Application of the Spatial Auto-Correlation Method for Shear-Wave Velocity Studies Using Ambient Noise

Abstract

Ambient seismic noise or microtremor observations used in spatial auto-correlation (SPAC) array methods consist of a wide frequency range of surface waves from the frequency of about 0.1 Hz to several tens of Hz. The wavelengths (and hence depth sensitivity of such surface waves) allow determination of the site S-wave velocity model from a depth of 1 or 2 m down to a maximum of several kilometres; it is a passive seismic method using only ambient noise as the energy source. Application usually uses a 2D seismic array with a small number of seismometers (generally between 2 and 15) to estimate the phase velocity dispersion curve and hence the S-wave velocity depth profile for the site. A large number of methods have been proposed and used to estimate the dispersion curve; SPAC is the one of the oldest and the most commonly used methods due to its versatility and minimal instrumentation requirements. We show that direct fitting of observed and model SPAC spectra generally gives a superior bandwidth of useable data than does the more common approach of inversion after the intermediate step of constructing an observed dispersion curve. Current case histories demonstrate the method with a range of array types including two-station arrays, L-shaped multi-station arrays, triangular and circular arrays. Array sizes from a few metres to several-km in diameter have been successfully deployed in sites ranging from downtown urban settings to rural and remote desert sites. A fundamental requirement of the method is the ability to average wave propagation over a range of azimuths; this can be achieved with either or both of the wave sources being widely distributed in azimuth, and the use of a 2D array sampling the wave field over a range of azimuths. Several variants of the method extend its applicability to under-sampled data from sparse arrays, the complexity of multiple-mode propagation of energy, and the problem of precise estimation where array geometry departs from an ideal regular array. We find that sparse nested triangular arrays are generally sufficient, and the use of high-density circular arrays is unlikely to be cost-effective in routine applications. We recommend that passive seismic arrays should be the method of first choice when characterizing average S-wave velocity to a depth of 30 m (V_s30) and deeper, with active seismic methods such as multichannel analysis of surface waves (MASW) being a complementary method for use if and when conditions so require. The use of computer inversion methodology allows estimation of not only the S-wave velocity profile but also parameter uncertainties in terms of layer thickness and velocity. The coupling of SPAC methods with horizontal/vertical particle motion spectral ratio analysis generally allows use of lower frequency data, with consequent resolution of deeper layers than is possible with SPAC alone. Considering its non-invasive methodology, logistical flexibility, simplicity, applicability, and stability, the SPAC method and its various modified extensions will play an increasingly important role in site effect evaluation. The paper summarizes the fundamental theory of the SPAC method, reviews recent developments, and offers recommendations for future blind studies.

1 Introduction

The microtremor method often also named the ambient noise method involves study of spectra and phase velocity dispersion of surface waves to deduce properties of the Earth’s surface at depths from a few metres to a few km. The frequencies in use are typically 1–30 Hz, attributable mainly to anthropogenic sources, and 0.1–1 Hz, attributable to natural phenomena such as wave action at coastlines and wind action on vegetation and topography. The majority of surveys use vertical component ground motion associated with Rayleigh waves, although surveys using three-component sensors have demonstrated that the properties of Love waves as well as Rayleigh waves can be used in the analysis. Array methods as reviewed in this paper measure the velocity of surface waves propagating across an array and thence derive shear-wave velocity–depth relationships. Single-station methods are also common, whereby the shape of three-component spectra and the ratio of horizontal to vertical spectral ratios (HVSR) of ambient noise provides similar or complementary information. Single-station methods are reviewed in a companion paper in this journal (Molnar et al. 2018).

The propagation velocity of surface waves is dispersive (frequency dependent), and in a vertically heterogeneous earth their sensitivity to properties of the earth at depth is a function of wavelength, where short wavelengths (high frequencies) are sensitive only to earth properties at shallow depths, and long wavelengths (low frequencies) are sensitive to earth properties at deep depths. In principle we seek to measure the dispersion curve of phase velocity versus frequency and by use of inversion modelling deliver a layered-earth model of acoustic properties. Figure 1 of Foti et al. (2011) is a useful illustration of the process.

In this review we consider the underlying assumptions of the SPAC method and their influence on field survey design and efficiency. We note the contrasting properties, strengths, and weaknesses of alternative array processing methods and argue that sparse arrays in passive seismic observations when processed with SPAC methods represent the most efficient first choice for shear-wave velocity site characterization. This does not negate the value of active seismic methods, or passive seismic methods with alternative array processing algorithms when a large number of instruments are available or when difficult geological or seismic noise features occur. For this purpose Sects. 9 and 10 provide an overview of frequency-wavenumber (FK) beamforming methods and seismic interferometry (SI) methods and the degree to which these alternative methods may be considered as complementary to SPAC methods.

Of the three elastic properties compressional-wave (P-wave) velocity V_p, shear-wave (S-wave) velocity V_s, and density ρ, only the V_s has sufficient effect on Rayleigh-wave phase velocities to be generally resolvable in an inversion process. V_p and ρ are usually fixed by assumption or by some empirical relation between the parameter and V_s such as those provided by Brocher (2005). See Sects. 3 and 5 for further discussion on instances where variations in V_p may influence Rayleigh-wave velocities. For Love waves, the phase velocity is independent of V_p (Haskell 1953).

The spatial auto-correlation method (SPAC—which can equally be termed spatially averaged coherency) contains two key assumptions. Firstly, the surface waves observed are far-field waves with plane wave fronts, and secondly that sources are distributed in azimuth sufficiently to provide spatial averaging of observed inter-station coherencies.

The method can be illustrated beginning with the concept of a single plane wave observed at two stations, whereby the inter-station coherency may be written:

$$c_{ij} (f,\theta ) \, = \, \exp \, \left[ {irk(f)\cos \, \left( {\theta - \phi } \right)} \right]$$

(1)

where c _ij (f, θ) is the coherency frequency spectrum between stations i, j; r is the station separation, k is the wavenumber, θ is the angular displacement of the pair of stations, and ϕ is the propagation direction of the wave front. Following Okada (2003) and Asten (2006a), for a single noise-free wavefront observed by multiple station pairs distributed around a circle, Eq. (1) becomes a spatial average around the azimuth θ,

$$\frac{1}{2\pi }\int_{0}^{2\pi } {c\left( {f,\theta } \right){\text{d}}\theta } = J_{0} \left( {rk} \right) = J_{0} \left( {{{2\pi fr} \mathord{\left/ {\vphantom {{2\pi fr} {v\left( f \right)}}} \right. \kern-0pt} {v\left( f \right)}}} \right)$$

(2)

where c(f, θ) is the SPAC spectrum, J₀ the Bessel function of zero order, and v(f) is the phase velocity dispersion curve of the observed wave. As shown by Capon (1973) the spatial averaging can equally be achieved by using a single pair of stations in the presence of an omnidirectional wavefield, i.e. the spatial average of Eq. (1) is achieved as an integration over a distribution of wave propagation directions ϕ.

In practice the spatial averaging necessary for SPAC methods to be useful is provided by both an angular distribution of stations in a seismic array, and by the usual angular distribution of propagating seismic noise. A series of studies exists attempting to quantify the effect of limited angular distribution of station azimuths and/or limited angular distribution of the wave field (e.g., Asten et al. 2004; Asten 2006a; Cho et al. 2008; Luo et al. 2016). Extensive use of the SPAC methods over the past 5 years where multiple arrays have been employed and where ground truth is available leads us to the view that the demands of spatial averaging are not onerous and relatively simple arrays can be used without degrading the method; the point is illustrated in Sect. 8 of this paper. The concept of minimal arrays, extending to use of a single pair of stations, has been demonstrated to work in some situations, discussed further in Sect. 6.

Passive seismic methods include an assumption that wave sources are in the far-field, i.e. wavefronts are plane across the array. Studies by Roberts and Asten (2006, 2008) and Maranò et al. (2017) show that this requirement is not onerous, and source distances of order two array radii from the centre of an array are sufficient for the SPAC method to perform acceptably.

Active surface-wave methods are also in regular use for engineering-scale projects involving depths typically to 30 m. The most common method is the multichannel analysis of surface waves (MASW) described by Park et al. (1999). Opinions differ as to whether passive or active methods should be the tool of first choice with the other being used as a supplementary tool. As discussed in Sect. 6, this review concludes that passive SPAC methods are preferable as the tool of first choice.

2 Major Comparisons of Passive and Active Seismic Survey and Interpretation Methodologies

Multiple studies exist seeking to characterize the strengths and limitations of passive and active surface-wave methods on depth scales from metres to kilometres. Asten and Boore (2005a, b), Boore and Asten (2008), and Stephenson et al. (2005) present comparisons by up to 14 separate groups designing and interpreting data in the Santa Clara Valley, California. Cornou et al. (2007) summarize results of an international blind study of both synthetic and field data from four synthetic and four field sites. The InterPACIFIC project funded by a consortium of European government sources obtained detailed non-invasive passive and active seismic field measurements, plus invasive borehole studies, at three European sites representing soft sediments, firm glacial sediments, and rock (Mirandola, Italy; Grenoble, France; Cadarache, France). A total of fourteen international groups interpreted the data with results summarized in Garofalo et al. (2016a, b). A further publication by Foti et al. (2017) distils the wisdom of the multiple groups into a practical guide for future surveys. Finally, a North American group encompassing both government and industry with interests in power generation, the Consortium of Organizations for Strong-Motion Observation Systems (COSMOS), is sponsoring a current project for development of guidelines for non-invasive site studies relating to seismic risk (see Acknowledgements below).

Recent results of the InterPACIFIC project, and results from a passive seismic array at the QEII site, Christchurch, New Zealand (Wood et al. 2014; Cox et al. 2014) quantify some of the uncertainties in V_s models when data are interpreted by different groups and/or with different methodologies. A strength and a limitation of these studies is that a large amount of data allows much redundancy, and analysts were able to interpret as much or as little of the data provided as they chose. This process facilitates maximum resolution of the V_s profile, but does not define benefits or limitations of different field arrays for routine efficient use. Sections 5, 7, and 8 of this review draw on data from these two studies.

3 Range of Frequencies and Detection Depths

While Wathelet et al. (2008) defined upper and lower bounds on resolvable depths based on station spacings of an array, we prefer to define bounds based on wavelengths of signals alone. The reason is that in direct fitting of observed and model SPAC curves, useful upper frequencies are not defined by a Nyquist relation applied to station spacings, but only by the highest usable frequency on a Bessel J₀ curve having (in general) multiple peaks and troughs. Likewise, it is the lowest usable frequency which defines the greatest depth of penetration, not the array diameter. Some authors such as Cho et al. (2013) claim useful estimates of phase velocity with a variant of the SPAC method for wavelengths up to 100 times the diameter of a miniature array. Such claims may be instrument and site dependent, and we treat them with caution, but they do underline the fact that limits on useful data with SPAC methods are set by the maximum wavelength, not intrinsically by the array size.

Shallow resolution is defined here as a fraction of the minimum observed wavelength λ, where the fraction is set from sensitivity studies on Rayleigh-wave modelling; as an example Asten (1976) found maximum sensitivity (the partial derivative of phase velocity with respect to depth) to be λ/3. Maximum depth of useful interpretation is about λ/2 for a fundamental-mode Rayleigh wave, so we propose the guidelines for minimum and maximum depths of investigation D_min, D_max, to be

$$\begin{aligned} D_{\hbox{min} } = \, \lambda_{\hbox{min} } /a_{\hbox{min} } = V/\left( {a_{\hbox{min} } \cdot f_{\hbox{max} } } \right),{\text{and}} \hfill \\ D_{\hbox{max} } = \, \lambda_{\hbox{max} } /a_{\hbox{max} } = V/ \, (a_{\hbox{max} } \cdot \, f_{\hbox{min} } ) \hfill \\ \end{aligned}$$

(3)

where λ is wavelength and V is the phase velocity of the Rayleigh wave, a_min, a_max are constants dependent on site conditions and data quality. In this study we find the values a_min = 3, and a_max = 2 as noted in the previous paragraph to be useful guidelines, but other values in the range 2–4 may be substituted based on locality and experience.

Table 1 provides some useful guidelines for common soil and rock types.

Table 1 Examples of minimum shallow resolution D_min and maximum effective depth penetration D_max for fundamental-mode Rayleigh waves, wavelength λ, at selected minimum and maximum frequencies (f_min, f_max) and selected soil and rock types

The vertical profile of S-wave velocity V_s with depth is the principal set of parameters resolvable from study of surface (Rayleigh) waves. Studies by Bloch et al. (1969) and McEvilly (1964) show that phase velocity is relatively insensitive to changes in compressional velocities V_p or in densities. An example study by Asten (1976) using a stiff sand layer (V_s= 500 m/s) over sedimentary rock (V_s= 2000 m/s) showed the maximum ratio of the partial derivatives of fundamental-mode phase velocity with respect to V_p and V_s for the sand was 0.25. For general applications of Rayleigh-wave methods it is impractical to resolve V_p from inversions, and the preferable approach is to set it at a fixed value for an “average” rock type, or set it via an empirical ratio relative to V_s, unless independent data from body-wave seismic studies are available. Brocher (2005, 2008) used a large database of laboratory and cross-hole measurements of V_p and V_s to derive results shown in Fig. 1. Regression fits give the relations for saturated sediments and rock in the form

$$V_{\text{s}} = \, 0.7858 \, {-} \, 1.2344V_{\text{p}} + \, 0.7949V_{\text{p}}^{2} - 0.1238V_{\text{p}}^{3} + \, 0.0064V_{\text{p}}^{4}$$

(4)

and

$$V_{\text{p}} = 0.9409 \, + \, 2.0947V_{\text{s}} \, - 0.8206V_{\text{s}}^{2} + 0.2683V_{\text{s}}^{3} - \, 0.0251V_{\text{s}}^{4}$$

(5)

where V_s, V_p are shear-wave (S-wave) and compressional-wave (P-wave) velocities in km/s. These equations are valid for 1.5 < V_p < 8 km/s, and 0 < V_s < 4.5 km/s (Brocher, 2005, Eqs. 6 and 9). The average standard deviation for V_s estimated from Eq. (4) versus independent measurements was 0.2 km/s for measurements in USA, Germany, and Japan, so the relation can be used globally with some confidence (Brocher 2008).

Fig. 1

Empirical relation between V_p and V_s measured on laboratory and borehole studies, from Brocher (2005). The long-dash curve (labelled “Brocher regression fit”) corresponds to this paper Eq. (4). For details of other annotations on the plot, see the original reference

Equation (5) thus provides a general relation for setting realistic values of V_p consistent with values of V_s resolved by surface methods over saturated sediments and rock. For unsaturated unconsolidated sediments estimation of V_p is more problematic. Prasad et al. (2005) provide detailed laboratory studies of V_p and V_s measurements on a series of sand-clay mixtures both dry and saturated. For confining pressures up to 20 MPa (corresponding approximately to depths of burial 0–100 m) the V_p/V_s ratio is broadly in the range 1.7–2.5, with higher values associated with higher values of clay/sand ratios. In the context of surface-wave modelling and interpretation without independent measurements of V_p, a V_p/V_s ratio of 2 is a satisfactory assumption for unsaturated sediments. Section 5 illustrates the difference in interpretation which results when SPAC data over a thickness of 30 m of soft sediments are interpreted using different assumptions of a high and low water table.

4 Choice of Array

A very large number of array shapes are reported in case histories, illustrated in Fig. 2, with comments on attributes of the different shapes in Foti et al. (2017). The circular array gives the most effective azimuthal averaging; examples using 5, 6, 7, and 9 stations on the ring appear in, for example, Wathelet et al. (2008), Cho et al. (2013), Roberts and Asten (2008), and Garofalo et al. (2016a).

Fig. 2

Five array geometries which have been used for microtremor studies. a Circular shape. b Nested triangular shape. c L-shape. d Sparse nested triangles (sizes increase ×3). e Common-base nested triangles (sizes increase ×3)

Where a large pool of recording instruments is available, circular or irregular arrays with five to 15 seismometers have been used successfully. Maintaining regular geometry with such large arrays is sometimes logistically challenging, and array processing with SPAC methods requires approximations which reduce high-frequency fidelity. For such arrays, frequency-wavenumber array processing methods are preferred, with Poggi et al. (2017) providing recent examples. Section 9 provides further discussion on such methods.

It is our experience that the sparse nested or common-base triangles are the most efficient geometry where array processing using SPAC methods is desired, since these geometries give sufficient azimuthal averaging in most cases, and logistical effort can be applied to a range of triangle sizes rather than a density of stations. In areas of restricted access (for example at a T-junction in a pair of roads) the common-base triangle or the L-shaped array is useful. In many cases useful SPAC data can be obtained with a linear array, at its most simple being a two-station array, with examples in Chavez-Garcia et al. (2005) and Hayashi et al. (2016).

5 Alternative SPAC Processing Methodologies

The classical approach to inversion of surface-wave data uses a first step where phase velocities are estimated from processing array data with SPAC or frequency–wavenumber methods, and a second step where layered-earth interpretation models are constructed by inversion of phase velocity dispersion data, illustrated by Foti et al. (2011, Fig. 1). Following estimation of SPAC coefficients c(f) from Eq. (2) the inverse solution to extract phase velocity v(f) can be done using estimates at multiple values of frequency for a single value of r (the modified SPAC method, MSPAC, (Bettig et al. 2001). This approach suffers from a limitation in that the transformation from array measurements of phase changes across an array into to phase velocities is highly nonlinear, and hence the error structure of surface-wave velocity estimates fed into an inversion algorithm may be subject to biases. The MSPAC method allowed generalization to allow use of moderately irregular array geometries and is probably the most popular method in use due to an accessible software implementation (Geopsy, 2017).

A variant of the SPAC method fits estimates of v(f) to multiple values of r for a selected set of frequencies (the extended SPAC method, ESAC, described by Ohori et al. 2002; Okada 2003; Foti et al. 2011; Galiana-Merino et al. 2016). This method has an advantage of obtaining v(f) directly from SPAC data, but it has the disadvantage of requiring separate SPAC plots for each frequency of interest, prior to compiling the dispersion curve needed for inversion to a layered-earth model. Foti et al. (2011, Figs. 15, 16) give an example of the method, and also compare it with processing via frequency–wavenumber methods. ESAC was demonstrated to be more robust than frequency–wavenumber methods at low frequencies, yielding useful velocity estimates for wavelengths as long as 10–20 times the largest inter-station distance of the array.

Within the SPAC group of array processing algorithms, a further approach developed by Asten et al. (2004) and Asten (2006a) utilizes direct fitting of observed and model SPAC curves, called the multiple-mode SPAC (MMSPAC) method due to its facility in identifying the presence of multiple modes of Rayleigh-wave propagation, if present. The technique has improved stability in that it employs only one inversion step, whereas ESAC and MSPAC use two. Direct fitting as used here uses an iterative approach and a least-squares criterion and perturbation methods in order to assess resolution of each chosen parameter. A forward model of the Rayleigh-wave dispersion modes for a trial model is computed using routines from Herrmann (2013) after which a model SPAC spectrum is computed using the relation

$$c\left( f \right) = J_{0} \left( {rk} \right) \, = J_{0} [(2\pi f \, r)/v(f)]$$

(6)

where c(f) is spatially averaged coherency at frequency f, J₀ is the zero-order Bessel function, r is the station separation, k is the Rayleigh mode wavenumber, v(f) is the computed phase velocity for the model. Where analysis needs to consider multiple modes of wave propagation, the method may be extended to use an effective phase velocity computed for an effective Rayleigh mode R_e using the theoretical energy partition for vertically directed impact sources as given by Ikeda et al. (2012). The direct fitting approach with a least-squares criterion provides as much information as an automated inversion and in some cases more information including estimates of error bounds on the important parameters (V_s values, or thickness, or total depth; see Sects. 5 and 7 for examples). The accuracy of the direct fitting method is illustrated in Schramm et al. (2012) where MMSPAC data show close overlap with interpretation of active seismic data. As an additional advantage, the direct fitting method provides a superior bandwidth of phase velocity data, demonstrated in Asten (2006b). The direct fitting approach has also been demonstrated on MSPAC data by Wathelet et al. (2005) and is available as an option in the Geopsy (2017) software.

The majority of published work on array processing of passive seismic data considers only the Rayleigh-wave fraction of propagating energy. However, Kohler et al. (2007) demonstrates that 3-component SPAC processing is possible and the resulting combination of Love wave in addition to Rayleigh-wave dispersion data has advantages especially where multiple-mode wave propagation occurs. Fah et al. (2008) and Poggi et al. (2017) describe the use of combined Rayleigh and Love wave dispersion analysis on passive seismic data processed with an alternative frequency-wavenumber array processing method. They likewise find some advantages, two being some reduction in model ambiguity, and some improved resolution of near-surface layers. The InterPACIFIC blind study (Garofalo et al. 2016a) discussed in Sect. 2 above shows that four out of fourteen of the contributing groups made use of Love as well as Rayleigh-wave analysis, but does not provide any measure of improvement in results where the combined analysis was used. It is fair to conclude that the use of three-component array processing on passive seismic data remains a subject for further study in terms of quantifying advantages in interpretation relative to processing complexity.

Figure 3 illustrates the processing stream for the alternative Rayleigh-wave MSPAC, ESAC, and MMSPAC methods, applied to a triangular array.

Fig. 3

Processing streams (at left) for ESAC, MSPAC and (at right) for MMSPAC (direct fitting) methodologies. A triangular array (top right) provides SPAC spectra for two station separations r₁, r₂, which can be inverted or fitted simultaneously

Figure 4 shows an example of MMSPAC direct fitting on data from Mirandola, Italy. The site is described in Garofalo et al. (2016a). The direct fitting method allows use of frequencies 2–30 Hz for this small triangle. By contrast, use of MSPAC at this site is unable to use with confidence frequencies above 7 Hz (see Fig. 29 of Foti et al. 2017) thus losing resolution of upper layers in the geological section. Use of MMSPAC direct fitting doubles the usable bandwidth and largely removes the need for complementary active surface-wave studies to provide shallow resolution. The point is emphasized by Fig. 4 of Garofalo et al. (2016a) which shows the usable frequency ranges as assessed by 12 independent groups who studied these data as part of the InterPACIFIC comparison project. That figure shows that Team 1, using MMSPAC direct fitting achieved the same useful frequency range with passive methods, as did the majority of other teams who used combined active and passive methods. The depth to basement of 116 m shown in Fig. 4d as interpreted using MMSPAC for the blind study described by Garofalo et al. (2016a) compares well with the depth of 113 m to Pliocene bedrock, as established after the study using invasive studies.

Fig. 4

MMSPAC method applied to a triangular array side length 12.7 m, Mirandola, Italy. Black: observed SPAC and HVSR spectra. Red, yellow, green: model curves for Rayleigh fundamental R₀, first-higher and second-higher modes. Blue: model curve for Rayleigh effective mode R_e (overwrites the plotted R₀ mode for most frequencies). σ(R₀) is standard deviation of fit of fundamental Rayleigh-wave model. σ(Re) is standard deviation of fit of effective mode of Rayleigh-wave model. a, b SPAC for r = 7.3 m (radii) and r = 12.7 m (sides). The standard deviation σ of the fit of observed SPAC with model SPAC for the alternative fundamental mode R₀ and Effective mode R_e is annotated. σ is computed for the frequency range 2–30 Hz (black bar at base of plot) c Thick blue line is the R_e model from Fig. 4a with least-squares best fit. Thin blue lines represent a sensitivity study on layer 3, where thickness of the single layer is varied such that σ increases by 10% d HVSR at array centre. e V_s profile from fitted SPAC (including larger triangles not shown here) and from HVSR fit. Horizontal red lines at depth 15 m represent error bounds on the base of layer 3 for perturbation models shown in Fig. 4c. Using definitions from Table 1, minimum resolution at surface is given by f_max= 30 Hz, D_min = 1.2 m. Maximum depth penetration (using data from a triangle with station spacings 201 m (not shown here) is given by f_min= 1.0 Hz, D_max= 350 m. The basement at depth at 116 m controls the HVSR peak at 0.7 Hz

Figure 4 also shows the presence of higher-mode energy in the band 10-13 Hz. The blue model curve represents the Rayleigh effective mode R_e, constructed as a superposition of modes where energy partition between modes is computed on the assumption that the wave field is generated by ideal surface point sources (Ikeda et al. 2012). This band of higher-mode energy is associated with the interpreted shallow, strong velocity contrast at depth 14 m in Fig. 4e.

The misfit of model curves relative to the observed MMSPAC spectrum provides a quantitative measure of sensitivity of the model curve to choice of layered-earth model parameters, and hence to the error bounds applicable on interpreted earth parameters. We use the term “error bounds” in a subjective sense rather than as a statistical term, because the misfit of model and observed curves depends on subjective choices of what bandwidth to use, how many different MMSPAC curves to use and the weighting to apply to that selection. Further subjective choices have to be made when combining Love wave data with Rayleigh-wave data and/or HVSR data in the fitting process Errors in the MMSPAC fit illustrated here are subject to observable systematic biases as well as random fluctuations so it is not possible to provide an objective definition based on the model and observed misfit, for error bounds on a specific model parameter. However, as a practical observation a change in the standard deviation of the misfit by 10 or 20% is a useful indicator of an “unacceptable” alternative model. Similar subjective decisions have to be made when accepting or rejecting alternative models based on inversion of dispersion curves in other SPAC methods. Figure 4c, e illustrates how this perturbation method can be used to test sensitivity of a model fit to layer parameters; however, issues of non-uniqueness in models inherent in surface-wave studies mean such error bounds are indicative but rarely definitive.

All SPAC methods, and in particular the MMSPAC method used here, benefit from simultaneous comparison of observed and model spectra of the horizontal/vertical particle motion ratio (HVSR) as determined from at least one 3-component recorder in the seismic array. Routinely the centre station of the array is used for such studies, since interpretation proceeds on an assumption of lateral homogeneity in geology below the array. However, review of HVSR data for all stations in an array is beneficial for the purpose of identifying variations where individual seismometers are subject to local cultural or wind noise, or where lateral variations in geology exist below the array. Claprood et al. (2012) demonstrate a methodology for interpreting HVSR variations in the latter case.

In Fig. 4d the peak in HVSR near 0.7 Hz is associated with the velocity contrast at basement depth 116 m. In this instance the HVSR data are interpreted only by qualitative fitting of observed and model spectral shapes, but Arai and Tokimatsu (2004, 2005), Ikeda et al. (2013) and Hobiger et al. (2013) provide examples where formal inversion fitting is employed. A separate paper Molnar et al. (2018) reviews HVSR methods in passive seismic studies. Where identifiable low-frequency peaks in HVSR exist, it is a general rule that the frequency of the peak provides information complementary to SPAC methods, and allows improved characterization of deep interfaces in some situations; two examples in Asten et al. (2014) illustrate the point.

An HVSR peak at low frequencies is especially useful due to a common practical limitation with temporary arrays, whereby inter-station coherencies at low frequencies (e.g., below 2 Hz) are reduced by the presence of wind or cultural noise with consequent degradation of array processing methods, but HVSR spectra retain sufficient shape to provide useful information on propagating Rayleigh waves. Macau et al. (2015) also demonstrate how secondary peaks in HVSR curves at high frequencies may be diagnostic of significant velocity contrasts within unconsolidated overburden (clay overlying gravels). Thus, combined use of SPAC and HVSR in routine surveys is highly desirable.

A fourth variant of the SPAC method is krSPAC, where direct fitting of SPAC spectra is carried out in the wavenumber domain instead of frequency (Asten et al. 2015). It has strong advantages in retaining high-frequency data (and hence depth resolution) when array geometry is grossly irregular.

The Mirandola data and model are also useful for demonstrating the importance of knowing the depth of water table at soft sites in order to get unbiased estimates of the near-surface V_s profile. Figure 5 shows SPAC observations and models for two alternative interpretations, of the upper 50 m of soft sediments; one where the water table is high (a nominal 2 m depth), and the other where a hypothetical deep water table at 30 m depth is used, as might be considered in an alternative desert environment. Figure 5a shows the alternative model curves fitting the observed SPAC data virtually identically, that is the Rayleigh-wave energy is not capable of differentiating between the two models. Figure 5b shows the alternative V_s profiles; for the deep water table the values of V_s yielded by the best fit model are 5% higher than those fitted under the for the high water table assumption. While the 5% difference in V_s is not large, it is probably significant since the direct fitting approach to MMSPAC interpretation is generally capable of yielding confidence limits of order 5% in the V_s of near-surface layers.

Fig. 5

MMSPAC method applied to a triangular array radius 7.3 m, Mirandola, Italy. Colours and symbols as for Fig. 4. a Thick blue line is for mode R_e for the preferred model of a high water table (WT) (depth 2 m). Thin blue line is for an alternative hypothetical model using a deep water table (depth 30 m); the thin lines are visible around 20 Hz but otherwise overlap the thick lines of the preferred model. b Thick, thin lines are the V_s profiles interpreted assuming the high, low water tables, respectively. The V_s profile for the alternative models differ by 5% in the upper 30 m

6 Integration of Passive and Active Surface-Wave Methods

A surface-wave method using active sources has become increasingly popular during the last several decades, for estimation of S-wave velocity to a depth of several tens of metres. Spectral analysis of surface waves (SASW) has been used for the determination of one-dimensional (1D) V_s profiles down to a depth of 100 m (Nazarian et al. 1983). SASW surveys employ a shaker or a vibrator as sources and calculate phase differences between two receivers via cross-correlation. Park et al. (1999) proposed a multichannel analysis of surface waves (MASW) method, which determines phase velocities directly from multichannel surface-wave data after transforming waveform data from the time-distance domain into the phase velocity–frequency domain. MASW enables us to perform surface-wave analysis using relatively inexpensive equipment, such as a sledge hammer, geophones, and an engineering seismograph, so that the method has been widely used for many engineering site investigations globally. A clear limitation of the active surface-wave methods is their penetration depth. Active sources, sledge hammers, weight drops, and shakers generally penetrate to a depth of 15–30 m depending on the site and source energy, and the depth of penetration is often not enough for investigation purposes. However, Stephenson (pers. comm. 2016) reports using a 220 kg weight drop to achieve 100 m depth penetration in the San Francisco Bay area. Integrating passive and active surface-wave methods is becoming increasingly popular. The passive method is mainly used to supplement the penetration depth or low-frequency phase velocities of the active method. SPAC is an ideal supplement to the active surface-wave method since the same equipment can be used for data acquisition, and the use of irregular shaped arrays, such as L-shaped or linear, works in many cases.

Figure 6 shows a typical example of integration of passive and active surface-wave methods. We can see that the maximum wave length obtained from the active method (MASW) was approximately 30 m, whereas one obtained from the passive method (SPAC, linear array) was approximately 150 m, although the same geometry and equipment were used for both methods. It clearly shows the advantage of passive methods over active methods in terms of the penetration depth achieved.

Fig. 6

Comparison of dispersion curves obtained from active (MASW) and passive (SPAC with linear and circular arrays) surface-wave methods at Mirandola, Italy

The zone of overlap between active and passive data is usually several Hz as shown in Fig. 6. A recent study by Hayashi et al. (2016) shows that SPAC provides phase velocities up to 20–40 Hz with wavelength of 2–3 m at most sites. The active methods require much more effort in field work compared with the passive method. If the passive methods provide phase velocities up to several tens of Hz, the active method might not need to be performed and the effort of field work may be dramatically reduced. It is obvious that, in general, performing both active and passive methods is the best for estimating the average V_s to a depth of 30 m (V_s30); however, these recent studies imply that performing SPAC may be better than performing MASW if we are restricted to performing only one of the methods.

MASW usually uses vertical impacts as sources of seismic energy, vertical component geophones as receivers, and analyses dispersion curves in terms of Rayleigh modes. Several researchers use horizontal impacts and receivers and analyse dispersion curves in terms of Love wave modes. Xia et al. (2012) investigated active surface-wave method using Love waves and found that MASW using Love waves has three advantages compared with that using Rayleigh waves; mode uncertainty and hence model uncertainty is reduced, signal-to-noise ratio is increased, and inversion stability is increased. These benefits are attributed in part to the fact that Love waves are independent of compressional-wave velocities; hence, there are fewer variables affecting the interpretation. In summary, and noting the discussion in the previous section on three-component methods on passive data, we conclude that each method has advantages and disadvantages and the selection of a method will depend on geological conditions and the purpose of the investigation.

To evaluate the accuracy and reliability of active methods, many researchers have compared V_s profiles obtained from MASW with that obtained from invasive methods such as a down-hole seismic method (e.g., Di Fiore et al. 2016). There is fairly general agreement that the average difference between MASW and invasive methods is 15–25%. As Di Fiore et al. (2016) observe, differences between invasive and non-invasive methods can be summarized as follows. Firstly, the invasive and non-invasive methods measure the subsurface on different scales and with different resolution and hence do not need to agree with each other; invasive methods measure local V_s down a borehole, whereas the non-invasive methods measure and average over a much larger volume of the earth. Secondly, despite the difference between the invasive and non-invasive methods, site responses calculated from both methods are generally consistent.

7 The Challenges of Low-Velocity Layers (LVL) and Multiple Modes

Surface-wave phase velocity curves have in general only a weak sensitivity to low-velocity layers (LVL). However, where low-noise SPAC data are available over a bandwidth of about two octaves or more, and the LVL has a thickness comparable with or greater than the depth of burial, examples of successful detection of an LVL exist. Roberts and Asten (2004) provide an example where MMSPAC resolves a 25-m layer of firm sediments (V_s= 700 m/s) below a 10 m thickness of basalt (V_s= 1800 m/s). Foti et al. (2011) give an example of a 13 m layer of soft sediments (V_s = 145 m/s) resolved below a harder layer only 3 m in thickness (V_s= 175 m/s). A challenging example was provided in the InterPACIFIC blind trials (Garofalo et al. 2016a, b) where the Grenoble site contained a 10-m-thick layer of soft clay (V_s= 260 m) under 25 m of sands and gravels. Only four of 14 groups analysing the surface passive seismic data were able to detect the LVL.

A series of very detailed passive and active surface-wave surveys are available from the Christchurch area, New Zealand (Teague et al. 2015). Four points in particular can be deduced from Fig. 7 (supported by other plots not shown).

Fig. 7

MMSPAC method applied to two 6-station circular arrays, Hagley Park, New Zealand. Colours and symbols as for Fig. 4. Array geometry is shown in Fig. 8d. a SPAC for r = 29.6 m. Thick red, blue lines: best fit model including LVL at 2–4.5 m depth, for R₀, R_e. Thin blue: model without LVL; the fit is degraded in the frequency band 7–10 Hz. The change in σ with/without LVL exceeds 10% and is considered significant. b SPAC for r = 195 m. Thick, thin, and dashed blue lines: best fit model with basement at 472 m (preferred model), 507 m, and 447 m, respectively, using the frequency band 0.5–1.2 Hz. c V_s profiles for the two models of (a), with and without a shallow LVL. f_max= 14 Hz, D_min= 2.2 m. d V_s profiles from all fitted SPAC data. Shows basement (layer 9) error bounds obtained from plot b and also layer 9 error bounds in V_s estimated similarly. f_min= 0.5 Hz, D_max= 1400 m. e HVSR at the array centre

• a. The high-frequency limit with MMSPAC at this site is 14 Hz, giving D_min= 2.2 m. This compares with a lower high-frequency limit obtained using MSPAC and inversion of dispersion curves, of 3 Hz (Teague et al. 2015). That study was also able to extract useful frequencies up to 10 Hz from the passive data, by an alternative method of frequency-wavenumber filtering. Complementary active surface-wave data in Teague et al. (2015) obtained useful frequencies up to 25 Hz.

• b. The variable frequency separation of peaks in the MMSPAC direct fitting allows resolution of a probable LVL at 2 m depth.

• c. Frequencies 0.5–1.2 Hz allow resolution of a basement interface at 472 m. Data at 0.5 Hz allow an estimate of D_max to be 1400 m.

• d. A band of frequencies 1–1.8 Hz (Fig. 7b) shows that the observed SPAC follows neither the R₀ mode nor the effective mode R_e. Furthermore, as shown in Fig. 8 the positioning of the observed SPAC relative to R₀ and R_e is not constant; thus, we conclude that while multiple modes clearly exist at this site, their composition cannot be estimated by the simple theoretical power partition ratio based on an assumption of vertically acting point sources.

  Fig. 8

  

  a MMSPAC for the 200–m-diameter array of Fig. 8d (blue squares), using six radial pairs to stations 4–9. Data length 50 min. The fitted model is same as Fig. 7c, d. Colours and symbols are as for Fig. 4. For frequencies above 2.5 Hz the model curve R_e (blue) overlies exactly the fundamental mode R₀ (red). b MMSPAC using a simple triangle array, with three radial pairs from centre to stations 4, 6, and 9. c MMSPAC using one pair only, stations centre and 4. Same data length and smoothing as a and b. d Map of layout of three circular arrays at Hagley Park (from Teague et al. 2015)

This last point underlines the challenge posed by higher modes, while their presence can usually be recognized in SPAC methods they cannot always be quantitatively modelled with confidence. A higher mode incorrectly interpreted as a fundamental mode will yield estimates of V_s biased to high values, which is undesirable when correct estimation of V_s for earthquake hazard studies is demanded. The problem is lessened when low-frequency (long wavelength) data are available, which increases the importance of survey design to gain such data.

The first point adds to the finding of Sects. 5 and 6, where we note that, when passive seismic processing is optimized to retain high-frequency data, the need for complementary active seismic surveys is reduced. A similar overlap of passive and active seismic data was observed in Schramm et al. (2012).

A comparison of layered-earth model interpretations for the Hagley Park site as shown in Fig. 7 and that shown in Teague et al. (2015) shows that while the model in this paper used a minimum number of eight layers, with high-contrast boundaries, the latter paper used a large number of 25 layers with gradational boundaries and monotonically increasing V_s. When the latter model is used to fit SPAC data in Figs. 7 or 8, the quality of the fit is insignificantly different. The two very different models demonstrate model equivalence in surface-wave dispersion studies. The comparison of interpretation methods in Teague et al. (2015) also shows an alternative model with multiple LVLs which in terms of fitting SPAC, frequency-wavenumber and active-source data is not distinguishable from the monotonically increasing V_s model. Thus, the use of error bounds obtained from perturbation studies as shown in Figs. 4, 5, and 7 is useful if the chosen mathematical model is close to the real geological model (e.g., if the hypothesis of a hard basement at a depth of hundreds of metres is geologically correct), but those error bounds will not be helpful if for example the correct geological hypothesis is a continuous gradational increase in V_s with depth, or a cyclic sequence of high and low V_s layers.

These last two observations illustrate the well-known limitation of surface-wave methods, that they suffer from a general lack of resolution of LVLs, and emphasize the importance of including all available geological constraints in modelling. Thus, any data on known high and low-velocity strata, and basement type, should be included as constraints in interpretations in order to reduce the likelihood of gaining mathematically acceptable but geologically untenable interpretations.

8 Practical Choice of Array

The high quality and high redundancy of the Hagley Park data allows an assessment of how simpler arrays may perform at the site. Figure 8 shows MMSPAC data fitted with the same model as was Fig. 7, firstly for a semicircular array of six stations, secondly for a simple triangular array, and thirdly for a single pair of stations. It is arguable that the triangular array presents no loss of information relative to the circular array, while the single pair (or two-station array) retains the essential shape of the SPAC spectrum although obviously degraded.

We conclude that triangular arrays present sufficient spatial averaging on such a site, and thus where layout time is important, sparse nested triangles are likely to be a cost-effective technical solution than the use of dense circular arrays. The fact that the single pair provides useful data with a data length of 50 min is encouraging. The additional effort of using two rather than one only station for HVSR surveys is not large, and the return on effort is likely to be high in terms of providing V_s information to supplement the more qualitative HVSR measurement. We note that Garofalo et al. (2016a) advocate “extreme caution” when using passive methods on linear arrays, but the results here and in Chavez-Garcia et al. (2005) and Hayashi et al. (2016) suggest that where logistical efficiency is a factor, such methods should be considered. Objective proof of our conclusions regarding the efficacy of sparse arrays and two-station arrays will require further blind tests on new sites, where minimal array data are supplied in a staged process to all participating groups.

9 Frequency-Wavenumber (FK) Processing Methods—Strengths and Weaknesses Relative to SPAC

As outlined in Sect. 1, the SPAC method relies for its accuracy on spatial (azimuthal) averaging of coherent plane waves propagating at a single scalar velocity at each frequency. Beamforming frequency-wavenumber (FK) array processing methods have four essential differences from SPAC; they perform at their optimum for unidirectional wave propagation, they remain effective when the array stations are irregularly spaced, they resolve (subject to limitations of the array response function) multiple velocities (or higher modes) of wave propagation, and they can provide robust estimates of wave velocity when incoherent noise is present in additional to the propagating wave signal. Foti et al. (2017) give an overview of these differences. Realization of these advantages does, however, come at the cost of the requirement for a larger number of stations in an array in order to have an array response function with sufficient resolution to gain useful velocity dispersion curves. Unlike SPAC methods, a sparse triangular array processed with FK methods is unlikely to be effective. Poggi et al. (2017) provide a detailed study of ten “difficult” hard-rock sites in Switzerland where topography places limitations on array design, and the use of FK methods on arrays of 10–14 seismometers was found to be the most effective approach for passive seismic characterization of site response. For less challenging sites the use of seven stations is usually sufficient for application of FK processing methods.

Figure 6 shows Rayleigh-wave phase velocities for the Mirandola site acquired with eight-station circular arrays (centre plus seven on the circumference) of radii 135 and 405 m, and processed by both SPAC and FK methods. Figure 9 shows a subset of the FK wavenumber plots produced using data from the 135 m array.

Fig. 9

Subset of beamforming wavenumber transform plots produced from an eight-station circular array, 135 m radius, at Mirandola, Italy (same site as Fig. 5, but larger array). Each cell of the plot is wave energy contoured on a k_x − k_y grid, where k_x, k_y are wavenumber in the east and north directions, respectively. The black circle on each plot has a radius equal to the scalar wavenumber k of the contoured wave energy maximum. Numbers at top left of each cell show (top) the frequency f (Hz) and (below) the deduced scalar wave velocity V of the contour maximum, obtained from the relation V = 2πf/k. The wavenumber transforms were computed using the high-resolution frequency–wavenumber method (Capon 1969)

The divergence in Fig. 6 of the observed dispersion curves at low frequencies for the SPAC and FK methods is obvious and provides a cautionary qualification on their respective limitations. A similar divergence was noted in a series of blind tests of interpretation of passive seismic array data for the Coyote Creek, California site (Asten and Boore 2005a). In particular Hartzell et al. (2005) compared dispersion curves, noted a similar divergence, and posed the question as to which method was more reliable. At that site borehole sonic velocity logs (both V_s and V_p) are available, and a comparison with layered-earth inversion models for the two array methods with the borehole V_s profile showed the FK data provided systematic bias to high phase velocities and high V_s values relative to the SPAC-derived values and the reference borehole V_s value. Asten and Boore (2005b) explained the sources of bias as follows.

The finite resolution of beamforming methods means that, where propagating waves are distributed in azimuth, the averaging effect of the array response function maximum results in 2D smoothing in wavenumber space and a consequent bias of the energy maximum towards smaller wavenumbers (equivalently, larger velocities). The effect is seen in Fig. 9 where frequencies of order 0.8 Hz produce unrealistically high phase velocity estimates in the range 1500–3500 m/s. The example also shows that frequencies of order 0.9, 1.0, and 1.15 Hz yield velocities greater than the estimates from SPAC but which would be credible in the absence of other information.

An important limitation of SPAC methods is that any incoherent noise (non-propagating signal) in array data reduces inter-station coherencies and will reduce SPAC coherencies. This results in phase velocity estimates biased downwards, and the effect of that bias is proportionately greater at the low-frequency end of the SPAC spectrum where SPAC coherencies lie in the range 0.9–1.0 (or equivalently, kr < 0.65); at such low frequencies any reduction in coherency associated with incoherent noise has a proportionately greater downward bias on estimation of the dispersion curve velocities. Examples of this effect can be seen as a low-velocity “tail” in SPAC-derived dispersion velocities for the frequency band 0.8–1.0 Hz for circular array data, and 1.3–2.0 Hz for linear array data, in this paper Fig. 6. A similar effect is seen in the Coyote Creek example by Hartzell et al. (2005, Fig. 3). Such problems of bias in SPAC data can be reduced by careful quality control on data from each station of the array, with firstly exclusion of stations suffering from anomalous incoherent noise, and secondly by using larger arrays. It should be noted that as a general rule FK tends to overestimate phase velocity, whereas SPAC tends to underestimate it. From the point of view of engineering site studies, overestimating phase velocity might be potentially hazardous, and hence, we need to keep in mind different biases inherent in FK and SPAC methods.

Table 2 summarizes the relative strengths and weaknesses of the two array processing methods.

Table 2 Relative strengths and weaknesses of SPAC and FK array processing methods applied to passive seismic data

10 Similarity and Difference Between Spatial Auto-Correlation and Seismic Interferometry

Recently, researchers have been studying ambient noise or microtremors in terms of seismic interferometry (Wapenaar, 2004). When we process surface waves, the seismic interferometry (SI) is the essentially same as spatial auto-correlation (SPAC). Tsai and Moschetti (2010) demonstrated that SI in the time domain is equivalent to SPAC in the frequency domain. We will briefly summarize the similarity and differences between SI and SPAC.

The SI is generally defined as the cross-correlation of microtremors obtained at two stations and is equivalent to the Green’s function between two stations. Unlike the SPAC method which requires isotropic arrays to allow processing which performs a spatial average around the azimuth θ, the SI appears not to require such a spatial average. However, the definition of SI essentially includes the spatial average for the following reasons. The SPAC defined in Eq. (2) can be only applied to isotropic arrays, such as circles or triangles, and cannot be applied to anisotropic arrays, such as linear arrays or L-shapes, in order to calculate the spatial average expressed as Eq. (2) when microtremors propagate only in a particular direction (Fig. 10a). However, if we assume that microtremors do not come from some specific direction and come from all directions equally, the spatial average in Eq. (2) can be calculated even if arrays are anisotropic (Fig. 10b). Generally, the direction of propagation of microtremors is not stable, and averaging over a sufficiently long time enables us to calculate the spatial average of Eq. (2) correctly. The SI uses the assumption of Fig. 10b and is equivalent to the SPAC in terms of the spatial average.

Fig. 10

Spatial average of microtremor data for anisotropic wave propagation with circular array (a) and isotropic wave propagation with linear array (b)

Figure 11 shows an example of SPAC in the frequency domain (Fig. 11a) and time domain (Fig. 11c) calculated from microtremors. The microtremor data were obtained from a linear array of 24 sensors with 1 m spacing. The coherency was calculated for each pair of sensors using Eq. (1), and the coherencies that have the same separation were averaged (Fig. 11a). Figure 11b shows the error between the coherencies and Bessel functions. Phase velocities that provide minimum error were defined as a dispersion curve. Frequency domain coherencies were transformed to time domain by inverse Fourier transform (Fig. 11c). The time domain data are what we call the SI and looks like a shot gather. Phase shift and stack (Park et al. 1999) transforms it to a phase velocity versus frequency domain (Fig. 11d), and phase velocities that provide maximum stacked amplitude were defined as a dispersion curve. Figure 11a–d corresponds to SPAC and SI, respectively, and we can see that the dispersion curves obtained by two methods are almost identical.

Fig. 11

a Example of SPAC in the frequency domain calculated from microtremor measurement. b Phase velocity image in the frequency domain calculated from a. c SPAC in the time domain calculated from a by inverse Fourier transform. d Phase velocity image in the frequency domain calculated from c

The main difference between SI and SPAC is in data acquisition rather than mathematical expression. Conventional passive surface-wave methods, SPAC, FK, and ReMi (Louie 2001), basically require spatially un-aliased data to calculate phase velocity, and sensors must be deployed with relatively small spacing. Unlike conventional methods, SI is generally applied to spatially aliased data and used to calculate group velocity. Researchers obtain and process data from relatively large spacing arrays. The group velocity, however, is not straightforward compared with phase velocity and generally requires a much longer record length. As a result, SI is still in the research and development phase and is not used in routine work for engineering purposes.

It should be noted that both SPAC and SI applied to surface waves essentially include spatial average and they are expressed as a Bessel function rather than trigonometric function. Phase velocity calculation has to take into account the spatial average. Data processing through Fig. 11a, b or SPAC takes into account the spatial average and is very superior to SI from a mathematical point of view although phase velocity images obtained from the SPAC (Fig. 11b) and the SI (Fig. 11d) look almost identical.

11 Recommendations

11.1 Model Equivalence

This will remain a challenge for all surface-wave array processing methodologies, especially where low-velocity layers exist. Inclusion of HVSR methods reduces uncertainty where large near-surface velocity contrasts occur in soft sediments, and where depth of a deep basin is weakly resolved. Use of Love-wave analysis may reduce some of these uncertainties. Guidance from geological logs on level of water table and types of strata and of basement is essential for accurate interpretation of a V_s depth profile, especially if LVLs are suspected or frequency bands exist where multiple modes dominate.

11.2 Array Geometry

We argue that passive-source microtremor or ambient noise methods should be the first method of choice for V_s30 and deeper studies. Joint use of passive and active-source methods is recommended for gaining maximum resolution near surface, but use of SPAC or MMSPAC direct fitting methods reduces the high-frequency advantage of active methods to the degree where the additional cost of active methods may not be justifiable. The key proposition arising from this study is that sparse nested triangular arrays are the most efficient use of instrumentation, except where strongly directional noise sources are expected. Supplementary use of at least one 3-component receiver usually provides additional depth information where strong velocity contrasts exist at basement. Where HVSR surveys are used as single stations for reconnaissance purposes, the use of a synchronized second station allows interpretation by 2-station SPAC, with significant advantages for quantitative estimation of V_s values.

11.3 Array Processing Algorithms

The use of SPAC interpretation by direct fitting of observed and effective-mode model SPAC spectra generally allows a wider frequency range (for both high and low frequencies) in interpretation than do the more conventional methods requiring estimation of dispersion curves prior to inversion to a layered-earth model. This facilitates detection of near-surface low-velocity layers if present. Multiple modes are a frequent occurrence, but generally restricted to frequency bands of an octave width or less. The use of effective-mode modelling with its theoretical assumption of perfect vertically directed sources is not a perfect solution since power partition between modes is source dependent and may also be time dependent. However, it is recommended to take steps to recognize multiple modes during processing and assess the likely validity of effective-mode modelling on a case-by-case basis. Three-component processing of array data to utilize Love wave as well as Rayleigh-wave energy provides some reduction in model uncertainty. Where logistical and/or cost considerations allow the simultaneous use of a large number of stations in an array (e.g., a minimum of seven, or preferably 10–14 seismometers), the use of FK methods complements the use of SPAC methods by their ability to quantify source direction, azimuthal spread and power partition of higher modes.

11.4 Opportunities in Future Studies

Blind comparative studies on interpretation methodologies using data from limited arrays such as sparse triangles, L-shaped arrays, and two-station arrays should be a goal during ongoing characterization of passive seismic methods. Use of a two-stage process where limited array data are provided first, followed by data from more comprehensive arrays, will facilitate objective assessment of whether sparse arrays can be demonstrated to be sufficient for general application. Inclusion of three-component processing for Love as well as Rayleigh modes should be included, with the goal of quantifying the reduction in model uncertainty achievable with such processing.