate residual sequence of the initial stage is used to construct a multi-path repetitive model, and it is used to correct the coordinate residual sequence of the uniform deformation stage; if the new stage is an accelerated deformation stage, the coordinate residual sequence of the uniform deformation stage is used to construct a multi-path repetitive model, and it is used to correct the coordinate residual sequence of the accelerated deformation stage.

[0070] If the number of landslide deformation critical points is 1, the coordinate residual sequence of the initial stage before the critical point is used to construct the multipath repetition model and calculate the multipath correlation delay, and is used to correct the coordinate residual sequence of the uniform deformation stage. If the number of landslide deformation critical points is 2, the coordinate residual sequence of the initial stage before the first critical point is used to construct the multipath repetition model and calculate the multipath correlation delay, and is used to correct the coordinate residual sequence of the uniform deformation stage, and the coordinate residual sequence of the uniform deformation stage before the second critical point is used to construct the multipath repetition model and calculate the multipath correlation delay, and is used to correct the coordinate residual sequence of the accelerated deformation stage.

[0071] S5, calculating the multipath related delay and constructing a multipath repetition model, including the following steps:

[0072] S51, calculate the correlation coefficient R based on the selected coordinate residual sequence xy (m), and use the correlation coefficient R xy (m) Estimate the lead time T(m) corresponding to the coordinate residual sequence of adjacent days, that is, the related delay of the coordinate residual sequence. The calculation formula is as follows:

[0073]

[0074] T(m)=mN

[0075] Among them, R xy (m) is the correlation coefficient, x n and n is the coordinate residual sequence (the selected coordinate residual sequence and the coordinate residual sequence to be corrected), E is the expected operator, * is the conjugate complex number, N is the length of the coordinate residual sequence, m = 1, 2, ..., 2N-1, if R = R xy(m) is the maximum value, then the offset T(m) corresponding to m is used as the advance amount.

[0076] The search range is 30 seconds before and after the nominal advance (multi-path nominal repetition period 236 seconds), and the correlation coefficients of adjacent coordinate residual sequences are calculated traversally. The maximum value of the correlation coefficient is used as the advance, and the traversal interval is 1 second. If the coordinate residual sequence of three days or more is used to calculate the advance, the search range is adjusted to (236 seconds × interval time ± 30 seconds, interval time = total number of coordinate residual sequences - 1).

[0077] S52, using a local mean decomposition algorithm to perform noise reduction processing on the selected coordinate residual sequence to construct a multi-path repetition model;

[0078] Among them, the method of constructing a multi-path repetition model is:

[0079]

[0080] in, For the multi-path repetitive model, PF i (t) is the decomposed PF component, u k (t) is the residual component after decomposition, k is the number of components obtained after performing local mean decomposition on the coordinate residual sequence, and j is the dividing point between the high-frequency PF noise component and the low-frequency PF multipath component, which is calculated by the following formula:

[0081]

[0082] Where CMSE() represents the mean square error function, Respectively represent PF l PF l+1 The reconstructed signal (calculated by equation (4)) is started, N is the length of the coordinate residual sequence, l = 1, 2, ..., k, and j is the minimum point of equation (5), i.e., the dividing point.

[0083] S6, using the multipath repetition model to correct the coordinate residual sequence to be corrected in S4, to obtain a corrected coordinate residual sequence. When performing the correction process, the translation step length of the coordinate residual sequence to be corrected is the maximum value of the correlation coefficient calculated in S4, retaining the landslide deformation information of the monitoring point and reducing the impact of the multipath error on the monitoring accuracy.

[0084] like Figure 2 As shown in the figure, the GNSS monitoring data of two consecutive days at a monitoring point in a landslide area are selected to perform GNSS multipath effect sidereal day filtering correction for landslide deformation monitoring:

[0085] In this embodiment, the data sampling rate is 1s, the satellite cut-off elevation angle is 10 degrees, and the coordinate residual sequence of the monitored point for two days is obtained by dynamic epoch-by-epoch solution, namely DOY090 and DOY091. The landslide critical point identification method in S3 shows that the coordinate residual sequence DOY091 has a critical point, which means that the landslide is in the initial stage and the uniform deformation stage (new stage). Therefore, the coordinate residual sequence DOY090 in the initial stage is selected to construct a multipath repetition model. The local mean decomposition algorithm is used to perform denoising on the coordinate residual sequence DOY090 in the initial stage and construct a multipath repetition model. The obtained multipath repetition model is as follows: Figure 2 As shown in the third sub-figure (from top to bottom) in the figure, the calculated advance is 242s. The multipath correction processing of the coordinate residual sequence DOY091 in the uniform deformation stage is performed using the advance and multipath repetition model, and the corrected residual sequence is obtained as follows Figure 2 As shown in the fourth sub-figure (from top to bottom) in FIG, the RMS value of the coordinate residual sequence DOY091 in the uniform deformation stage after correction is reduced by 26%.

[0086] In addition, if it is identified that a certain continuous coordinate residual sequence (not necessarily on the same day) contains two critical points, that is, this sequence is divided into an initial stage, a uniform deformation stage, and an accelerated deformation stage, the coordinate residual sequence of the initial stage is used to construct a multipath repetitive model, and the coordinate residual sequence of the uniform deformation stage is corrected; then the coordinate residual sequence of the uniform deformation stage is used to construct a multipath repetitive model, and the coordinate residual sequence of the accelerated deformation stage is corrected.

[0087] Therefore, the GNSS multipath effect sidereal day filtering correction method of the present invention selects GNSS monitoring data of a suitable stage to construct a multipath repetition model under the premise of considering the deformation stage of the landslide monitoring point. Not only does it not lose the important landslide displacement information contained in the subsequent monitoring data, but it also makes the multipath correction result more accurate, effectively reducing the impact of the multipath effect on the monitoring accuracy.

Claims

1. A multipath effect sidereal day filtering correction method suitable for landslide deformation monitoring, characterized in that: The following steps are involved: S1, GNSS monitoring data collection: Deploy GNSS equipment at a monitoring point in the landslide area to obtain GNSS monitoring data of the monitoring point for multiple consecutive days; S2, preprocessing of GNSS monitoring data: Eliminate gross errors in GNSS monitoring data, interpolate missing data in GNSS monitoring data, and obtain a set of coordinate residual sequences; S3, identification of landslide critical point: According to the coordinate residual sequence set, the number and position of the critical points of the landslide deformation stage of the monitoring point are identified, and the stage of the landslide deformation is determined by the number of critical points. The landslide deformation stage includes the initial deformation stage, the uniform deformation stage and the accelerated deformation stage; The parameters of the ischange function are set, including the linear change detection method and the maximum number of change points, where the maximum number of change points is 2; the ischange function determines the number and position of critical points of each coordinate residual sequence in the coordinate residual sequence set X according to the sudden change of the slope and the intercept; S4, determine the selected coordinate residual sequence and the coordinate residual sequence to be corrected: According to the number and position of the critical points of landslide deformation, when the landslide deformation enters a new stage, if the new stage is a uniform deformation stage, the coordinate residual sequence of the initial stage is used to calculate the multipath correlation delay and construct a multipath repetition model, and is used to correct the coordinate residual sequence of the uniform deformation stage; if the new stage is an accelerated deformation stage, the coordinate residual sequence of the uniform deformation stage is used to calculate the multipath correlation delay and construct a multipath repetition model, and is used to correct the coordinate residual sequence of the accelerated deformation stage; S5, calculating the multipath related delay and constructing a multipath repetition model, including the following steps: S51, calculate the correlation coefficient R based on the selected coordinate residual sequence xy (m), and use the correlation coefficient R xy (m) Estimate the multipath correlation delay T(m) corresponding to the coordinate residual sequence of adjacent days. The calculation formula is as follows: T(m)=mN Among them, R xy (m) is the correlation coefficient, x n and n is the selected coordinate residual sequence and the coordinate residual sequence to be corrected, E is the expected operator, * is the conjugate complex number, N is the length of the coordinate residual sequence, m = 1, 2, ..., 2N-1, if R = R xy (m) is the maximum value, then the offset T(m) corresponding to m is taken as the multipath correlation delay; S52, using a local mean decomposition algorithm to perform noise reduction processing on the selected coordinate residual sequence to construct a multi-path repetition model; Among them, the method of constructing a multi-path repetitive model is: in, For the multi-path repetitive model, PF i (t) is the decomposed PF component, u k (t) is the residual component after decomposition, k is the number of components obtained after performing local mean decomposition on the coordinate residual sequence, and j is the dividing point between the high-frequency PF noise component and the low-frequency PF multipath component, which is calculated by the following formula: Where CMSE() represents the mean square error function, Respectively represent PF l PF l+1 The signal after starting reconstruction, N is the length of the coordinate residual sequence, l = 1, 2, ..., k, j is the minimum point of the above formula (5), that is, the dividing point; S6, using a multipath repetition model to correct the coordinate residual sequence to be corrected in S4, to obtain a corrected coordinate residual sequence.

2. The multipath effect sidereal day filtering correction method according to claim 1, characterized in that: The method for eliminating gross errors described in S2 is: The triple interquartile range method is used to calculate the outlier limit of GNSS monitoring data, and the data exceeding the outlier limit in the GNSS monitoring data are removed as gross errors. The outlier limit calculation formula is: [Q1-3×IQR,Q3+3×IQR] Where Q1 and Q3 are the first quartile and the third quartile of the GNSS monitoring data, respectively, IQR is the interquartile range, and IQR = Q3-Q1.

3. The multipath effect sidereal day filtering correction method according to claim 1, characterized in that: The interpolation process in S2 adopts the spline interpolation method.

4. The multipath effect sidereal day filtering correction method according to claim 1, characterized in that: In S2, a coordinate residual sequence is the data within 24 consecutive hours.

5. The multipath effect sidereal day filtering correction method according to claim 1, characterized in that: In S3, when the following equation holds true, it means that the coordinate residual sequence X contains critical points: C(S1)+C(S2)+ε<C(X) Among them, C is the cost function, ε is the set positive threshold, and S1 and S2 are two sequences of coordinate residual sequences.

6. The multipath effect sidereal day filtering correction method according to claim 1, characterized in that: In S4, if the number of critical points of landslide deformation is 1, the coordinate residual sequence of the initial stage before the critical point is used to construct a multipath repetition model and calculate the multipath correlation delay, and is used to correct the coordinate residual sequence of the uniform deformation stage; if the number of critical points of landslide deformation is 2, the coordinate residual sequence of the initial stage before the first critical point is used to construct a multipath repetition model and calculate the multipath correlation delay, and is used to correct the coordinate residual sequence of the uniform deformation stage, and the coordinate residual sequence of the uniform deformation stage before the second critical point is used to construct a multipath repetition model and calculate the multipath correlation delay, and is used to correct the coordinate residual sequence of the accelerated deformation stage.

7. The multipath effect sidereal day filtering correction method according to claim 1, characterized in that: In S51, the search range is 30 seconds before and after the multipath nominal repetition period of 236 seconds, and the correlation coefficients of adjacent coordinate residual sequences are traversed and calculated. The maximum value of the correlation coefficient is used as the advance amount, and the traversal interval is 1s; if the coordinate residual sequence of three days or more is used to calculate the advance amount, the search range is adjusted to: 236 seconds × interval time ± 30 seconds, interval time = total number of coordinate residual sequences - 1.

8. The multipath effect sidereal day filtering correction method according to claim 1, characterized in that: In S6, when the correction process is performed, the translation step length of the coordinate residual sequence to be corrected is the maximum value of the correlation coefficient calculated in S4.

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