An automatic algorithm for surface wave dispersion curve picking based on Hessian matrix attributes

Abstract

A new automatic method for dispersion curve picking based on Hessian matrix attributes is proposed, on which, an algorithm is developed in this paper. The algorithm is based on dispersion power spectra transformed from surface waves. It fulfills the automatic picking of the surface wave dispersion curves from the fundamental to high orders, by ridge searching and extraction, ridge line segment connection, dispersion curve selecting and order sorting. The algorithm does not have to rely on model training and there is no manual interaction requirement. This inherent efficiency advantage of the algorithm provides an efficient tool for surface wave dispersion curve picking for large projects such as oil and gas exploration. The algorithm is tested using two conceptual models. On the synthetic data of the test models, surface wave dispersion curves up to the 8th order are successfully picked, which shows the excellent picking ability of the method. Further comparative experiments show that the algorithm is superior to other methods in terms of picking accuracy, completeness, anti-noise performance and computation efficiency. In order to evaluate the practicability of the algorithm, we applied it to surface wave data from an industrial prospecting project. Three dispersion curves, including the fundamental order, first order and second order, are successfully obtained. The dispersion curve inversion result matches perfectly with the vertical seismic profile. This application case verified the effectiveness and practicability of the algorithm.

Introduction

Surface wave method is widely used in engineering and oil and gas explorations^1–5. Traditionally, surface wave dispersion curves need to be picked manually. Because of low efficiency, it is difficult to meet the production requirements of large-scale exploration projects. Therefore, automatic picking methods are desired for efficiency purpose. Galiana-Merino et al.⁶ designed a method, based on color processing and morphology techniques, for automatic picking of the fundamental order dispersion curve for Louie’s⁷ refraction micromotion method. Xie et al.⁸ proposed another picking method using virtual shot gather constructed by cross-correlation. However, this method can only pick the fundamental order dispersion curve.

Zheng and Miao⁹ proposed a semi-automatic picking method by combining the binarization method and the thinning method. Taipodia et al.¹⁰ proposed an automatic dispersion curve picking method by using an energy threshold. However, both methods have the problem of missing weak energy, which affects the picking of high-order dispersion curves.

In recent years, artificial intelligence has been introduced into dispersion curve picking. Alyousuf et al.¹¹ use a three-layer neural network model to pick the fundamental order dispersion curve successfully. Dai et al.¹² treat the picking of dispersion curves as an instance segmentation task and proposes a dispersion curve automatic picking network, i.e., DCNet, which can pick up both fundamental order and high-order dispersion curves. Kaul et al.¹³ assumes the automatic picking process of dispersion curve as a binary segmentation task, adopt the Unet network structure of Ronneberger et al.¹⁴ with an replacement of the Unet’s convolution blocks by residual blocks¹⁵ to extract the fundamental dispersion curve. Zhang et al.¹⁶ adopted the Unet network of Ronneberger et al.¹⁴ to pick up the fundamental order and first-order dispersion curves. Song et al.¹⁷ developed an automatic method by adoption of the Unet++ network structure of Zhou et al.¹⁸, which can successfully pick up both fundamental and high-order dispersion curves. The DisperNet method of Dong et al.¹⁹, developed by combining the Unet¹⁴ and hierarchical cluster analysis (HCA), can also pick up both fundamental and high-order dispersion curves. However, these methods belong to supervised learning. As these methods rely on model training. New model training is required for every new prospecting project.

In order to get rid of the dependence on models, unsupervised learning methods have emerged. For example, Rovetta et al.²⁰ proposed a dispersion curve automatic picking method based on Density-Based Spatial Clustering of Applications with Noise (DBSCAN). By determining the maximum energy position, DBSAN method is used to clustering and then the dispersion curves are picked. But when the dispersion power spectrum is messy, this method cannot pick up accurate dispersion curve. Wang et al.²¹, by combining Gaussian Mixture Model (GMM) and DBSCAN, proposed a new Unsupervised Machine Learning Clustering algorithm (UMC). Because the GMM algorithm is sensitive to initialization²², the result of UMC method is subject to the accuracy of the initial value.

To sum up, traditional dispersion curve automatic picking methods solve the picking efficiency problem partly, but the picking ability of high order dispersion curve is poor. Some automatic picking methods based on neural networks can pick up multi-order dispersion curves, but they are limited by prior model training. Unsupervised learning methods, although not model-dependent, are often affected by the signal-to-noise ratio of the data (such as the DBSCAN algorithm) or initial value (such as the UMC method). In order to overcome these limitations, we propose a data-driven multi-order dispersion curve automatic picking algorithm, which is based on Hessian Analysis and independent of model and has strong adaptability.

Method

Seismic surface wave dispersion curve picking is usually done in the domain. Front-end data preprocessing and transformation of surface wave data from domain to domain belongs to conventional processing, which is not to be discussed here. Therefore, the method presented in this paper is introduced directly with the dispersive power spectrum as a start.

Basic principles

In a dispersion power spectrum, the surface wave energy is distributed in the shape of strips (as shown in Fig. 1a). Through three-dimensional top view, the energy band looks like a range of hills (Fig. 1b), and the trajectory distribution of the ridge in the figure is the dispersion curve we need to extract. Therefore, we can complete the extraction of the dispersion curve by searching all ridge points with hill characteristics in the graph and joined them together to form curves.

Fig. 1.

Surface wave dispersion energy diagram. It shows the characteristics of the planar banded distribution of dispersion energy (a). (b) is the corresponding three-dimensional top view, showing the three-dimensional mountain-like features of the energy belt.

This basic principle is essentially the same as traditional manual picking. The key point here is about automation. The flow chart of the algorithm is shown in Fig. 2, and the specifics of which will be introduced in detail below.

Fig. 2.

Algorithm flow chart.

Scaling filter

The dispersion power spectrum contains not only the surface wave dispersion energy, but also other information of different scales. In order to focus on the surface wave dispersion energy of interest, it is necessary to perform two-dimensional filtering on the power spectrum :

1

where is the convolution operation symbol, and the filter is a two-dimensional Gaussian function:

2

Let and , where and is the grid spacing of discrete space of , the above formula becomes

3

This is equivalent to the corresponding scaling transformation of the two coordinates, so that the uniform scale can be used to filter the features in the graph. Generally, when is small, the feature details of the graph are preserved more. Conversely, when is large, smaller scale features in the image are suppressed or filtered out. Therefore, by adapting to the scale of the surface wave dispersion energy feature that we need to preserve, the above scale filtering can play a role in suppressing the non-surface wave dispersion energy. In addition, the above transformation is a smooth filtering. It is essentially useful for providing a good data basis for the identification and extraction of the ridge of the dispersion curve by Hessian matrix analysis which will be introduced below.

Ridge point search

We use the Hessian matrix²³ to analyze the characteristics of the surface wave dispersion power spectrum after Gaussian filtering. We know that the eigenvalues of the Hessian matrix

4

approximate the curvature of the image function at ²⁴. Let and be the two-eigenvalues corresponding to the maximum and minimum module of the Hessian matrix (as shown in Fig. 3), i.e., . Since the dispersion curve generally has the smallest curvature change along the tangent direction of the ridge line, the corresponding eigenvalue . On the other hand, the curvature along the vertical direction of the dispersion curve is negative and typically with a large absolute value, we have . Therefore, if the plane point is located on dispersion curve, the maximum and minimum eigenvalues of the Hessian matrix of generally satisfy the following relation:

5

Fig. 3.

Hessian analysis diagram. The arrows in the figure show the maximum and minimum absolute eigenvalues and of the Hessian matrix at , and the corresponding maximum and minimum eigenvectors and . Where the eigenvector is perpendicular to the ridge line, while the eigenvector is in the same direction along the ridge line.

It is noted that in the direction perpendicular to the ridge, there can be several points satisfying this condition in the distance adjacent to a ridge point. Especially in the case of wide dispersion bands, the number of points meeting this condition can even be quite large. In this case, we may consider the point of maximum energy as the true ridge point. To do this, we need to add a local extreme condition to determine a ridge point.

Since the eigenvalue corresponding to the eigenvector of the Hessian matrix is orthogonal to the ridge line, the extreme value point of the image function can be searched along the direction, so as to determine the corresponding ridge point in the image.

It is noted that in the discrete space, the extreme point is not necessarily to locate exactly at a discrete grid point, and we can only approximate it by a grid point that is closest to the extreme point. In other words, whether a discrete grid point is the best approximation of an extreme point can be determined by the distance between the two points.

Here, we take Steger’s²⁵ ridge point search idea as reference and use a second-order Taylor expansion to approximate the continuous function of grid point in the dispersion power spectrum:

6

For the sake of simplicity, the above expression uses the comma operator to represent the partial derivative operation.

Let be the distance between and , which means . Let , we have , where and are the two components of a unit direction vector from to . In particular, in the direction of the gradient, and . Insert it into Eq. (6), and let , we can get the distance from extreme point to :

7

If

8

is the grid point closest to the extreme point. In other words, is a discrete point with the best approximation for this extreme point.

Combined with the above conclusions, we obtain the sufficient and necessary conditions for discrete grid point to be a ridge point

9

These can be concluded to become the following ridge search algorithm.

1. Given an image grid point , the eigenvalue of the Hessian matrix is calculated;

2. If the two eigenvalues of the Hessian matrix of the grid point satisfy Eq. (5), the distance t between the grid point and the extreme value point is further calculated according to Eq. (7);

3. If the distance t satisfies formula (8), it is a ridge point;

4. Repeat the above three steps to traverse all image grid points and complete the extraction of all ridge points.

Ridge line connection

The ridge point search algorithm can be used to extract a discrete set of ridge points of a dispersion power spectrum. These discrete ridge points need to be joined to form the dispersion curve. The joining can be fulfilled in the following two steps. First, the adjacent ridge points are joined into ridge segments, and then the ridge segments with adjacent ends and similar trends are connected into a dispersion curve. The connection method is as follows.

Step 1: ridge points connection

If a ridge segment is part of a dispersion curve, we may assume that it satisfies the following three conditions:

1. This ridge segment is not forking;

2. The distance between two adjacent points on the ridge segment is not greater than the oblique distance of the discrete pixel distance of the image, i.e., ;

3. The eigenvector angle of two adjacent points on this ridge segment is not greater than .

According to these assumptions, we can complete the ridge point connection to form ridge segments according to the following ridge point connection algorithm:

1. Sort the ridge points in the ridge set according to the eigenvalue ;

2. Take the ridge point with the smallest eigenvalue as the starting point for connection tracking, and remove it from the ridge point set;

3. On the side where frequency of the current tracking point increases, that is , the 4 grid points closest to the current tracking point are investigated:

4. If none of the four grid points are in the ridge point set, it indicates that the current ridge point is a right endpoint. Go to Step 4;

5. If only one grid point is in the ridge point set, connect the ridge point to the current ridge point, and then this ridge point is the new current point, remove it from the ridge point set, and then continue to the start of Step 3;

6. If there are multiple grid points in the ridge point set, select the point with the smallest eigenvector angle to the current eigenvector to connect, and then use this ridge point as the new current point, remove “multiple grid points” from the ridge point set, and proceed to start of Step 3;

7. On the side of decreasing of the current point, that is , investigate the 4 grid points that are closest to the current point:

8. If none of the 4 grid points are in the ridge point set, the current point is the left endpoint or isolated point of a ridge segment. The current ridge segment is connected complete, and turn to Step 2 to start the connection of a new ridge segment until the ridge point is integrated into an empty set;

9. If only one grid point is in the ridge point set, connect the ridge point to the current ridge point, and then this ridge point is the new current point, remove it from the ridge point set, and then continue to the start of Step 4;

10. If there are multiple grid points in the ridge point set, select the point with the smallest eigenvector angle to the current eigenvector to connect, and then use this ridge point as the new current point, remove “multiple grid points” from the ridge point set, and proceed to the start of Step 4.

In theory, the variation trend of the dispersion curve of the horizontal layered model is a monotonically decreasing function, which is monodromic at any frequency and satisfies the condition that the average slope is less than zero.

Before connection reaches the end of a segment, the search direction would not shift from the direction of frequency increasing to the direction of frequency decreasing, and vice versa. The ridge segments formed by this connection can ensure that no frequency reversing occurs and satisfy the property of the dispersion curve as a single value function of frequency.

However, the above connection algorithm cannot guarantee that the detected ridge segments can satisfy the monotonically decreasing property. If this condition is not met, it may not be a true dispersion curve, and theoretically the ridge segment can be removed accordingly. However, considering the untypicality of actual geological conditions and other interference factors, a real dispersion curve may deviate from the above monotonic decreasing condition. To this end, we may wish to set a slope fault tolerance threshold, for example , by eliminating ridge segments whose average slope is greater than this fault tolerance threshold, to ensure that the next ridge connection would not form an unreasonable dispersion curve.

Step 2: ridge segments connection

The ridge trace connection method above is based on a basic assumption: ridge points are closely connected to each other. If the dispersion energy distribution is disturbed, there would be discontinuity between the ridge points, and the above dispersion curve tracking will be interrupted, resulting in a dispersion curve staggered into several ridge segments. Therefore, after tracking the set of ridge segments, it is also necessary to connect the broken ridge segments to form a dispersion curve.

If the two ridge segments are broken from the same dispersion curve, the two ridge segments should have certain correlations, including (1), the staggered fracture distance is not too large; and, (2) The directionality of the two ridge segments will not change too much. Therefore, it may be assumed that the transverse and longitudinal distance of the two ridge segments at the breakpoint is not greater than and , and the difference in direction angle is not greater than . Then, if there are two ridge segments, of which the distance and directional angle difference between the left end point of one and the right end point of the other meet the above assumptions, we consider them to belong to the same dispersion curve, and they can be joined together to form a larger ridge segment.

To do this, let’s first define a few concepts:

Strength of ridge segment

10

End direction angle

11

where the curve integral notation and represent the ridge segment curve and the endpoint adjacent segment of the curve, respectively. Since the length and energy of the dispersion curve usually decreases gradually with the dispersion curve order, the intensity of the ridge defined above generally reflects the order change characteristics of the dispersion curve it belongs to, and the intensity of the ridge segment caused by noise is generally smaller than that of the ridge segment of the low-order dispersion curve. Therefore, we can select the initial connection segment according to the order of the intensity of the ridge segment from large to small, which is equivalent to selecting the ridge segment connection from the fundamental order to the higher order dispersion curve in turn.

Specifically, the ridge segments can be connected to form a dispersion curve according to the following ridge segment connection algorithm:

1. Calculate the intensity of each ridge segment in the ridge segment set according to (10), and calculate the direction angle at two endpoints of each ridge segment according to formula (11);

2. Set the transverse and longitudinal distance thresholds and , and the difference threshold of the direction angle of the end of the line segment . Take the ridge line with maximum strength as the current ridge segment, and search the broken ridge segment on its right side. In other words, to find the ridge segment from the ridge segment set, in which the current ridge segment is excluded, that its left end point and the right end point of the current ridge segment satisfy the given distance and direction angle difference threshold. There are three possibilities, (1) is that there are multiple ridge segments that satisfy the above conditions; (2), there is only one ridge segment that satisfies the above conditions; (3), no ridge segment satisfies the above conditions. In case (1), the segment with the smallest difference in direction angle is selected and merged with the current ridge segment through the connection as part of the current ridge segment. In case (2), it can be directly merged with the previous ridge segment through the connection as part of the current ridge segment.

Dispersion curve selection and order calibration

The above selected dispersion curve is generally mixed in quality, and the order is not distinguishable. Therefore, it is necessary to carry out the spurring screening and order calibration.

In general, a true dispersion curve has a certain length, and its energy is normally more prominent; The pseudo-dispersion curve formed by random noise is shorter in length and lower in energy. From this we can use the intensity defined by the curve integral (10) as a measure of the authenticity of the dispersion curve. A direct perception identification idea is to give an intensity threshold, if the intensity of the dispersion curve is greater than this threshold, it is considered as a true dispersion curve, otherwise it is eliminated as a pseudo-dispersion curve caused by noise. However, the determination of intensity threshold is difficult and challenging.

In fact, we can also use another idea to bypass the difficulty introducing from the strength threshold. We know that the curve intensity defined by formula (10) is from large to small, which basically reflects the order of the dispersion curve from the fundamental order to the high order, and then further to the pseudo-dispersion curve caused by noise. Therefore, we can complete the screening of the dispersion curve by sorting the intensity of the selected dispersion curve, and then select the corresponding number of curves with the intensity in the front as the effective dispersion curve according to the order of the dispersion curve we want to pick up, and discard the curve with the least intensities.

According to the general relationship between the intensity and order of the dispersion curve, the above screening process also determines the order of the filtered dispersion curve, that is, the pseudo-fundamental dispersion curve with the highest intensity, the pseudo-first-order dispersion curve with the second largest intensity, and so on.

Given the atypical nature of general geological scenario, for example, if the intensity of a first-order dispersion curve is lower than that of a higher-order dispersion curve due to some interference, the above ranking by intensity may be wrong. In this case, we can further accurately determine the order of the dispersion curve by the following method.

We know that the dispersion curve has a clear distribution nature in the domain: the order of the dispersion curve gradually increases from the low frequency and low speed region to the high frequency and high speed region. Accordingly, we can order the detected dispersion curve by the following order sorting algorithm:

1. Assume n dispersion curves are detected. Firstly, the ridge points of each dispersion curve are sorted from low to high frequency, and its length midpoint coordinates of the curve are determined;

2. The coordinates of this midpoints are fitted linearly by the least square method to obtain a straight line, and the orthogonal projection points coordinates of each midpoint on this line is obtained;

3. Sort the orthogonal projection points coordinates in rank from small to large. This order obtained represents the order of the dispersion curves gradually increases from the fundamental order to higher orders.

This sorting method avoids the errors of sorting by strength.

Determination of velocity range of dispersion curve

The velocity range of the dispersion curve obtained in the previous step yet needs to be determine. Let’s consider a simple case of Rayleigh surface waves propagating in a uniform elastic half space, where the Rayleigh wave equation is²⁶.

12

where , ; and , and v represent the P-wave velocity, shear wave velocity, and Rayleigh wave phase velocity, respectively, Eq. (12) has at least one real root and in the interval , and thus , the Rayleigh wave phase velocity is always less than the shear wave velocity . When the medium is a Poisson solid, the phase velocity in a semi-infinite space is ²⁶. When the propagation wavelength of Rayleigh waves in the layered medium is smaller than the layer thickness, the propagation velocity of Rayleigh waves approaches ( is the layer S-wave velocity). When the Rayleigh wave grows larger than 30 times the layer thickness, the propagation velocity of Rayleigh wave is approximately equal to ( is the S wave velocity in half-space)²⁷. Xia²⁸ also explained through practical examples that the low-frequency and high-frequency parts of the Rayleigh wave phase velocities of the fundamental order approach ~ 0.92 times of the uniform half-space S-wave velocity and the surface S-wave velocity, respectively; while the low-frequency and high-frequency parts of the phase velocities of the high-order approach the uniform half-space S-wave velocity and the surface S-wave velocity, respectively. Therefore, we could use the velocity range of the fundamental order dispersion curve to calculate the upper and lower limits of phase velocity, i.e., and , where and represents the maximum and minimum velocity, respectively, of fundamental order dispersion curve.

Algorithms

According to the method introduced above, an algorithm is summarized as follows.

1. Data preparation: The surface wave exploration data are transformed into the frequency-velocity domain to obtain the dispersion power spectrum . Then, according to the dispersion curve energy characteristics, the filtering scale parameter σ is determined, and the Eq. (1) is used to scale filter the dispersion power spectrum , and after scale filtering the power spectrum image is obtained.

2. Ridge point search: (a) Given an image grid point , calculate the Hessian matrix eigenvalue of the power spectrum function ; (b) If the two eigenvalues of the Hessian matrix of the grid point satisfy Eq. (5), the distance between the grid point and the extreme value point is further calculated according to Eq. (7); (c) If the distance satisfies formula (8), is a ridge point. Repeat the above three steps (a), (b) and (c) to traverse all image grid points and complete the extraction of all ridge points.

3. Ridge line connection: First, according to the ridge search algorithm introduced in Sect. “Ridge line connection”, the extracted ridge points are connected into ridge segments; Then, given the transverse and longitudinal distance thresholds and , and the direction angle difference threshold of the line segments end points, the ridge segments are connected to form a ridge set according to the ridge segment connection algorithm introduced in Sect. “Ridge line connection”.

4. Screening and sorting of dispersion curves: Given the order parameter n of the dispersion curve that is expected to be extracted, n ridge lines with the highest intensities are extracted according to the ridge intensity defined by formula (10) as the dispersion curve to be extracted. Then the order of the dispersion curve is determined according to the order sorting algorithm of the dispersion curve introduced in Sect. “Dispersion curve selection and order calibration”.

5. Determination of velocity range: Calculate the picked velocity effective range according to the method described in Sect. “Determination of velocity range of dispersion curve”, and output the dispersion curve within the range.

Algorithm testing

Test data

Geological model

To verify the method, we designed two conceptual geological models for numerical test experiments. Model 1 is a two-layer structure cited from Xia et al.²⁹, with the upper layer being the topsoil layer and the lower layer being the bedrock filled with half-space. Specific parameters of the velocity and density of the longitudinal and horizontal waves are shown in Table 1. Model 2 is formed by adding a layer of weakly weathered bedrock at the bottom of Model 1, and the specific parameters are shown in Table 2.

Table 1.

Medium parameters of Model 1²⁹.

Layer	Thickness (m)	P-wave velocity (m/s)	S-wave velocity (m/s)	Density (kg/m3)
1	10	800	200	2000
2	Half-space	1200	400	2000

Table 2.

Medium parameters of Model 2.

Layer	Thickness (m)	P-wave velocity (m/s)	S-wave velocity (m/s)	Density (kg/m3)
1	10	800	200	2000
2	20	1200	400	2000
3	Half-space	2500	800	2200

Acquisition system

The model and surface wave acquisition system are shown in Fig. 4. The size of the model is 26000 m × 200 m. The shot locates at (12600 m, 0 m). The observation array of 596 receivers locate from (12605 m, 0 m) to (13200 m, 0 m), with an interval of 1 m, and the minimum offset of 5 m. The source signal is a 20 Hz Riker wavelet (Fig. 5a) with a spectrum range of 0-60 Hz (Fig. 5b).

Fig. 4.

Model 2 and the acquisition system. The red star marks the location of the shot. The black inverted triangles mark the receivers.

Fig. 5.

Riker wavelet (a) and its spectrum analysis (b) used for source signal, whose effective frequency band is between 0 and 60 Hz.

Modeling

The widely used SPECFEM2D seismic wave simulation software³⁰ was used to synthesize the test data. The upper boundary of the model is a free surface boundary, while the bottom boundary and the left and right boundaries are attenuated by CPML absorption boundary conditions³¹. The thickness of the CPML boundary is of 30 m. Gmsh software³² was used to digitize a grid mesh model. According to the partitioning rules of Li et al.³³, the maximum mesh size of the first layer is 2.5 m × 2.5 m, the second layer is 5 m × 5 m, and the third layer is 10 m × 10 m. The calculation time interval is 10^–4 s. The output is resampled at 10⁻³ s, and the record length is 6 s. The record data by forward modeling are shown in Fig. 6. It can be seen that the seismic records mainly contain the first arrival waves, reflected waves and surface waves, with the surface waves stand out with strong energy.

Fig. 6.

Modeling record; (a) forward modeling record corresponding to Model 1; (b) forward modeling records of Model 2; First arrival waves, reflected waves, and surface waves can be observed in the record, with surface waves being the strongest in energy.

Automatic detection of dispersion curves

The picking of dispersion curves from the data are demonstrated step by step following the procedures of the automatic algorithm introduced in Section “Method”.

Algorithm step 1: data preparation

The Frequency-Bessel Transform Method (FJ)³⁴ was adopted to transform the data from the domain to domain as shown in Fig. 6. The Gaussian filtering parameter σ of 6a and 6b takes 3 and 4 sampling intervals respectively. The effective frequency band range is 2-50Hz and velocity calculation range is 150m/s-800m/s. The results are shown in Fig. 7. On the spectrum corresponding to Model 1 (Fig. 7a), five orders of dispersion energy distribution can be seen. On the spectrum corresponding to Model 2 (Fig. 7b), 8 orders of dispersion energy distribution can be seen.

Fig. 7.

spectrum obtained by FJ transform and scale filtering; Figure a and b are the spectra corresponding to Model 1 and Model 2, from which we can see the dispersion energy distribution with 5 orders and 8 orders, respectively.

Algorithm step 2: ridge point search

Hessian analysis is performed to calculate the maximum absolute eigenvalues and corresponding eigenvectors at each grid point. The algorithm mentioned above is used to screen the calculated grid points one by one to determine whether there are extreme values along the direction of the maximum eigenvector at the grid point. If there are extreme values, the grid point will be kept; or otherwise, it will be discarded. The ridge point distribution results are shown in Fig. 8. It can be seen that the ridge points are distributed regularly along the dispersive energy band, while when it is away from the dispersive energy band, the ridge points are distributed chaotically.

Fig. 8.

The ridge point distribution of Models 1 (a) and 2 (b).

Algorithm step 3: ridge segment connection

After sorting, the ridge points are connected to form ridge lines. The results are shown in Fig. 9. It can be seen that only the ridge segments distributed along the dispersive energy band have good continuity.

Fig. 9.

Ridge segments after ridge segment connection. The results for Models 1 and 2 are shown in Figures a and b, respectively.

Algorithm step 4: dispersion curve selecting and sorting

Due to the limitation of sampling along the frequency direction, a frequency of a ridge may correspond to multiple velocities. This is particularly the case at the low frequency end. This would result in step shape of the curve. To improve this, we carry out median filtering along the curve. The result after this processing is shown in Fig. 10. It can be seen that the stepped change at the low frequency end is alleviated. The curve changes smoother.

Fig. 10.

Ridge distribution after median filtering. The results of Models 1 and 2 are shown in Figures a and b, respectively.

Among the ridges obtained, there are not only ridges along the trend of the dispersion energy, but also ridges in other random directions. From Fig. 10, it can be seen that the slope of the ridge line distributed along the dispersive energy is < 0, and the slope of the ridge line distributed along the non-dispersive energy is ≥ 0. We need to screen out this part of the line segment with the slope ≥ 0, so as not to affect the subsequent connection of the dispersive curve. In order to avoid the dispersion curve being mistakenly deleted, we calculated the average slope along each ridge line and discarded the ridge lines with average slope higher than . As shown in Fig. 11, most of the random ridge lines are filtered out.

Fig. 11.

Ridge lines after filtering. The results of Model 1 and 2 are shown in Figures a and b, respectively.

After filtering, the ridge segments are connected using the method introduce in Sect. “Ridge line connection” to form a dispersion curve. For the connection of the dispersion curve, the frequency search range for the adjacent line segments was 0-2Hz, the velocity difference range was 10m/s, and the angle difference was < 30°. The result is shown in Fig. 12. It can be seen that ridge lines belonging to the same order dispersion curve are successfully joined together after the ridge connection processing.

Fig. 12.

The dispersion curves connected of Models 1 and 2 are shown in Figures a and b, respectively. When connected, the frequency search range of adjacent line segments is 0-2 Hz, the velocity difference is 10 m/s, and the angle difference is < 30°.

After the connection, the methods of Sections “Dispersion curve selection and order calibration” and “Determination of velocity range of dispersion curve” are used to screen and classify the dispersion curves, and the results obtained are shown in Fig. 13. For Model 1 and 2, 5 and 9 dispersion curves were identified, respectively.

Fig. 13.

Automatic picking and classification results of Models 1 and 2 are shown in Figures a and b, respectively. The black circle in the figure represents the center of the dispersion curve, and the black dotted line is their linear fitting; the red solid circle is the projection point of the black hollow dot on the fitting curve, and the red dotted line in the figure is the calculated velocity range.

To sort the identified curves, the midpoint of each curve (black hollow circle in Fig. 13) is determined. And its projection on the fitted line (the solid red circle in Fig. 13) is calculated and sorted in frequency order. This order is used as that of the dispersion curves. For Model 1, 5 dispersion curves were picked, ranging from the fundamental-order to the 4th order. For Model 2, 9 dispersion curves were picked, ranging from the fundamental-order to the 8th order. The red dotted line in Fig. 13 is the velocity range calculated according to the method in Sect. “Determination of velocity range of dispersion curve”; The velocity ranges are [190, 382] for Model 1, and [190, 686] for Model 2.

Velocity inversion

In order to verify the accuracy of the extracted dispersion curve, the neighborhood algorithm of Wathelet et al.³⁵ was used for the inversion of velocity from the extracted dispersion curve. For the inversion of Model 1, the velocity ranges for the P- and S-waves are 500–1500 m/s and 100–1000m/s, respectively. And the density is 2000 kg/m³, and the layer thickness varies within 1–50 m. For the inversion of Model 2, the velocity ranges for the P- and S-waves are 500–2500 m/s and 100–1000 m/s, respectively. The density is 2000 kg/m³, and the layer thickness ranges from 1–50 m. The inversion results are shown in red in Fig. 14. It can be seen that the inversion models agree with the original models (black) well. The errors of Models 1 and 2 are shown in Tables 3 and 4, respectively. It can be seen that the depth and velocity errors of the first layer of Model 1 are 0.6% and 0.34%, respectively, and the velocity error of the second layer is 0.68%. For Model 2, the depth error and velocity error of the first layer are 0.6% and 0.34%, respectively; the depth error and velocity error of the second layer are 0.00% and 0.32% respectively; and the velocity error of the third layer is 0.02%. The low errors demonstrate that the picking quality is in very high precision.

Fig. 14.

Inversion result of Models 1 and 2 are shown in Figures a and b, respectively. The black dotted line is the geological model of shear wave velocity used in forward modeling, and the red solid line is the inverse shear wave velocity.

Table 3.

Model 1 inversion error statistics.

Layer	Thickness (m)			S-wave velocity (m/s)
	Model	Inversion	Error	Model	Inversion	Error
1	10	10.06	0.6%	200	200.68	0.34%
2				400	402.71	0.68%

Table 4.

Model 2 inversion error statistics.

Layer	Thickness (m)			S-wave velocity (m/s)
	Model	Inversion	Error	Model	Inversion	Error
1	10	10.06	0.60%	200	200.68	0.34%
2	20	20.00	0.00%	400	398.72	0.32%
3				800	800.14	0.02%

Application

In order to test the practicability of the method, we processed a passive source surface wave data from an industrial project in north-west Sichuan Basin.

Experimental design

Two surface wave observation points is combined to form a data set for the experiment. A vertical seismic profile (VSP) exploration was performed at the middle to provide a velocity structure for comparison purpose.

VSP data

The well depth of VSP is 15 m. 12 receivers were used in a trace interval of 1 m. The sampling interval is 0.25 ms, and the record length is 128 ms. The hammer source locates 1 m away from the well on the ground surface. Two sets of data were collected. The receivers were employed at the depth range from 4 to 15 m for the first set (Fig. 15a), and 1 to 12 m for the second set (Fig. 15b), with an overlap of 8 receivers.

Fig. 15.

VSP records. Figures a and b are the records of the two arrays with their first receiver at the depths of 4 and 1 m, respectively.

The first arrivals were picked from the VSP data. The relationship between the first arrival and the depth is shown in the black solid circles in Fig. 16a. As can be seen from the figure, the first arrival can be divided into two sections, of which the first section is at a depth of 1–4 m and the second is at a depth of 5–15 m. The slope of the two sections is the layer velocity, and the intersection of the two sections of data is the depth position of the velocity interface change. Based on this interpretation, it is determined that the site is at a two-layer structure, with a layered interface lying at a depth of 4.4 m. The first layer is the quaternary topsoil layer, with a P-wave velocity of 662 m/s. The second layer is weakly weathered bedrock with a P-wave velocity of 2026 m/s (Table 5). The specific velocity stratification is shown in Fig. 16b.

Fig. 16.

VSP results. The black solid dots in Figure a represent the first arrivals from the VSP records, and the red straight line is the velocity profile obtained from the fitting first arrival. Figure b is the subsurface velocity structure obtained by VSP.

Table 5.

VSP result.

Layer	Depth(m)	S-wave velocity (m/s)
1	4.4	662
2		2026

Surface wave data

The surface wave data comprise 30 traces, with a trace interval of 3 m, array length of 87 m, and time sample interval of 5 ms. The recordings are shown in Fig. 17.

Fig. 17.

Surface wave records, of which shown in Figures a and b are the data from the first and second arrays, respectively.

In order to enhance signal-to-noise ratio, each recording is cut into separate segments of 10 s long, with an overlap of 80%. One of the virtual shot records of 10 s long is shown in Fig. 18. These result in 742 recording gathers. The spectra of the gathers are stacked and shown in Fig. 19a. 3 dispersion energy strips can be clearly seen in the figure.

Fig. 18.

Virtual shot records. Shown in Figures a and b are the records from the data of the first and second arrays as shown in Fig. 17a and b, respectively.

Fig. 19.

Stacking spectra (a) and its automatic picking result (b). In (b), the black is the fundamental dispersion curve, red is the first-order dispersion curve, and blue is the second-order dispersion curve.

The automatic picking algorithm is used to pick dispersion curves. The scaling transformation parameter σ is 8 sample points. The ridge segment whose average slope is greater than is discard. When the dispersion curve is connected, the frequency search range is 0–2 Hz, the velocity difference range is 10 m/s, and the angle difference is < 30°. The results of the automatic picking are shown in Fig. 19b. It can be seen that three dispersion curves were picked, ranging from the fundamental order dispersion curve (black), the first-order dispersion curve (red), and the second-order dispersion curve (blue).

Dispersion curve inversion

The neighborhood algorithm of Wathelet et al.³⁵ was used for the inversion of the shallow surface velocity model from the picked dispersion curves The inversion parameters given are: the total number of layers is 5; The thickness of each layer varies in the range of 1–15 m; The shear wave velocity of each layer varies in the range of 100–1200 m/s. Poisson’s ratio and P-wave velocity are given with reference to the experience of Keceli³⁶, where Poisson’s ratio varies in the range of 0.33–0.49, and P-wave velocity varies in the range of 500–2200 m /s. Since the shear wave velocity structure is not sensitive to density change¹, the inversion parameter is fixed at 2000 kg/m³. Inversion results are shown in Fig. 20. It can be seen that there are three layers above the bedrock, the velocity and depth of each layer are listed in Table 6.

Fig. 20.

Inversion results. The red curve in the figure shows the best fit inversion model with the minimum inversion error.

Table 6.

Inversion Result.

Layer	Depth(m)	S-wave velocity (m/s)
1	1	143
2	4.4	335
3	14.5	407
4		1063

Experimental conclusion

The comparison of surface wave data results with VSP data shows that, because the measuring range of VSP is 1-15m, it is impossible to obtain the velocity information shallower than 1m, and the information for the depth of ~ 15m is also not effectively usable as it located at the bottom edge of data measurement. Therefore, the stratified interface of shallow and deep is missed by VSP. In contrast, the surface wave data effectively revealed the three-layer geological structure of the area, and the stratified interface at 4.4 m, which is consistent with the VSP.

The comparison shows that the dispersion curve automatic picking algorithm is feasible for industrial field data processing.

Discussion

The experiments above demonstrate successful achievements of the Hessian method in both numerical tests and practical applications. However, the limitation and advantage of the method compared to other methods still need to be further analyzed.

Comparison with other methods

Comparative experiments were performed of the Hessian method with other three methods, including manual picking based on the Geopsy software³⁵, semi-automatic picking based on the Geogiga software, and fully automatic picking based on the unsupervised learning clustering method UMC²¹. The experiments are based on Model 2 (Refer to Sect. “Test data” for details), and the picking results are shown in Fig. 21.

Fig. 21.

Picking results of Model 2 by UMC (a), Geogiga (b), Mannual (c), and Hessian (d). The hollow circles represent the picked data, and the solid lines are the dispersion curves of the model.

As can be seen from Fig. 21, both the manual method (Fig. 21c) and the UMC method (Fig. 21a) can pick up 4 dispersion curves, from the fundamental order to the 3rd order. The semi-automatic method of Geogiga software can pick up 5 dispersion curves up to the 4th order (Fig. 21b), while the Hessian method can pick up 9 dispersion curves from the fundamental order to the 8th order (Fig. 21d). This high order dispersion curve picking ability of the Hessian method demonstrates its superior advantage over the other 3 methods.

To characterize the picking accuracy quantitatively, we define

13

14

where , the root mean square error (rms) of velocity between the picked dispersion curve, , and the model, , represents the picking accuracy; , the ratio of the segment range of the picked dispersion curve between the maximum and minimum frequencies ( and respectively) to the total effective range of the relative model dispersion curve between the effective maximum and minimum frequencies ( and , respectively). From (13) we know that . The smaller is, the closer the picked dispersion curve is to the model curve. As it characterizes the accuracy of the picking, we will refer to as the picking accuracy in the following context. From (14) we know that . The larger the value, the more complete the pick dispersion curve is. We will refer to as the completeness of the dispersion curve picked in the following context.

The accuracies and completeness of the pickings using different methods are shown in Fig. 22. It shows that, Hessian method can detect high-order dispersion curves that other methods fail to detect, and the curves picked by the Hessian method are systematically more complete across different orders than the other methods.

Fig. 22.

The picking accuracy (a) and completeness (b) of different methods.

Shear wave velocity inversion of the dispersion curves were performed using the neighborhood method³⁵. The inversion results are shown in Fig. 23. And the statistics of the picking accuracy and completeness are shown in Table 7. We can see that the inversion of dispersion curves by Hessian methods can recover the velocity model with high accuracy, while the inversion using dispersion curves by manual picking, Geogiga picking and UMC fail to recover the velocity of the bedrock layer with acceptable accuracy.

Fig. 23.

Shear wave velocity inversion results using dispersion curves by different methods. the dotted line and solid line are the geological model and inversion, respectively.

Table 7.

Inversion errors.

Model parameter	Model	UMC		Geogiga		Manual		Hessian
		Inversion	Error	Inversion	Error	Inversion	Error	Inversion	Error
1st layer thickness (m)	10	10.06	0.60%	10.06	0.60%	10.16	1.6%	10.06	0.60%
2nd layer thickness (m)	20	22.82	14.1%	21.85	9.25%	20.20	1.0%	20.00	0.00%
1st layer velocity (m/s)	200	200.68	0.34%	200.68	0.34%	200.68	0.34%	200.68	0.34%
2nd layer velocity (m/s)	400	394.77	1.31%	406.74	1.69%	410.80	2.7%	398.72	0.32%
3rd layer velocity (m/s)	800	995.95	24.49%	995.94	24.49%	668.93	16.38%	800.14	0.02%

Evaluation of program running efficiency

To evaluate the computational efficiency of the Hessian method, a comparative experiment with the UMC method was performed. As introduced earlier in Sect. “Comparison with other methods”, UMC only picked up 4 dispersion curves from the fundamental order to the 3rd order, while the Hessian method picked up 9 dispersion curves up to the 8th order. To compare the calculation efficiency of the two methods under the condition of the same detection rate, we reduced the velocity sampling rate of the spectrum from 4 to 6 m/s. At this rate the Hessian method picked up four dispersion curves from the fundamental order to the 3rd order. This detection result was the same as that of UMC. By comparing the running times of the two, it is found that the time taken by UMC is 1.33 times of the Hessian method. In other words, the program running efficiency of the Hessian method is 25% higher than that of the UMC method.

Anti-noise performance test

We further investigate the anti-noise performance of the Hessian method and the UMC method on data with alias and of different signal-to-noise ratios.

Anti-alias experiment

In surface wave exploration, alias would occur on the dispersion power spectrum due to insufficient spatial sampling. This would interfere the picking of dispersion curves at the high-frequency end. To test the anti-alias-noise ability of the Hessian method, we carry out a passive source surface wave simulation. The experiment adopts the model of Xia³⁷, of which the parameters are shown in Table 8. The observation system is shown in Fig. 24a. It has 48 receivers, with a spacing interval of 5 m, a sampling rate of 2 ms, and a recording length of 60 s. 400 seismic sources are randomly distributed within a circular ring belt radius of 1–2 km from the midpoint of the receiver array. The amplitude of random seismic source follows a uniform distribution ranging from 0 to 1. The time delay is randomly generated within 0 to 60 s, and the main frequency is randomly generated within 10 to 60 Hz. The simulation was performed using the method of Park et al.³⁸. 50 sets of random data were generated for each shot. And the virtual source records were synthesized by cross-correlation followed by stacking.

Table 8.

Passive source surface wave simulation parameter³⁷.

Layer thickness (m)	P-wave velocity (m/s)	S-wave velocity (m/s)	Density (kg/m3)
25	1350	200	1900
∞	2000	1000	1900

Fig. 24.

Passive source surface wave simulation. (a) is the observation system. The red dots represent the source distribution and the blue triangles are the receivers. (b) is the noise records. The dispersive power spectra and the automatic picking results are shown in (c and d), respectively.

Figure 24b shows one of the 50 sets of synthetic data. Dispersion energy analysis of the virtual source records was carried out using the phase shift method of Park³⁹, and the dispersion power spectra obtained are shown in Fig. 24c, on which the fundamental dispersion energy and alias distribution can be clearly seen. The dispersion curves detected by the Hessian method and the UMC method are shown in Fig. 24d. It can be seen that at the low-frequency end, the pickup result of the Hessian method fits the model better than UMC, while at the high-frequency end, the Hessian method picks up more dispersion information than UMC. This indicates that the Hessian method is superior to UMC on anti-alias performance.

Anti-noise experiment

To test the picking ability of the Hessian method in the case of low signal-to-noise ratio, we designed an anti-noise experiment. The experiment considered 18 different signal-to-noise ratio levels, with 9 ratios ranging from 0.2 to 1.0 with an interval of 0.1, and the other 9 ratios ranging from 2.0 to 10.0 with an interval of 1.0. Gaussian random noise were added to the forward modeling records from Model 2 (see Sect. “Test data”) using the add-noise module of Seismic Un*x⁴⁰, and 50 sets of random data were synthesized for each signal-to-noise ratio level.

Figure 25a1 and a2 show the synthetic source with signal-to-noise ratios of 5 and 0.5, respectively. The FJ method³⁴ was adopted to conduct dispersion analysis on these data, and the results are shown in Fig. 25b1 and b2. The dispersion curves picked using the Hessian and the UMC methods, are shown in Fig. 25c1, c2 and d1, d2 respectively. We can see that when the signal-to-noise ratio is 5, both the UMC method and the Hessian method can pick up 4 dispersion curves, but the dispersion curves picked up by the Hessian method is longer and more complete. When the signal-to-noise ratio is 0.5, the effective signal is severely interfered with noise. The UMC method can only pick up one dispersion curve, while the Hessian method can pick up two dispersion curves. This indicates that the Hessian method’s anti-noise performance is superior to UMC method.

Fig. 25.

Two noise data with signal-to-noise ratios of 5.0 and 0.5 (a1 and a2), the corresponding dispersion power spectra (b1 and b2), the dispersion curves picked up by the Hessian method (c1 and c2) and the dispersion curves picked up by the UMC method (d1 and d2).

To obtain a more comprehensive comparation results, we used formulas (13) and (14) to calculate the accuracy and completeness of the dispersion curves from each set of data. The statistical results are shown in Fig. 26.

Fig. 26.

Statistics of picking accuracy and completeness. When the UMC method fails to detect in the case of low signal-to-noise ratio, the Hessian method can still successfully detect the dispersion curves (see a1–e1). In addition, the completeness of the dispersion curves picked up by the Hessian method is also superior to that of UMC (see a2–e2).

From the perspective of picking accuracy (Fig. 26a1–e1), the UMC method experiences detection failure in the case of low signal-to-noise ratio, while the Hessian method can still detect the dispersion curve successfully. When the signal-to-noise ratio is good, the error fluctuation range picked up by Hessian is smaller than that of the UMC method. From the perspective of the completeness of the dispersion curve pickup (Fig. 26a2–e2), the Hessian method is systematically and significantly higher than that of the UMC method.

This experiment demonstrates that the anti-noise performance of the Hessian method is significantly better than that of the UMC method.

Conclusions

In this paper, a new automatic picking method for surface wave dispersion curves is proposed based on Hessian matrix attributes and an algorithm is developed. On the dispersion power spectrum transformed from the surface wave data, the algorithm automatically picks the surface wave dispersion curve from the fundamental order to the high orders, by ridge point searching and extraction, ridge line segment connection, and dispersion curve sorting. The algorithm does not have to rely on model training and there is no manual interaction requirement. This inherent efficiency advantage of the method can provide a highly efficient tool for surface wave picking for large prospecting projects such as industrial oil and gas exploration.

The algorithm is tested using two conceptual models. The seismic record data is synthesized by wave field forward modeling, and the spectra is obtained by transformation. Then the dispersion curve is automatically extracted by our method, and the surface wave dispersion curve up to the 8th order is obtained successfully, which shows the powerful picking ability of the method. Finally, the inversion results are compared with the original model, and the results show that the two are highly consistent, which verifies the correctness of the method to pick up the dispersion curve.

Comparative experiments were performed to evaluate the Hessian method with other methods including the machine learning method UMC. The results show that the picking accuracy and completeness, computational efficiency and anti-noise performance of the Hessian method are all better than the other methods.

In order to evaluate the practicability of the method, we applied it to a surface wave data processing in a practical production area. Three dispersion curves are successfully picked, including the fundamental, first-order and second-order. The inversion results of the dispersion curves are highly consistent with the VSP results, which verified the effectiveness and practicability of the proposed method.

Author contributions

HXP: data acquisition, writing-review and editing, writing-original draft. YJS: writing–review and editing, methodology, funding acquisition. YJL: writing–review and editing, funding acquisition. FXB: writing–review and editing. FXR: coding. HC: experiments. LZG: data acquisition. QG: data processing. ZQ: data acquisition.

Data availability

The datasets used and analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.