Sir Rayleigh first proved the existence of surface waves in 1885. There are two kinds of surface waves related to near-surface earthquakes, namely Rayleigh wave and Loew wave. Stoneley wave is the third kind of surface wave, but it can only be observed at the underground interface, but not at the surface free interface. Grant and West gave a simple mathematical derivation of surface waves in 1965. Generally speaking, we think that the two waves defined above exist independently, but their horizontal components can be transformed into each other under certain boundary conditions and distances and observation systems. This problem is beyond the scope of this course, so I won't discuss it in detail here.
In fact, the amplitude of surface wave decreases exponentially with the increase of depth. This property that the amplitude decreases sharply with the increase of depth is why they are called surface waves. At the interface, their amplitudes decrease approximately with the increase of propagation distance.
The results of the following two surface wave studies show that the earth is layered and uneven.
1) Theoretically, the existence of Loew wave must meet one of the following two conditions: the velocity must increase monotonically or there must be a low velocity layer above the interface. Figure 2-3- 1 shows the well-developed Loew wave propagation in the low-velocity layer. The heavy hammer is used as the seismic source and the horizontal geophone is used as the receiving device. Shear wave refraction shows that this low velocity layer is only a few feet thick. Love wave's first arrival is the most remarkable in the second half of the record.
Figure 2-3- 1 Loew surface wave in low-velocity layer
2) Actually, both Loew wave and Rayleigh wave can have dispersion phenomenon. Because the wave with the largest wavelength penetrates deeper and the velocity in the deep layer is usually higher, the wave with the largest wavelength reaches the detector first. When the velocity increases with the depth, the dispersion phenomenon becomes more serious. For example, in Figure 2-3-2, the deviation of the left half is smaller than that of the right half. By observing the left half of the diagram, we find that there are two clear horizons in the surface wave. It can be proved that when the recording time is 30 ms at an offset of 1 m and 230 ms at an offset of 20 m, the upper layer is a low-speed layer with a phase speed of about100 m/s.. However, the arrival time of about 100 ms is recorded at 1 m offset and that of 150 ms is recorded at 25 m offset, which proves that the lower layer is a high-speed layer with a phase velocity of about 480 m/s, and these two wave trains are at 5 ~ 15 m offset.
Figure 2-3-2 Comparison of Wave Dispersion of Different Surfaces
It should be noted that these phase velocities are about 90% of the velocities of direct wave and refracted S wave, respectively. Sometimes these two kinds of different surface wave trains will appear on the record, each of which is produced in different strata. The left half of Figure 2-3-2 is a good example to illustrate this problem.
According to previous examples, we found that useful geological information can sometimes be obtained by studying surface waves on seismic records and simple calculations. However, surface waves are usually considered as useless noise by exploration seismologists. But in any case, civil engineers have begun to use surface waves (especially Rayleigh surface waves) to study the engineering mechanical properties of shallow foundation. Using the frequency spectrum analysis (SASW) of surface wave, the stiffness coefficient profile of near-surface materials can be obtained by forward simulation or inversion of surface wave velocity. Using these wide-band Rayleigh waves, we can get different depth results.
Generally, it is generally believed in scientific literature that the velocity of surface wave in medium is about 0.92 times that of shear wave, but the dispersion phenomenon closely related to surface wave is ignored. To some extent, for media with Poisson's ratio of 0.25 (typical hard rocks, such as granite, basalt and limestone), the relationship of 0.92 times is established, but in fact this layer does not exist; For the medium with Poisson's ratio of 0.0, the surface wave velocity should be 0.874 times of S wave velocity; For the medium with Poisson's ratio of 0.5, it should be 0.955 times; For loose materials, Poisson's ratio is often between 0.40 and 0.49. It is generally assumed that Rayleigh wave velocity is 0.94 times of S wave velocity, and this assumption error is less than 65438 0%.
Although we often think that Rayleigh wave velocity has nothing to do with P wave velocity, don't forget that P wave velocity is one of many factors that determine Poisson's ratio. Rayleigh wave velocity is less dependent on Poisson's ratio, so it is less dependent on P wave velocity.
The frequency of surface wave is generally lower than that of body wave, especially in near-surface research, because the propagation path of body wave in deep layer is short, the high-frequency component is not attenuated. Results The surface wave in near-surface reflection can be eliminated by a simple low-cut filter. Figure 2-3-2 is an excellent example of the frequency difference between shear wave and surface wave. The main frequency of direct wave and refracted shear wave is above 60Hz. The main frequency of the shallow Loew wave is below 40Hz, and the main frequency of the Loew wave penetrating the lower high-speed layer is below 25Hz.
Just as an organ has many modes, so do surface waves. However, the basic mode is usually the most important. Rix et al. (1990) proved through experiments that at 16Hz, 73% of the particle displacement in the measurement area is provided by the fundamental mode, while at 50Hz, 87% is provided by the fundamental mode.
Second, the types of surface waves.
1. Rayleigh surface wave
Rayleigh wave is vertically polarized, and the trajectory of its particles is an inverted ellipse on the polarization plane. That is to say, at the top of the elliptical path, the direction of particle displacement points to the source. For an observer who is hundreds of meters away from the explosion, the Rayleigh wave generated by high-energy explosives loaded with dozens of kilograms will make people feel "ground rolling". So Rayleigh waves are often called "ground rolling waves", but in fact most waves are like this.
In most cases, the propagation of surface waves on the ground is limited to a wavelength range. At a certain depth, the amplitude of Rayleigh wave is zero. When it is greater than this depth, the particle will move in the opposite direction, and it will move in the clockwise ellipse direction. A plane with zero amplitude is called a nodal plane, and its depth depends on Poisson's ratio. For example, when Poisson's ratio is 0.25, the nodal plane is located at the wavelength of 0. 19 times below the surface, while when Poisson's ratio is 0.45, the nodal plane is located at the wavelength of 0. 15 times below the surface (from Grant and West, 1965).
It is generally believed that the motion of Rayleigh wave is mainly vertical, because it is related to the ground roll wave that can be observed by vertical geophone in the field. But the horizontal motion component also exists, which vibrates back and forth in the plane perpendicular to the plane where the shot point and the detection point are located, and propagates outward. The ratio of horizontal motion to vertical motion at all depths also depends on Poisson's ratio. For example, we often use surface or near-surface geophone. For the medium with Poisson's ratio of 0.25, the amplitude ratio of Rayleigh wave is 1.25, and for the medium with Poisson's ratio of 0.45, it is 1.7.
The figures given in the first two paragraphs are obtained when the medium is assumed to be an elastic half-space medium. In fact, even media can still be used when their thickness reaches 4 ~ 5 times of the maximum wavelength in seismic records. When the embedment of geophones is consistent and the directional adjustment devices of these geophones work normally, Poisson's ratio can be directly determined by the relative amplitudes of the horizontal and vertical components of Rayleigh waves. The case of uneven surface layer and small uniform layer thickness is more complicated, so I won't discuss it in detail here.
In seismic records, the amplitude of Rayleigh wave at zero offset is not zero. In 1904, Lamb proved that Rayleigh waves can be generated by the diffraction of the curved wavefront of the body wave on the free interface. As a result, Rayleigh waves cannot propagate outward until the bulk wave reaches the surface and begins to diffract on a small volume above the excitation point. So one way to reduce Rayleigh wave is to increase the depth of the source. Similarly, because a curved initial wavefront is needed, Rayleigh waves will not appear in the solution of the plane wave equation.
Figure 2-3-3 Rayleigh Wave Dispersion Example
In an infinite half-space homogeneous medium, the Rayleigh wave velocity only depends on the properties of the medium, and there is no dispersion phenomenon at this time. When there is a layered medium or a velocity gradient under the ground, the velocity of Rayleigh wave varies with the wavelength. Therefore, the dispersion of surface waves means that the underground is a layered medium or there is a velocity gradient.
Figure 2-3-3 is an example of dispersive Rayleigh waves propagating in the low-velocity layer, from which direct waves and longitudinal waves can also be seen. It is worth noting that the penetration depth of Rayleigh wave increases with the increase of wavelength when the cannon is offset.
We have noticed that some useful geological information can be found by looking at the seismic records. In Figure 2-3-3, the first break of the refracted wave at the right third of the seismic record is disturbed, and this interference also affects the Rayleigh wave, which is shown as an additional example in the figure. Although the ground roll wave will show obvious disturbance near the area with severe geological changes, sometimes it will appear obvious disturbance even without obvious changes, because the static correction caused by topographic changes will sometimes produce the same effect. The importance of these interferences in the data can sometimes be determined by examining the topographic survey data along the survey line.
Figure 2-3-4 is an example of Rayleigh wave with relatively no dispersion. The wave propagates at a distance of 24m from the shot point, and the recording time starts from 15ms and ends at145 ms. Note that the medium through which the wave propagates is uniform.
2.love wave
Loew wave is like "trough wave", it only moves in the horizontal direction, and the direction of movement is perpendicular to the direction of wave propagation. The essence of Loew wave is diverse, which comes from the total reflection of S wave when the surface layer is low velocity layer. Loew waves cannot propagate without the low velocity layer. The right half of the seismic record in Figure 2-3-5 was collected on the spillway of TuttleCreek Reservoir near Manhattan, Kansas, and the geophone was placed on the limestone just exposed by the flood. The limestone layer is about 2m thick, and it is covered with thick layers of rocks, shale and limestone. It is noted that there is no coherent Loew surface wave chain in the whole record. The seismic record at the left of the picture was collected in the alternating layers of shale and limestone with similar thickness near Lawrence, Kansas, and the geophone was placed on the weathered shale at the top. Pay attention to the dispersion direction of Loew wave.
Figure 2-3-4 Examples of Non-dispersion in Homogeneous Media
Figure 2-3-5 Schematic diagram of Loew surface wave and its dispersion characteristics in low-velocity layer
In the past, natural seismologists used Loew wave extensively to measure the crustal structure. At present, some people have tried to apply Loew wave to near-surface static correction (Mari,1984; Song et al., 1989).Lee and McAchan (1992) use the backscattered echo of Loew wave to image the inhomogeneous medium near the surface.
Loew wave, like Rayleigh wave, has non-zero amplitude at non-zero offset. Because the Loew wave is reflected from the bottom of the low velocity layer, it takes a certain time from the shot point to the interface and finally received by the ground detector. This characteristic of Loew wave can be used to evaluate the near-surface geological conditions, but as far as we know, there is little research in this field.
Generally, Loew waves can be seen in every part of seismic records, which can well prove that the earth is layered, and the speed of Loew waves increases with the increase of depth in many places. Because Loew wave must propagate in layered media and has dispersion phenomenon, information such as thickness, velocity and number of layers of overlying layer can be extracted according to this property. The velocity of Love wave with the shortest wavelength is proportional to the velocity of S wave in the lowest velocity layer, while the velocity of Love wave with the longest wavelength is proportional to the velocity of S wave in the deepest medium. The dispersion phenomenon makes the amplitude of Loew wave decay slightly faster with the increase of distance, which is about 0.
Third, the deviation curve
The direct result of Rayleigh wave exploration is Rayleigh wave dispersion curve. The characteristics and changes of dispersion curve are closely related to underground conditions, such as the thickness and wave velocity of each layer. In this paper, the general law of this change is given, and the factors affecting Rayleigh wave dispersion and the causes of several common abnormal curves are discussed.
1. Characteristics of frequency dispersion curve in layered media
For an infinite half-space homogeneous medium, the Rayleigh wave velocity only depends on the properties of the medium. At this time, there is no dispersion phenomenon, and the Rayleigh wave velocity changes linearly with wavelength (or frequency), as shown in Figure 2-3-6.
When the underground is a layered medium or there is a velocity gradient, the velocity of Rayleigh wave changes with the change of wavelength (or frequency), that is, there is dispersion phenomenon. Figure 2-3-7 shows the dispersion curve of Rayleigh wave propagating in two-layer medium, and Figure 2-3-8 shows the dispersion curve in multi-layer medium. It is not difficult to see from the figure that the curve changes monotonously as a whole, that is, the phase velocity increases with the increase of wavelength and decreases with the increase of frequency, but there are also "local" changes, and often these local changes contain rich horizon information.
Figure 2-3-6 Uniform Media in Infinite Half Space
Figure 2-3-7 dispersion curve of double-layer medium
2. Factors affecting dispersion curve
As mentioned above, the direct result of Rayleigh wave exploration is Rayleigh wave dispersion curve, and the quality of dispersion curve affects the inversion result, so it is necessary for us to discuss the factors affecting dispersion curve here.
Generally speaking, the dispersion curve is obtained by extracting surface wave information from field seismic records. Therefore, the quality of seismic records in field surface wave exploration directly affects the quality of dispersion curve. For a certain survey area, vR has nothing to do with acquisition methods and parameters, but only with medium characteristics, and its frequency characteristics are related to the inhomogeneity of the earth's medium, which is close to shear wave velocity numerically. Therefore, generally speaking, the variation range of vR is certain, and the factors that affect the wavelength largely depend on the frequency components of surface waves. The propagation characteristics of low-frequency surface waves reflect deep information, while the characteristics of high-frequency components reflect shallow information. This shows that the frequency component is the decisive factor affecting Rayleigh wave exploration, and different excitation methods and acquisition parameters should be selected as far as possible for different exploration target layers to enhance the surface wave energy in the corresponding frequency band. If the exploration depth is very shallow (such as highway pavement detection), the frequency should be as high as possible (about several hundred weeks), and if the exploration depth is large (above 10 m), the low-frequency components should be kept as much as possible. In transient Rayleigh wave exploration, the factors that affect the frequency components of surface waves mainly include the following aspects.
Figure 2-3-8 Dispersion Curve of Multilayer Media
Excitation frequency of (1) source
It is best to use broadband pulse source, especially when detecting deep target layer, which requires that it can excite particularly low frequency energy.
(2) Frequency response characteristics of the receiver detector
In an ideal situation, the frequency response characteristics of geophones used in surface wave exploration should be a wide band from zero to hundreds or even thousands of cycles, which is beyond the reach of general seismic exploration geophones, so a wide band geophone suitable for surface wave exploration should be developed.
(3) Record the frequency response of the system.
At present, seismic data acquisition systems generally have frequency response characteristics of several weeks to several thousand weeks, which can basically meet the requirements of surface wave exploration, but attention should be paid to the selection of filter files when collecting.
(4) the influence of time sampling rate
According to the sampling theorem
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The higher the time sampling rate, the higher the high-frequency components satisfying the false frequency theorem, the lower the frequency resolution in the frequency domain after Fourier transform, the smaller the δ T in the instant domain and the larger the δ F in the frequency domain. As we have said, the variation range of phase velocity is fixed in a certain depth range, which often does not exceed an order of magnitude, but the frequency component of surface wave ranges from several weeks to hundreds of weeks, even reaching more than a thousand weeks in extremely shallow exploration. Therefore, from the formula (2.3. 1), it can be seen that when f increases at equal intervals Δ f, the λR values corresponding to different f in low frequency band are very different, while the λR values corresponding to different f in high frequency band are very different, which leads to the curve characteristics of extremely uneven dispersion point distribution in general transient Rayleigh wave exploration: that is, the high frequency band points are very dense, while the low frequency band points are particularly rare.
This requires determining the time sampling rate according to different exploration target layers. For shallow and extremely shallow exploration, a higher time sampling rate should be adopted, while for deep exploration, a lower sampling rate should be adopted, so as to increase the frequency points in the low frequency band on the dispersion curve line and improve the resolution of deep exploration. In addition, another way to solve this problem is to increase the number of points of FFT transform, so as to increase the number of frequency points of the low frequency band F on the dispersion curve, or to refine it specifically.
In addition to the above factors directly related to frequency, the following factors also have great influence on transient Rayleigh wave exploration.
(5) the influence of spatial sampling rate
As we all know, in reflection seismic exploration, the spatial sampling rate is not only related to the horizontal resolution, but also to the vertical resolution. In Rayleigh wave exploration, the dispersion effect reflects the average effect of the medium between two receiving points, indicating that the smaller the spatial sampling rate, the more detailed the representation of the lateral change of the medium, that is, the higher the lateral resolution; On the other hand, the spatial sampling theorem requires that
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If the above formula is not satisfied, spatial aliasing will occur in wave number domain data processing; Furthermore, even without data processing in wave number domain, it is only required to satisfy δx≤λR from the reliability of phase shift calculation, otherwise the phase shift between two paths is not the phase difference between surface waves with the same frequency, and an erroneous dispersion curve will be obtained. This shows that the spatial sampling rate has an influence on the longitudinal resolution, so we should pay special attention to this point when designing the acquisition parameters, especially when detecting shallow targets (such as highway pavement detection), the detection depth may be only tens of centimeters, and the speed is high, so it is easy to meet the situation of (2.3.2) or Δ x ≤λ r, and then Δ x should be determined according to the following principles. According to the half-wavelength and spatial sampling theorem (2.3.2) or the empirical basis of δx≤λR, δx is required to satisfy δx≤h or δx≤2h, so as to distinguish the stratum with depth (or thickness) of h..
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(6) The influence of channel consistency in multi-channel reception.
According to the principle of transient Rayleigh wave exploration, only the signals received by adjacent geophones have good correlation can we achieve good exploration results. Therefore, the receiving geophone is required to have good amplitude and phase consistency, otherwise, the difference of correlation (including amplitude and phase) between channels will cause errors in the calculation and interpretation of dispersion curve.
(7) Influence of non-exploration target objects
For example, buildings around the site and shallow obstacles below the topsoil (such as wall foundation) will produce reflected surface waves, which will affect the calculated value of dispersion curve.
The above factors may lead to poor correlation between traces (including amplitude and phase) in multi-channel surface wave recording, and this inconsistency between traces will cause calculation errors when calculating dispersion curves.
3. Analysis of several abnormal curves
1) In the dispersion curve shown in Figure 2-3-9, λR is equal to or close to a constant, which is obviously an abnormal situation. From λR=vR/f, because λR is a constant, f becomes a linear function of vR, and because δ φ =, δ φ is a constant for all f of the dispersion curve in A section. It can be seen that the reason for the abnormality of section A in the dispersion curve is that the phase shift Δ φ is equal to a constant, which is obviously incorrect.
2) In the dispersion curve shown in Figure 2-3- 10, the vR value decreases rapidly with the decrease of frequency, which is the result of serious interference all the time. Its characteristic is that the surface wave velocity is obviously lower than the normal formation wave velocity. The reason for this result must be the calculation error of phase shift Δ φ, which is caused by the serious interference of surface waves or the inconsistency between the two detectors.
3) Oblique straight line segment in IF dispersion curves of Figure 2-3- 1 1a and B. The dispersion curve of Figure 2-3-1a is a curve dominated by oblique straight lines; Fig. 2-3- 1 1b is composed of normal dispersion curve and oblique straight line segment. Let's analyze the reasons for this situation. We can use the following functional relationship to describe the diagonal segment:
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Where k and vR0 are constants and λR=vR/f, then vR=vR0+k, after transformation =vR0, and according to VR =.
Figure 2-3-9 Abnormal Dispersion Curve Section A
Figure 2-3- 10 Abnormal Deviation Curve
Figure 2-3- 1 1 dispersion curve caused by regular interference
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Equation (2.3.5) shows that Δ x is directly proportional to f. According to Fourier analysis theory, we know that if signal f2(t) is only a delayed form of f 1(t), then in their cross-power spectrum, the phase difference between the same frequency components is directly proportional to their frequencies, and their amplitudes are the same. According to formula (2.3.5), Δ φ is also proportional to the frequency, that is, the two records that produce oblique straight line segments are the same and do not disperse. Because the direct wave and refracted wave in seismic records are non-dispersive, the oblique direct dispersion curve appears because the energy of the direct wave and refracted wave is too strong, so attention should be paid to eliminating and weakening these waves when collecting data.
Other methods can also be used to improve the quality of dispersion curves, including f-K filtering (Al-Husseini et al., 198 1) and narrowband filtering (Mari,1984; Herrmann, 1973), and p-ω method (McMechan and Yedlin,1981; Mochtar et al., 1988).
Four, surface spectral analysis (SASW)
The most promising application of Rayleigh wave is to evaluate engineering geological sites through surface spectrum analysis (Stokoe et al., 1994). This method has been used in highway quality evaluation and civil engineering to measure the stiffness of materials within a few meters underground. By using different ranges of wavelengths, the media with different depths can be sampled.
SASW method is developed from the steady Rayleigh wave method. This steady-state Rayleigh wave takes an exciter with a given frequency as the source, moves a single vertical geophone outward step by step from the source point, and finally is continuously buried in the same phase. At this time, the distance between seismic wave and geophone is one wavelength. If we know the frequency and wavelength, we can find the speed corresponding to this frequency.
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Because different wavelengths reflect the properties of different depths, the velocity profile can be established by constantly measuring wavelengths by changing the frequency. But the disadvantage of this technology is that it is time-consuming.
1994 adopts frequency-swept multi-channel receiving technology. The signal is transformed into frequency domain by fast Fourier transform, and the phase difference of each frequency is calculated in frequency domain. The propagation time difference is given by the following formula:
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For various frequencies, where φ(f) is the phase difference expressed in radians; F is the frequency in hertz;
When the distance d between detectors is known, the Rayleigh wave velocities of various frequencies can be calculated by the following formula.
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The wavelength of Rayleigh wave is:
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For various frequencies, these calculation results will be plotted as v-λ diagram.
By comparing and matching with the theoretical curve obtained by the forward model, and through a certain inversion program, the stiffness parameter model is extracted.
Five, multi-channel surface wave analysis technology (MASW)
Multi-channel surface wave analysis is a relatively new technology. Miller et al. and Xia et al. have successfully solved some production examples by using this technology. This technology includes the following advantages.
1) The seismic source is portable and reusable, and can generate Rayleigh surface waves with wide-band effective energy (2 ~ 2~ 100Hz).
2) The processing program for extracting and analyzing one-dimensional Rayleigh wave dispersion curve is stable, flexible, easy to use and accurate.
3) The one-dimensional near-surface shear wave velocity profile obtained by the generalized linear iterative inversion method with minimum hypothesis has the characteristics of stability and flexibility (Tian G et al., 1997, Xia J et al., 1999).
4) Establish two-dimensional shear wave velocity field.
5) Its observation system is similar to CDP method, which provides a basis for using both body wave reflection and surface wave information in one exploration (Okada et al., 2003).
It is easy to obtain surface waves by using scanning sources (such as vibroseis) or pulse sources (such as heavy hammers). For multi-channel analysis, the original irrelevant data is the most suitable, therefore, if the frequency and amplitude can meet the needs of exploration purposes, it is preferable to use scanning seismic source. On the other hand, it is necessary to decompose the pulse source data into scanning frequency format to show the relationship between phase velocity and frequency of dispersion ground roll wave. The basic field equipment and acquisition program of MASW method are consistent with the common center point measurement in traditional reflected wave exploration, and are consistent in some principles. MASW and traditional Rayleigh wave exploration are the same or similar in principle, but different equipment is used in field work and different calculation and interpretation methods are used in indoor processing. The following is a brief introduction to the selection principles of some parameters in MASW method.
1. Near migration
Good seismic wave recording requires that the field equipment and acquisition parameters are suitable for recording Rayleigh waves in the basic mode, but not for recording other types of sound waves. Due to the influence of the near field, Rayleigh wave can only be regarded as a plane wave in horizontal transmission after propagating a certain distance from the source.
Surface wave propagation in plane form is impossible under any circumstances, and it must satisfy that the minimum offset distance (x 1) is more than half of the required maximum wavelength (λmax), that is
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In the multi-channel recording displayed in the scanning frequency format, the near-field effect makes the phase coherence at low frequency poor, and this coherence decreases with the increase of frequency, as shown in Figure 2-3- 12(b). Different researchers have given different proportional relationships between x 1 and zmax. It is generally believed that the penetration depth of surface waves is approximately equal to the wavelength (λ), and the maximum exploration depth zmax (from which a reasonable vS can be calculated) is considered to be half of the maximum wavelength (λmax). Therefore, the formula (2.3. 10) should be changed to
Fig. 2-3- 12 surface wave records of different qualities obtained by vibroseis
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It can be seen that the formula (2.3. 1 1) provides a good principle for selecting small offset.
2. Far migration
With the propagation of various sound waves in the ground, the high frequency part of surface waves decays rapidly. If the maximum deviation is too large, the high frequency part of surface wave energy will not dominate the spectrum. Especially when there are body waves, the body wave interference caused by the attenuation of high frequency surface waves at the cannon offset is called far field effect. This effect limits the measurement of phase velocity at the highest frequency. When the initial layer number model is determined according to the half-wavelength principle, the maximum frequency (fmax) component usually displays the top image for a specific phase velocity.
Equation (2.3. 12) can be used to roughly estimate the minimum thickness of the shallowest layer. If you want to find a smaller h 1, you need to reduce the detector arrangement or offset (reduce the offset x 1 or reduce the track spacing dx). In order to avoid spatial aliasing, dx should not be less than half of the shortest wavelength.
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Where: vRmin and λmin represent the minimum phase velocity and the minimum wavelength, corresponding to the maximum frequency fmax. Although the final inversion vS profile may be shallower than h 1, it is generally considered that the vS values of these layers are unreliable (Rix and Leih, 19 1).
They are: ① good coherence; ② Near field effect; ③ Far field effect. The offset distances are ①27m, ② 1.8m and ③89m.
3. Sweep frequency records.
Sweep frequency records can be obtained directly or indirectly. When preparing for scanning recording, there are three parameters to be considered: the lowest recording frequency f 1, the highest recording frequency f2 and the length t or stretching function of frequency-time coordinate. The selection of these parameters must meet certain principles.
The lowest frequency f 1 determines the maximum exploration depth, i.e.
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Where vR 1 is the phase velocity corresponding to the frequency f 1.
The lowest frequency (f 1) is usually limited by the natural frequency and source type of geophone. If zmax can't meet the needs of exploration depth, it is necessary to use a seismic source that can produce rich low-frequency components or a geophone with low natural frequency.
Generally, the highest frequency (f2) is several times of the ground roll video rate, but less than the optimal value of the frequency obtained by noise analysis.
The length (t) of the scanning record must be long enough. When the near-surface properties change sharply with depth, a longer recording length is needed. Generally speaking, when f 1 and f2 are selected, the record with the length of 10 can meet the processing needs.
4. Stretch function
The pulse record r(t) obtained by a heavy hammer or a drop hammer can be converted into a swept frequency record Rs(t) by convolution operation of stretching functions S (t) and r(t), namely
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Where: "*" stands for convolution operation. The stretching function is a sine function, which is a function of time. S(t) usually adopts a linear scanning function similar to controlled source exploration:
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Where f 1, f2 and t represent the lowest frequency, the highest frequency and the length of s(t) respectively.
In practical work, these parameters can be reasonably selected through some pre-designed programs.
5. Dispersion curve
In order to obtain accurate vS profile, obtaining dispersion curve is the most critical step. Draw the dispersion curve in the phase velocity-frequency coordinate system (Figure 2-3- 13), and establish the relationship between them by calculating the phase velocity in the linear range of each frequency component on the swept frequency record. By analyzing and removing noise in surface wave data, the accuracy of dispersion curve can be improved. Multi-channel consistency can well separate various frequency components from surface wave seismic records, convert pulse data into frequency domain for calculation, and then get dispersion curves.
Fig. 2-3- 13 dispersion curve of surface wave records of a dam in Kansas
Step 6 turn upside down
Inversion of vS curve by iterative method (Figure 2 3 14) needs to know dispersion curve data, Poisson's ratio and density. The generalized least square method makes the inversion method automatic. In the whole inversion process, Poisson's ratio, density, layers and P-wave velocity can be constant, and only the S-wave velocity can be variable, so iteration is carried out. In iterative inversion, it is necessary to concretize the initial model as the starting point of inversion. The initial model consists of shear wave velocity, longitudinal wave velocity, density and layers. Among these four parameters, the shear wave velocity has the greatest influence on the convergence of iterative method, and there are several methods to ensure the reliability and accuracy of the convergence after the initial vS profile calculation. In the vS profile, the relationship between shear wave velocity (vS) and phase velocity (vR) at a certain frequency (vS= 1.09vR) must be explained in detail. The relationship between the depth and wavelength corresponding to this frequency is as follows.
Figure 2-3- 14 vS curve iterative inversion
Figure 2-3- 15 Variation of Coefficient A with Frequency
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Where a is a coefficient with little change with frequency, and it is based on the generalization model shown in Figure 2-3- 15.
Using a series of one-dimensional vS curve values with different distances obtained by inversion, a two-dimensional vS profile can be obtained by using a drawing program (such as Surfer). Fig. 2 3 16 is the inverse shear wave velocity profile obtained by the author on the dam of the university of Kansas.
Fig. 2-3- 16 shear wave velocity profile obtained on the dam