Optimal Recovery of Rayleigh‐Wave Overtones by Multi‐Directional Acquisition

Rayleigh waves are ubiquitously used for subsurface characterization through dispersion curve inversion, whose quality depends on the number of useable overtones. Traditional analysis is based on vertical receivers and, for active surveys, sources. However, for layered media, eigenfunction theory shows that optimal recovery of any given dispersion mode and frequency can be achieved by either vertical source‐vertical receiver or radial source‐radial receiver configurations, with source directivity being more dominant. Multi‐directional near‐surface wave‐equation modeling and field examples, including distributed acoustic sensing data, validate these predictions. Statistical analysis of compaction‐type near‐surface models shows that overtones are better recovered for radial‐radial surveys in large portions of the useable spectrum, extending beyond 60% for the second and above modes. We conclude that incorporating radial‐radial data acquisition is beneficial and should become standard procedure in active surveying, as well as analyzing the radial component in ambient noise and earthquake‐induced Rayleigh waves.

Many authors have recognized the importance of properly recognizing and inverting overtones, as they increase the inversion's sensitivity to the velocity in deeper regions, stabilize the non-linear inversion problem, and minimize uncertainty (Luo et al., 2008;Maraschini et al., 2010;Park et al., 1999;Wu et al., 2020;Xia et al., 2003;Xu et al., 2006).Nonetheless, the factors that govern the existence of recoverable Rayleigh wave overtones in active surveys have remained largely unexplored.Whereas new data processing techniques aim at better overtone recovery, the acquisition stage in Rayleigh-wave surveys is still principally conducted with vertical geophones recording a vertical source such as a weight drop or hammer shot.We report that in multi-component surveying, by varying source and receiver directivity to include in-line components, optimal excitation and recovery of overtones can be achieved.This phenomenon has been observed in several cases (Dal Moro, 2019;Dal Moro et al., 2015) but not systematically analyzed.We explain and exemplify this phenomenon through surface wave eigenfunction theory, wave-equation modeling, and field data examples using both geophone and DAS data.We also conduct a statistical analysis of the influence of geology on the usefulness of in-line surveying, showing that multi-directional acquisition yields improved recovery of all modes, with the biggest advantage in the second and higher overtones.The methodology we develop is critical to understand DAS data, dominated by in-line recording.We conclude that multi-directional surveying and data analysis would benefit almost any active Rayleigh-wave study, and comment on the impact of our analysis on ambient-noise and earthquake-driven surface wave analysis.

Theoretical Background
A detailed derivation of the eigenfunction theory is in Text S1 in Supporting Information S1, following Aki and Richards (2002).Its main result is that the radial (u R ) and vertical (u Z ) particle motion of Rayleigh waves that propagate through a layered medium, provided that the sources and receivers are along a line at the surface, are given by: ( where F R and F Z are radial and vertical force sources, respectively, n is the mode, c is the phase velocity, U is the group velocity, k n is the wavenumber of the specific mode, r is the propagation distance, ω is radial frequency, and d is the first energy integral.r 1 and r 2 are the eigenfunctions of the radial and vertical components, respectively, and they are calculated numerically.For the surveys we describe, only their values 10.1029/2023GL105885 3 of 9 at the surface (z = 0) are needed.Equation 1 can shed light on the relation between source/receiver directivity and the resulting particle motion.The amplitude difference between u R and u Z is due to each term in the sum being multiplied by  √ 1( = 0) for u R and by  √ 2( = 0) for u Z .On the source side, assuming pure F Z and F R , F Z is multiplied by r 2 (z = 0) and F R by r 1 (z = 0).Assuming F R = F Z = 1 in each case, the amplitude of a certain mode n is where A is a quantity which is constant for all cases.We mark In general, the eigenfunction values can change as a function of both dispersion mode and frequency.For a given mode and frequency range, the maximal amplitude will always be obtained either for radial-radial acquisition (if E R ≥ E Z ) or for vertical-vertical acquisition (E R ≤ E Z ).However, we note that the source directivity is more important than receiver directivity, because the source term scales as ∝E R or ∝E Z , instead of √  for the receiver.

Wave-Equation Modeling
We conduct a numerical simulation in order to demonstrate the theoretical eigenfunction results.In the synthetic experiment, we model the subsurface as a four-layer structure overlaying an infinite half-space (see Text S2 in Supporting Information S1).This model coarsely represents the results of a multi-mode surface-wave inversion performed on real data and presented later.The synthetic data are acquired along a line, but the wave equation is solved in 3-D.We vary source and receiver directivity to create four virtual surveys, in which all other acquisition parameters are identical.The source is a directional point force with broad-band wavelet with near-flat spectrum between 5 and 70 Hz, identical for all surveys.Receivers measure particle velocity.To generate dispersion images we use the phase-shift (Park et al., 1999) method, which is commonly used for near-surface surveys, throughout this paper.
In Figure 1, we show the strong effect of the source and receiver direction on the recorded data.As expected from the theory, the largest difference in relative prominence of different modes is between   , ̇ and  , ̇ .They are proportional to   3∕2  and   3∕2  , respectively, and are thus mutually exclusive compared to mixed-direction acquisitions that depend on both E Z and E R .We also compute R = log(E R /E Z ) to estimate the ideal source-receiver directivity for each mode and frequency.If it is positive, radial-radial acquisition is better than vertical-vertical, and vice versa.We emphasize that the energy partitioning between the different propagation modes depends on the subsurface structure, over which we have no control.Therefore, we focus on the relative strength of each mode in a radial/radial versus vertical/vertical comparison, since we control the acquisition.In this context, it is clear why the eigenfunction analysis is useful in predicting the frequency range in the dispersion images for which we can expect a better recovery of the dispersion curves in one acquisition or the other.

Field Data Examples
We exemplify the benefits of directional acquisition through two field examples.The first is a near-coastal environment in which approximately 15 m of Kurkar dunes (aeolian quartz sandstones with carbonate cement) overlay the water table.The synthetic example shown above utilizes a velocity model (see Text S2 in Supporting Information S1) estimated for this area.We conduct a survey with 51 3-C 5-Hz geophones spaced 2 m apart, in which we control source directivity.The source is a mechanical vibrator applying a single sweep in the frequency range of 10-100 Hz, and hence the low-frequency part of the spectrum is not shown.We cross-correlate recorded data with the known sweep before generating dispersion images in Figure 2. The similarity to the synthetic example is evident, and it shows that all overtones are significantly better recovered for the radial-radial acquisition.In the mixed acquisitions, the overtones are visible albeit being weaker than the fundamental mode, as expected 4 of 9 from theory.A noticeable difference is that for the field data, the fundamental mode is visible in the radial-radial case, whereas in the synthetic case it is not.The reason is that for the synthetic case, despite R ∼ 0 for the fundamental mode, the overtones overpower it.In the field data, the overtones are likely less dominant compared to the fundamental, and we can thus partially see the fundamental mode.We conduct a survey with the same parameters in a different environment.The lithology of the area is a sand-shale mixture with occasional sandstones.The area can be characterized by a compaction model driven by overburden pressure, representative of unconsolidated sediments without clear boundaries between units, leading to velocity increasing with depth.We also obtain a velocity model of the area from inversion of Rayleigh waves (see Text S2 in Supporting Information S1).In Figure 3, we show that for this velocity structure the radial acquisition is overall less effective compared to the previous example.Nonetheless, the second mode is visible in the radial-radial case.The eigenfunctions calculated using the velocity model indeed imply that both the fundamental and the first mode are consistently weaker in the radial-radial case than in the vertical-vertical case, but the second is stronger and hence visible in the data.

Dependence on Subsurface Structure
As we show in the field data section, it is clear that the subsurface velocity model influences the relations between u R and u Z , and thus the added benefit in mode recovery from a radially directed survey.We try to quantify it by first randomly generating 250,000 different 1-D velocity models, consisting of 10 layers and an infinite half-layer below them.Details about the model generation are in Text S2 in Supporting Information S1.A subset of the sampled models is in Figure 4a.For each model, we compute the values of E R , E Z for the first five dispersion modes in a frequency range of 10-70 Hz, which is representative of the recoverable frequency band in the field example.Then, for each model and dispersion mode, we compute the percentage of frequencies for which E R > E Z .This computation takes into account the cutoff frequencies of the overtones.Figure 4b shows the histogram of this range for the different modes, along with the mean range of E R  ∕E z for the frequency range in which E R > E z .A similar analysis for deeper models and lower frequencies is discussed in Text S3 in Supporting Information S1.The eigenfunction analysis explains this result-the second mode is slightly stronger in the radial case between 20 and 40 Hz, which is the range in which we can observe it in (d).In any other mode and frequency range, the radial acquisition is not beneficial as log(E R /E Z ) < 0.
Overall, this analysis corroborates the main conclusion of this study-the addition of radial-radial acquisition yields a significant improvement of overtone dispersion curve recovery compared to only using the traditional vertical-vertical survey.The benefit is particularly noticeable for the second overtone and above, as about 70% of the frequencies are better recovered and the mean improvement in recorded amplitudes is by more than an order of magnitude, as is it proportional to   3∕2  .

Implications for DAS
In fiber-optic sensing using DAS (Hartog, 2017), the measurement is conducted along the axis of the fiber.For near-surface Rayleigh-wave surveys using DAS, the sources are generally located along the fiber.As a consequence, the receivers are by definition recording purely radial Rayleigh wave propagation.As discussed in the theoretical background section, the source directivity term is more important than receiver directivity, so varying the source has merit even for DAS cases where the receiver directivity cannot be modified.Nonetheless, a DAS acquisition will never be ideal because it cannot include the vertical-vertical component.
There are two fundamental differences between DAS and geophones.First, DAS measures strain, or strain-rate, instead of particle displacement or velocity.However, the scaling between them amounts to the apparent phase velocity (Lellouch et al., 2019;Lior et al., 2021), a quantity that is independent of the receiver directivity.A second difference is the DAS gauge length (GL)-instead of point-wise measurement, DAS measures strain (or strain-rate) over a spatial window whose size is the GL.Effectively, the GL effect acts as a moving average along the fiber.In the wavenumber domain, it is equivalent to a sinc filter with a width of  1 GL (Dean et al., 2016).Consequently, it is generally accepted that wavenumbers shorter than the GL are not recovered by DAS due to aliasing.This particularity of DAS data can strongly limit the analysis of Rayleigh waves, especially for high temporal frequencies and low phase velocities (Abukrat et al., 2023).We thus expect the fundamental mode, which has the lowest phase velocities, to be most affected by the GL.
We repeat the field data experiment in the coastal environment using a portable land-streamer DAS fiber.The fiber is a 150 m long semi-tactical cable weighed down by a fiber hose to improve coupling to the surface.While this deployment method yields lower-quality data than buried fibers, its ease of use allows for surveys in almost any environment.We record DAS data at a 1 m resolution and 10 m GL for vertical and radial sources, with the same source spacing as for the geophone experiment.In Figure 5, we show dispersion images obtained from DAS data.By comparison of the dispersion curves picked over radial-radial geophone data, we see that the DAS data for a radial source, despite lower signal-to-noise ratio, is consistent.However, the fundamental mode is practically invisible.For a vertical source, the fundamental mode is better recovered, but the gauge-length effect limits its recovery at high frequencies.Overtones, as is the case for geophones with the same directivity, are significantly weaker. ∕E z computed over those frequencies.The analysis is conducted for each mode separately (fundamentalblue, first-orange, second-green, third-red, fourth-purple) and shows the benefit of radial-radial surveys for overtone recovery.

Discussion and Conclusions
This study shows that the maximal amplitude of each frequency and propagation mode of Rayleigh waves will be recorded by either vertical source-vertical receiver or radial source-radial receiver acquisition.The statistical analysis of both near-surface compaction-type and kilometer-deep linear gradient models shows that the second to fourth overtones are better recovered by radial-radial acquisition in about 70% of the cases.For these frequencies, the improvement in recorded amplitudes is by more than an order magnitude.For the fundamental and first overtone, traditional vertical-vertical acquisition is preferable for about 85% of tested scenarios.As the optimal directivity depends non-linearly on the geological structure, and cannot be unequivocally predicted before the acquisition, we recommend acquiring both vertical-vertical and radial-radial data wherever possible.For DAS recordings, the radial nature of the receiver inherently limits the quality of recorded data, as vertical-vertical surveying is impossible.In addition, GL effects mostly hinder the fundamental mode.We thus recommend acquiring DAS data with both vertical and radial sources to mitigate these negative effects.
Nonetheless, there remain some practical questions related to subsequent data analysis.First, working with two different dispersion images can be practically difficult.Further methodological developments that would allow for the creation of a unified dispersion image would be highly useful.In addition, overtone identification is often a challenge, compared to the fundamental mode that is usually unambiguous.Wrong identification can have a severe detrimental effect on inversion results.This difficulty can be methodologically mitigated by first conducting the inversion based on the unambiguous fundamental mode and potentially the first overtone.Based on the inverted model, predicting the remaining overtones is straightforward, and they can guide the classification process.Naturally, conducting the procedure using a single dispersion image is preferable.
We have shown that the source directivity term is more dominant than the receivers'.For surface waves generated by ambient noise or earthquakes, we have control only over the receivers.However, as there can still be a difference between dispersion images, we recommend analyzing both vertical and radial receivers.In addition, for earthquake seismology, choosing earthquakes with certain moment tensors can prove beneficial.Whereas this study uses a formulation based on point forces, there is a similar derivation for moment tensor components (Aki & Richards, 2002).As such, given an earthquake catalog and an initial subsurface velocity model, we can choose events that will maximize u R or u Z , and analyze them with the appropriate receiver polarity.

Data Availability Statement
Data acquired for this study are publicly available at Lellouch (2023).We utilize the evodcinv package (Luu, 2023) for eigenfunction calculation, dispersion mode computation, and inversion.

Figure 2 .
Figure 2. Dispersion images of a field test with different source and receiver orientations.(a) Vertical source, vertical receiver (b) Vertical source, radial receiver (c) Radial source, vertical receiver (d) Radial source, radial receiver.Overtones, as manually picked over (d), are marked in yellow dashed lines in all subplots.Overtones are most noticeable for the radial-radial case, in which case the fundamental tone disappears beyond 30 Hz.

Figure 1 .
Figure 1.(a-d) Dispersion images computed using the phase-shift method for a synthetic data set.(a) Vertical source, vertical receiver (b) Vertical source, radial receiver (c) Radial source, vertical receiver (d) Radial source, radial receiver.Predicted dispersion curves are in dashed-dotted lines, and marked by their appropriate mode number.(e) Logarithm (base 10) of the ratio of radial to vertical eigenfunction, computed for the first six modes.There is an excellent match between high values of log(E R /E Z ) and visible overtones in (d), which depends on   3∕2  .The fundamental mode in (d) is masked because of the relative prominence of the overtones, but is still visible.

Figure 3 .
Figure 3. Dispersion images computed for a field test in a different area, with (a) Vertical source, vertical receiver (b) Vertical source, radial receiver (c) Radial source, vertical receiver (d) Radial source, radial receiver.We can primarily observe only the fundamental mode.Predicted dispersion curves are in dashed-dotted lines, and marked by their appropriate mode number.Only in (d), we can see a faint second mode.(e)The eigenfunction analysis explains this result-the second mode is slightly stronger in the radial case between 20 and 40 Hz, which is the range in which we can observe it in (d).In any other mode and frequency range, the radial acquisition is not beneficial as log(E R /E Z ) < 0.

Figure 4 .
Figure 4. (a) 100 out of 250,000 random models tested for statistical analysis.The models are ordered by their average shearwave velocity (black-low, pink-high).(b) Histogram depicting percentage of frequencies for which E R > E Z , along with the mean value of E R  ∕E z computed over those frequencies.The analysis is conducted for each mode separately (fundamentalblue, first-orange, second-green, third-red, fourth-purple) and shows the benefit of radial-radial surveys for overtone recovery.

Figure 5 .
Figure 5. Dispersion images for distributed acoustic sensing (DAS) data-(a) Vertical source (b) Radial source.We repeat the overtones, picked using 3-C geophones and shown in 2, in yellow dashed lines, showing good agreement between DAS and geophones for radial-radial acquisition.In dashed red, we mark the theoretical aliasing limit imposed by a 10-m gauge length.Below the line, recorded wavefronts are heavily distorted and cannot be accurately recovered, as is most obvious in the abrupt cut-off of the fundamental mode in (a).