# A Procedure for Developing Uncertainty-Consistent Vs Profiles from Inversion
of Surface Wave Dispersion Data

Joseph P. Vantassel  The University of Texas at Austin  Brady R. Cox  The
University of Texas at Austin

###### Abstract

Non-invasive surface wave methods have become a popular alternative to
traditional invasive forms of site-characterization for inferring a site’s
subsurface shear wave velocity (Vs) structure. The advantage of surface wave
methods over traditional forms of site characterization is that measurements
made solely at the ground surface can be used routinely and economically to
infer the subsurface structure of a site to depths of engineering interest
(20–50 m), and much greater depths (>1 km) in some special cases. However, the
quantification and propagation of uncertainties from surface wave measurements
into the Vs profiles used in subsequent engineering analyses remains
challenging. While this has been the focus of much work in recent years, and
while considerable progress has been made, no approach for doing so has been
widely accepted, leading analysts to either address the propagation of
uncertainties in their own specialized manner or, worse, to ignore these
uncertainties entirely. In response, this paper presents an easy-to-implement,
effective, and verifiable method for developing uncertainty-consistent Vs
profiles from inversion of surface wave dispersion data. We begin by examining
four approaches presented in the literature for developing suites of Vs
profiles meant to account for uncertainty present in the measured dispersion
data. These methods are shown to be deficient in three specific ways. First,
all approaches are shown to be highly sensitive to their many user-defined
inversion input parameters, making it difficult/impossible for them to be
performed repeatedly by different analysts. Second, the suites of inverted Vs
profiles, when viewed in terms of their implied theoretical dispersion data,
are shown to significantly underestimate the uncertainty present in the
experimental dispersion data, though some may appear satisfactory when viewed
purely qualitatively. Third, if the uncertainties in the implied theoretical
dispersion data were to be examined quantitatively, which has not been done
previously, there is no obvious remedy available to the analyst to resolve any
inconsistency between the measured and inverted dispersion uncertainty.
Therefore, a new approach is proposed that seeks to remedy these shortcomings.
First, beyond appropriate considerations that must be given to all inversions,
the method is governed by only one user-defined input parameter, to which it
is not overly sensitive. Second, it is shown to produce suites of Vs profiles
whose theoretical dispersion data quantitatively reproduce the uncertainties
in the experimental dispersion data. Third, the final step of the procedure
requires the analyst to compare the measured and inverted dispersion
uncertainties quantitatively, and should the analyst find the results to be
lacking, clear guidance is provided on the actions necessary to produce Vs
profiles whose theoretical dispersion data better account for the experimental
uncertainty. Using two synthetic tests and a real-world example, the procedure
is shown to produce suites of Vs profiles that accurately capture the site’s
Vs structure, while rigorously propagating the dispersion data’s uncertainty
through the inversion process.

##  1 Introduction

Surface wave methods have become a popular alternative to traditional invasive
forms of site characterization for measuring a site’s subsurface shear wave
velocity (Vs) structure. Vs, which is directly related to the site’s small-
strain shear stiffness ($G_{max}=\rho Vs^{2}$, where $\rho$ refers to mass
density), is a critical parameter in many seismic hazard analyses, such as
ground response analyses (Foti et al., 2009; Rathje et al., 2010; Teague and
Cox, 2016; Passeri et al., 2019) and liquefaction-triggering (Andrus and
Stokoe, 2000; Kayen et al., 2013; Wood et al., 2017). The performance of
surface wave testing is traditionally broken into three stages: acquisition,
processing, and inversion (Foti et al., 2015). The acquisition stage involves
non-invasively (i.e., from the ground surface) measuring surface waves as they
propagate through a site. This can be accomplished using active-source
methods, where surface waves are generated by the experimenters, and/or
passive-wavefield methods, where sensors are left undisturbed to record
ambient surface waves. The processing stage transforms these time-domain
recordings to measurements of the site’s dispersive properties, which in this
context describes how the site’s surface wave phase velocity changes as a
function of frequency (or equivalently wavelength). Importantly, this
measurement of the site’s dispersive properties should include site-specific
estimates of frequency-dependent uncertainty, discussed in detail later. For
clarity of expression, we will refer to the measurement of the site’s
dispersive properties from the processing stage as the experimental dispersion
data. Importantly, when surface wave processing is performed by an experienced
analyst, the experimental dispersion data is robustly determined, with
uncertainties that typically range from 5% – 10% coefficient of variation
(COV) between analysts (Cox et al., 2014; Garofalo et al., 2016). This
consistency led Griffiths et al. (2016a; 2016b) and Teague et al. (2018a) to
aptly refer to the experimental dispersion data as part of a site’s
“signature”. The final stage of surface wave testing is inversion. In this
stage, numerical search algorithms are used to identify one-dimensional (1D)
layered earth models whose theoretical dispersion curves, determined through
an analytical forward problem, best fit the experimental dispersion data. It
is important for the reader to understand that, while throughout this paper we
focus primarily on Vs, as it has the greatest impact on the theoretical
dispersion curve (Wathelet, 2005) and is of primary importance to subsequent
engineering analyses, the computation of a theoretical dispersion curve (and
therefore surface wave inversion) requires the definition of an entire 1D
ground model. These ground models are composed of a stack of layers described
by their thickness (H), Vs, compression-wave velocity (Vp), and $\rho$. The
models whose theoretical dispersion curves best fit the experimental
dispersion data, as determined by a misfit function, are considered to be the
most likely representations of the site’s subsurface conditions.

As the primary focus of this paper is on the consistent propagation of
experimental dispersion data uncertainty into the Vs profiles resulting from
inversion, it is important to first discuss the types and potential sources of
these uncertainties. Uncertainties can be broadly grouped into two categories
commonly used in probabilistic seismic hazard analysis (PSHA): epistemic and
aleatory. Epistemic uncertainty describes those unknowns which stem from a
lack of knowledge. In surface wave testing, sources of epistemic uncertainty
include, among others, the selected dispersion processing wavefield
transformation and the number of 1D subsurface layers used during inversion.
Aleatory uncertainty, or sometimes referred to as aleatory variability, is the
result of inherit randomness within the quantity being measured. When
discussing site characterization in particular, aleatory uncertainty is
classically used to define how the material properties in the subsurface
change in three-dimensional space (i.e., randomness with space, or spatial
variability). Of course, this could also be considered epistemic uncertainty
because it stems from a lack of knowledge rather than from any inherit
randomness within the subsurface (i.e., randomness with time). Being able to
consider subsurface variability as either epistemic or aleatory in nature
hints that, for most real-world applications, the two uncertainty “categories”
are not as distinct as their names may indicate. Regardless, it is to be
expected that all experimental dispersion data contains some amount of
uncertainty and that this uncertainty is in part epistemic (dependent largely
on the quantity and quality of the data acquired) and part aleatory (dependent
on the complexity of the subsurface and the spatial extents of the surface
wave arrays). And, while it is difficult-to-impossible to separate epistemic
and aleatory uncertainties in the experimental dispersion data, it is of
paramount concern that the surface wave analyst attempt to quantify the
combined uncertainty on a site-by-site basis (discussed briefly next) and then
propagate that uncertainty through the inversion process in order to develop
uncertainty-consistent Vs profiles.

As this study begins under the assumption that site-specific estimates of
dispersion uncertainty have been established, how this might be accomplished
deserves a brief discussion. However, before doing so, it is important to
acknowledge that while many previous researchers have developed procedures for
accounting for uncertainty in experimental data (Lai et al., 2005; Foti et
al., 2009; Cox and Wood, 2011; Teague et al., 2018a) no single procedure has
been accepted widely into practice, and therefore the example presented here
is but one of many potential alternatives that may be used. To facilitate a
more practical discussion, the example presented here is shown in Figure 1.
Figure 1a illustrates a measurement of a site’s dispersive properties at some
location termed “Location A”. This measurement is directly linked to a given
location (i.e., Location A), experimental setup (e.g., array configuration),
wavefield recording (e.g., source offset or noise time window), and wavefield
processing method (e.g., frequency-wavenumber transformation). The light-
colored portions of Figure 1a show high surface wave power and the dark-
colored portions low surface wave power. The maximum power at each frequency
is selected as the representative experimental dispersion data for that
location, setup, wavefield recording, and processing method. To then develop
meaningful statistics that incorporate the uncertainties previously discussed,
the procedure shown in Figure 1a should be modified and/or repeated in a
systematic manner to encompass reasonable combinations of these contributing
sources. As shown schematically in Figure 1b, these combinations may include
performing the test at various locations across a site (i.e., Location B),
using various source locations for active-source experiments (i.e., Offset X),
various time-windows for passive-wavefield experiments (i.e., Window Y), and
multiple wavefield processing methods (i.e., Method Z). Finally, once these
sources of uncertainty have been accounted for in the form of experimental
dispersion data (i.e., Figure 1b) they may be summarized into a statistical
representation at each frequency (or wavelength), as shown in Figure 1c. A
previous study by Lai et al. (2005) showed that the uncertainty in
experimental dispersion data for active-source experiments based on multiple
source impacts was normally distributed, therefore, it is common to represent
the Rayleigh wave velocity (Vr) at each frequency with a mean and standard
deviation. However, it is important to note that at some sites with
significant lateral variability, such as the Garner Valley site examined by
Teague et al. (2018b), it may be more appropriate to develop alternative
dispersion data sets (one per location) with their own experimental
uncertainty and invert them separately rather than trying to represent all of
the dispersion data with a single statistical distribution. As a more in depth
discussion of developing experimental dispersion data is beyond the scope of
this paper, the reader is encouraged to carefully consider the impact of how
defining their experimental dispersion uncertainty will affect the resulting
distribution of Vs profiles and whether those profiles will correctly
communicate the measured uncertainty.

Figure 1: Schematic illustrating one possible procedure for developing
experimental dispersion data with measures of uncertainty in terms of the
site’s Rayleigh wave velocity (Vr). This involves: (a) the processing of
recorded waveforms from a single location, experimental array setup, source
offset/noise time window, and wavefield transformation method, (b) the
modification and repetition of the procedure illustrated in (a) to produce
estimates of the site’s dispersion uncertainty, and (c) the synthesis of these
estimates of uncertainty into a statistical representation called the site’s
experimental dispersion data.

While not the primary focus of this work, it is important to briefly mention
some of the methods currently being used to account for Vs uncertainty in
seismic hazard studies and highlight why developing uncertainty-consistent Vs
profiles from experimental dispersion data is of such critical importance. At
present, most important seismic design projects utilize some form of Vs
randomization to account for Vs uncertainty in ground response analyses, where
a baseline Vs profile is randomized (i.e., manipulated) using a statistical
procedure. The most commonly used of these randomization approaches is that
proposed by Toro (1995), leading to its adoption in the guidelines for hazard-
consistent one-dimensional ground response analyses (Stewart et al., 2014) and
the design of critical facilities (EPRI, 2012). However, multiple studies
(Griffiths et al., 2016a, b; Teague et al., 2018b; Teague and Cox, 2016) have
expressed serious concern that the blind application of this type of Vs
randomization can result in highly unrealistic ground models that do not fit
the experimental site signature and an inability of such models to accurately
predict ground response. The reader will note that a new Vs randomization
approach has recently been proposed by Passeri et al. (2020). However, the
authors have not yet evaluated it rigorously to determine if it produces Vs
profiles that are more consistent with the site signature. Importantly, this
paper, while not discussing Vs randomization directly, offers an alternative
to Vs randomization by propagating measured, site-specific uncertainty into
suites of Vs profiles obtained from surface wave testing. These suites of
profiles can then be used either directly to address Vs uncertainty in
subsequent engineering analyses (e.g., ground response analyses) or, if the
use of Vs randomization is still desired, to better inform its many unknown
input parameters with site-specific and uncertainty-consistent values. While
this study will focus primarily on the development of the former, it is
important that the reader be aware of the wider implications of the proposed
method on greater engineering practice.

This study begins by examining common approaches found in the literature for
developing uncertainty-consistent Vs profiles from the inversion of
experimental surface wave dispersion data. Using a synthetic dataset, these
approaches from the literature are shown to yield suites of subsurface models
whose theoretical dispersion curves severely underestimate the experimental
dispersion data’s uncertainty, resulting in Vs profile which underestimate the
site’s uncertainty. This is then followed by the presentation of a new
procedure for rigorously propagating measured experimental dispersion
uncertainty through the inversion process to obtain suites of Vs profiles that
more accurately represent Vs uncertainty at the site. This new procedure is
applied to the same synthetic dataset as the literature-based approaches and
is shown to be quantitatively superior, and able to precisely propagate the
experimental dispersion uncertainty into the resulting Vs profiles. The new
procedure is then extended to incorporate epistemic uncertainty in the
inversion’s layering parameterization using the same synthetic example, and
again shows excellent results. The study concludes with the application of the
method at a real site where it is shown to produce suites of Vs profiles which
agree favorably with a borehole Vs profile while simultaneously capturing the
experimental dispersion data’s uncertainty.

##  2 Common Approaches used to Account for Vs Uncertainty in Surface Wave
Inversion

To illustrate the problem this paper proposes to solve, we first examine
several approaches from the literature that have been used to account for Vs
uncertainty in surface wave inversion. All of the approaches mentioned below
have a basic commonality, in that they first search through large numbers
(often tens-of-thousands to millions) of trial layered-earth models to find a
significant number of models with an acceptable fit to the experimental
dispersion data. The acceptability of a model is typically judged using a
misfit function that quantifies the goodness-of-fit between the theoretical
dispersion curve for a given model and the experimental dispersion data. Due
to the non-uniqueness of the inverse problem and the experimental dispersion
data’s uncertainty, it is often possible to find thousands- to tens-of-
thousands of acceptable models that range from subtly to significantly
different. From these acceptable models, a subset are selected to account for
uncertainty in Vs. The discussion below will focus primarily on how different
studies obtained suites of acceptable models, and how they then selected from
those to estimate Vs uncertainty.

Before presenting the various approaches that have been used to account for Vs
uncertainty in surface wave inversion, it is important to briefly discuss the
details surrounding surface wave inversion, as these details will be important
for understanding the results presented later. First, all of the approaches
presented in the literature utilize large numbers of trial models from a
global-search inversion algorithm to find suites of acceptable models. Global-
search algorithms vary in their implementation and may search for acceptable
models in several different ways, including randomly (i.e., pure Monte-Carlo),
with the aid of some optimization algorithm (i.e., pure optimization), or by
using a combination of the two. By far, the most popular tool for performing
global-search surface wave inversion is the Dinver module (Wathelet et al.,
2004) of the open-source software Geopsy (Wathelet et al., 2020). As a
testament to its popularity, Dinver was the inversion algorithm of choice in
all but one of the works discussed below. Of particular relevance to the
discussion below, and worth discussing here, is the misfit function proposed
by Wathelet et al. (2004) and implemented in Dinver. The Wathelet et al.
(2004) misfit function can be described as a root-mean-square error normalized
by the experimental dispersion uncertainty (Yust et al., 2018). This gives the
misfit function a useful physical interpretation, as it represents on average,
across all frequencies/wavelengths, how far (in number of standard deviations)
a theoretical dispersion curve strays from the experimental dispersion data.
For example, a theoretical dispersion curve with a misfit of 1.0 can be
understood as a curve that on average is one standard deviation away from the
mean. With these details in mind, it is now possible to discuss the approaches
presented in the literature to account for Vs uncertainty in surface wave
inversion.

To develop suites of acceptable layered-earth models, Wathelet et al. (2004)
and Wathelet (2008) performed pure-optimization inversions considering many
(over 100,000) trial models. For convenience of expression, and in order to be
consistent with common vernacular, we will refer to a pure optimization
inversion as a minimum misfit of zero (M0) inversion. M0 inversions are named
as such because the goal of the inversion algorithm is to find a model with a
misfit equal to zero (i.e., one whose theoretical dispersion curve perfectly
matches the mean experimental dispersion data). In the studies noted above,
from the many trial models searched, all models with a dispersion misfit value
less than 1.0 (tens of thousands in most cases) were selected as a means to
propagate the experimental dispersion data’s uncertainty into the resulting Vs
profiles. This approach has three shortcomings: (1) it may result in a highly
variable number of acceptable profiles, depending on the number of trial
models attempted in the inversion and the subjective user-defined quality
threshold (i.e., misfit less than 1.0 in this case), (2) for inversions with
many models below the threshold (e.g., tens-of-thousands) it may readily
become computationally unmanageable to again propagate the Vs uncertainty into
subsequent engineering analyses (e.g., ground response analyses), and (3) the
use of an M0 inversion will likely cause the majority of the theoretical
dispersion curves to be clustered around the mean of the experimental
dispersion data (i.e., a misfit of zero) rather than following the
distribution of the experimental dispersion data uncertainty.

Foti et al. (2009) used a pure Monte-Carlo global-inversion algorithm (i.e.,
not Dinver) to develop suites of acceptable models (more than 50,000 in the
examples presented) whose theoretical dispersion curves fit the experimental
dispersion data. To select a more manageable subset of models, they sampled
from these large suites using a statistical test to select models which could
be considered equivalent in term of their fit to the experimental dispersion
data, given its uncertainties. While the selection process reduced the
computational burden of having to consider ten-of-thousands of Vs profiles, it
still maintained the disadvantage of the previous technique that the number of
profiles varied between application, ranging between 6 and 270 Vs profiles for
the cases presented. Furthermore, the theoretical dispersion curves from the
selected models can be seen to underestimate the experimental dispersion data
uncertainty, even for the most favorable case where 270 profiles were selected
by the statistical test.

Hollender et al. (2018), when developing statistics for the time-averaged
shear wave velocity in the upper 30 m (Vs30), used an “acceptable-misfit”
approach [i.e., an inversion where the minimum misfit was not zero (M0+)] to
develop a large suite of acceptable models (over 50,000) to represent the
experimental dispersion data uncertainty. From those models, they randomly
extracted at least 6000 Vs profiles to develop statistics on Vs30. They then
selected a smaller set of 33 representative Vs profiles, whose uncertainty in
Vs30 matched that of the randomly selected 6000 profiles, for use in
subsequent analyses. While this approach ensured an equal and manageable
number of profiles (i.e., 33) for each site, and permitted the propagation of
the estimated uncertainty in Vs30, it does not guarantee that the initial
large suite of acceptable models properly accounted for the experimental
dispersion data’s uncertainty. Thereby, potentially allowing the subsequent
uncertainties (i.e., the Vs30 uncertainty) upon which they are based to not
fully capture the site-specific uncertainty in the experimental dispersion
data.

This final section groups a number of studies that are similar, in that they
all use a M0 inversion to develop their acceptable suites of models from which
to select some fixed number of lowest misfit models to represent Vs
uncertainty. Di Giulio et al. (2016) selected the 100 lowest misfit profiles
out of over a million trial models. Griffiths et al. (2016a; 2016b) and Teague
and Cox (2016) chose 50 randomly selected models from the 1000 lowest misfit
models obtained from hundreds-of-thousands of trial models, as the 50 randomly
selected models were shown to have similar statistical properties as the 1000
best, but were more computationally manageable. Cox and Teague (2016), Teague
et al. (2018a) and Deschenes et al. (2018) presented the 1000 lowest misfit Vs
profiles from hundreds-of-thousands of trial models. Vantassel et al. (2018),
Cox and Vantassel (2018), and Yust et al. (2018) presented the 100 lowest
misfit Vs profiles from hundreds-of-thousands of trial models considered to
account for Vs uncertainty, as these 100 lowest misfit Vs profiles were shown
to be statistically similar to the 1000 lowest misfit Vs profiles, but were
more manageable. This approach (i.e., selecting some number of the lowest
misfit/“best” models) resolves the issues of variability between analyses,
avoids potentially large numbers of profiles, and is relatively simple.
However, this approach is still deficient in two specific ways: (1) it relies
on an M0 inversion which, as mentioned previously, will likely produce
profiles with theoretical dispersion that is clustered around the experimental
dispersion data’s mean rather than being properly distributed, and (2) the
uncertainty accounted for in the 100 or 1000 best models is indirectly tied to
the number of trial models attempted by the analyst (i.e., different apparent
uncertainty will results whether the analyst uses 1000, 10,000 or 1,000,000
trial models).

In summary, all of the presented approaches follow a basic two-step process.
First, a large number of acceptable models are obtained using a global-search
algorithm, and second, some subset of profiles are selected from the
acceptable models as a basis for assessing Vs uncertainty. In all of the
presented approaches we observe a dependence on two basic assumptions: (1)
that the large suite of acceptable models (i.e., from the first step) properly
accounts for the experimental dispersion data’s uncertainty, and (2) that the
selection process will guarantee a set of Vs profiles which rigorously
propagates the uncertainty into subsequent analyses. The remainder of this
section is devoted to quantitatively assessing the veracity of these
assumptions by examining the effectiveness of these approaches.

To quantitatively assess the ability of the approaches from the literature to
propagate experimental dispersion data uncertainty into the resulting Vs
profiles, four variations were applied to an experimental dispersion dataset.
The experimental dispersion data is taken from a large synthetic study focused
on the performance of surface wave inversion (Vantassel and Cox, 2021). The
data itself has been published as one of twelve surface wave inversion
benchmarks, which is publically available on the DesignSafe-CI (Vantassel and
Cox, 2020). While a full detailed discussion is provided in the previous
references, in short, the synthetic data was developed by taking the
theoretical dispersion curve from an assumed ground model, resampling it in
log-wavelength, and assuming a normal distribution in Rayleigh wave velocity
(Vr) with a coefficient of variation (COV) of 0.05. A COV of 0.05 was based on
typical experimental dispersion data uncertainty values from several blind
analyst studies (Cox et al., 2014; Garofalo et al., 2016). For reference, the
experimental dispersion data is the same as that shown in Figure 1c. The
proposed methods to be considered were selected to address two primary
questions of interest, namely: (1) what effect, if any, does the type of
inversion (pure optimization, pure random, or some combination) have on the
resulting acceptable models, and (2) how should the analyst sample from the
acceptable models. With regard to the type of inversion preformed, we consider
two inversion alternatives: (a) a pure-optimization (i.e., M0) inversion, as
this approach is the most commonly used in the literature, and (b) a combined
approach that is partly optimized and partly random. This second approach will
be referred to as a minimum misfit of 1.0 (M1) inversion because the inversion
algorithm is forced to search randomly in those regions of the model space
where the calculated misfit is below 1.0. A pure-random search was not
explicitly considered here, as it is expected to produce similar results to
the M1 inversion at greater computational expense. With regard to how the
representative Vs profiles should be sampled from the acceptable inversion
models, we consider two alternatives: (a) the 100 lowest misfit/“best” models
(b100), as this was the most popular approach in the literature, and (b) 100
randomly selected models from all models with a misfit less than 1.0 (n100).
Note that the n100 alternative was a necessary adaptation to the approach of
selecting all profiles below the misfit threshold of 1.0 to ensure a
computationally manageable number of profiles and a fair comparison to the
b100 profiles.

For illustration purposes and to simplify initial discussions, the
experimental dispersion data was inverted using only a single, three-layer
parameterization (i.e., a Layering by Number (LN) = 3 parameterization), as
this layering parameterization was shown to perform the best when inverting
the example dataset in the previous study by Vantassel and Cox (2021). The
experimental dispersion data were inverted using the Neighborhood Algorithm
(Sambridge, 1999) as implemented in the Dinver module of Geopsy (Wathelet et
al., 2020). For both the M0- and M1-style inversions, an initial 10,000 random
trial models followed by 50,000 neighborhood-algorithm trial models were
considered. Note that the number of neighborhood-algorithm models used here is
less than those used in previous studies from the literature, which tended to
use a hundred-thousand or more. We do this to mitigate bias in the resulting
suite of acceptable models caused by the inclusion of many (potentially tens-
of-thousands) very similar models with misfits close to zero. We believe the
use of a smaller, but sufficient (Vantassel and Cox, 2021), number of models
provides these techniques the best possible chance of successfully propagating
the experimental dispersion data’s uncertainty into the resulting models. The
selected b100 and n100 models are shown in terms of their Vs profiles and
associated uncertainty in Figure 2. Note, to better illustrate their
concentration, the Vs profiles have been discretized in terms of depth and Vs,
binned into cells, and color mapped in terms of the number of profiles in each
cell. Vs uncertainty is expressed in the form of the lognormal standard
deviation of Vs ($\sigma_{ln,Vs}$), which is commonly used in seismic hazard
studies, and is similar to the COV for values less than approximately 0.3. The
theoretical dispersion curves for the b100 and n100 models are presented with
the experimental dispersion data in Figure 3 so the reader can view how the
apparent uncertainty in the inversion-derived Vs profiles relates to the
apparent uncertainty in the theoretical dispersion curves.

Figure 2: Shear wave velocity (Vs) profiles obtained from a synthetic
experimental dispersion dataset using four common approaches from the
literature for accounting for Vs uncertainty in inversion. In order to better
illustrate their concentration, the profiles have been discretized in terms of
depth and Vs, binned into cells, and color mapped in terms of the number of
profiles in each cell. The presented approaches include: (a) a minimum misfit
of 0 (M0) inversion with the selection of the 100 lowest misfit/“best” models
(b100), (c) a M0 inversion with 100 random profiles selected from all models
with a misfit less than 1.0 (n100), (e) a minimum misfit of 1.0 (M1) inversion
with the selection of the 100b models, and (g) an M1 inversion with the
selection of the n100 models. The Vs profiles from these approaches are shown
alongside the solution profile in red. Adjacent to each suite of profiles is
their corresponding lognormal standard deviation of Vs ($\sigma_{ln,Vs}$). The
sharp spikes in $\sigma_{ln,Vs}$ are due to the uncertainty in the profile’s
layer boundaries, reflecting a shortcoming in how $\sigma_{ln,Vs}$ has been
calculated historically, and are not the result of actual uncertainty in Vs
directly.

Figure 2 illustrates that the resulting Vs profiles are most sensitive to how
they are sampled (b100 vs n100), and significantly less dependent on how the
inversion is performed (M0 vs M1). This is shown both qualitatively when
viewing the range of Vs profiles [i.e., panels (a), (c), (e), and (g)] and
quantitatively when viewing $\sigma_{ln,Vs}$ [i.e., panels (b), (d), (f), and
(h)]. For example, the Vs profiles obtained from the b100 approach show
minimal scatter/uncertainty, while those obtained from the n100 approach show
significantly more scatter/uncertainty. Note that the spikes in
$\sigma_{ln,Vs}$ are due to the uncertainty in the layer boundaries,
reflecting a shortcoming in how $\sigma_{ln,Vs}$ has been traditionally
calculated, and are not the result of uncertainty in Vs directly. The trends
in Vs scatter/uncertainty shown in Figure 2 are tied directly to visible
trends in the scatter/uncertainty of the theoretical dispersion curves shown
in Figure 3. The color scale in Figure 3 indicates the range of dispersion
misfit values, which varies widely between approaches. The b100 models are
primarily composed of models with low misfit dispersion misfit models (<0.2)
which fit the mean trend of the experimental data very well, but severely
underestimate the experimental uncertainty. In contrast, the n100 models
contains essentially all higher misfit models (>0.6) that qualitatively appear
to fit the experimental dispersion data’s uncertainty, although some biases
can be observed, particularly towards lower Vr values at low frequencies
(e.g., in Figure 3b at frequencies below 4 Hz). To better illustrate these
problems, the vertical blue dashed line at 4 Hz in each panel of Figure 3
corresponds to a “slice” shown in each of the corresponding panels of Figure
4. The mismatch between the measured and inverted distributions of Vr in
Figure 4 clearly show that none of the methods from the literature are able to
capture the mean, standard deviation, and distribution of the experimental
dispersion data. To illustrate this quantitatively across the entire frequency
range, Figure 5 shows the normalized residual mean and residual COV for the
dispersion data for the four approaches considered. The vertical dashed blue
line again indicates the location of the “slices” shown in Figure 4. Figure 5a
confirms that using b100 sampling will tend to result in a suite of models
that well-fit the mean trend of the experimental dispersion data (i.e.,
normalized residual mean near zero). Whereas, using n100 sampling may result
in significant errors in the mean (e.g., overestimation of Vr at 4 Hz). Figure
5b illustrates that using b100 sampling, which was shown to fit the mean Vr so
precisely in Figure 5a, severely underestimates the true dispersion
uncertainty at all frequencies. Recall that the dispersion uncertainty for
this synthetic example was set using a COV of 0.05 at all frequencies; meaning
that the b100 sampling with a residual COV of nearly -0.05 underestimates the
true dispersion uncertainty by nearly 100%. In contrast, the n100 models that
did not well-fit the mean trend tend to provide a better representation of the
experimental dispersion uncertainty, although in this case they still
underestimate the measured uncertainty at both high and low frequencies by
approximately 0.02, or 40%. These results clearly indicate that the currently
available methods of accounting for Vs uncertainty in surface wave inversion
are unable to propagate the uncertainty in the experimental dispersion data
into the resulting suites of Vs profiles.

Figure 3: Comparison of the experimental dispersion data from a synthetic
dataset with the individual theoretical dispersion curves obtained from
inversion using four common approaches from the literature for accounting for
Vs uncertainty. Each of the four panels (a)-(d) illustrate one of the four
common approaches from the literature. They include: (a) a minimum misfit of 0
(M0) inversion with the selection of the 100 lowest misfit/“best” models
(b100), (b) a M0 inversion with 100 random profiles selected from all models
with a misfit less than 1.0 (n100), (c) a minimum misfit of 1.0 (M1) inversion
with the selection of the b100 models, and (d) an M1 inversion with the
selection of the n100 models. The vertical dashed blue line at approximately 4
Hz, shown on all panels, indicates the location where a “slice” is shown in
Figure 4 for each of the four approaches. Figure 4: Distributions of the
measured (i.e., experimental) and inverted (i.e., theoretical) dispersion data
at a frequency of approximately 4 Hz using four common approaches from the
literature to account for Vs uncertainty in surface wave inversion. The four
panels have the same caption as those shown in Figure 3. Figure 5:
Quantitative assessment of four common approaches from the literature for
propagating uncertainty in experimental dispersion data to the inverted Vs
profiles. The comparison is made on the basis of: (a) the dispersion residual
mean ($\mu_{res}$) [i.e., the difference between the mean of the inverted
theoretical dispersion curves ($\mu_{inv}$) and the mean of the measured
experimental dispersion data ($\mu_{meas}$)] normalized by $\mu_{meas}$,
expressed in percent, and (b) the dispersion residual coefficient of variation
[i.e., the difference between the coefficient of variation of the inverted
theoretical dispersion curves ($\delta_{inv}$) and the coefficient of
variation of the measured experimental dispersion data ($\delta_{meas}$)]. The
vertical dashed blue line at approximately 4 Hz in panels (a) and (b)
indicates the location where a “slice” was shown in Figure 4 for each of the
four approaches.

##  3 A New Procedure to Account for Vs Uncertainty in Surface Wave Inversion

The four example approaches from the literature considered in the previous
section had three specific shortcomings: (1) the profiles tended to capture
either the mean trend or the variance of the dispersion data, but not both
(refer to Figure 5), (2) theoretical dispersion curves from the suites of
inverted models did not follow the distribution of the synthetic experimental
dispersion data (refer to Figure 4), and (3) when the results were found to be
unacceptable, as was the case for the synthetic example discussed above, there
was no clear procedure to remedy the inconsistency. This section presents an
alternative procedure to account for Vs uncertainty in surface wave inversion.
The new procedure is presented schematically in Figure 6. Figure 6a shows
experimental dispersion data with site-specific measurements of uncertainty.
This uncertainty is quantified in terms of surface wave phase velocity at each
frequency using a mean, standard deviation, and, very importantly, correlation
coefficients between all frequency pairs. All of these statistics can be
easily quantified by the analyst using the approach to developing dispersion
data with site-specific uncertainty discussed previously (recall Figure 1).
Consider the experimental dispersion data in Figure 6a, which contains 13
frequency points. Its statistics are completely described using 13 mean values
(shown with a circle), 13 standard deviations (shown with error bars), and a
matrix of 13x13 correlation coefficients (not shown) relating the phase
velocity at each frequency to the other 12 values. With the experimental
dispersion data described in terms of its statistics, it’s now possible to
simulate a realization from that experimental dispersion data, shown
schematically in Figure 6b. It is during this simulation stage that the
inclusion of the correlation information is so critical. Without such
information one is left to assume the correlations, or worse, independence.

Figure 6: A new procedure for developing uncertainty-consistent shear wave
velocity (Vs) profiles from surface wave dispersion data. The procedure
involves: (a) describing the site’s experimental dispersion data in terms of
its statistics, (b) generating a realization of the experimental dispersion
data, (d) inverting the realization to obtain a single, best theoretical fit,
repeating (b) and (d) for N realizations, and (c) comparing the statistics of
the resulting theoretical dispersion curves (i.e., the implied dispersion
data) to that of the measured experimental dispersion data.

However, doing so would likely result in simulated dispersion curves that were
erratic from frequency-to-frequency and inconsistent with the dispersion data
that was used to estimate the uncertainty. In essence, the inclusion of the
correlations between dispersion points imposes a relational constraint that
helps to encourage simulated dispersion curves that are consistent with the
measured data (i.e., in this case, smooth and continuous). Once a curve has
been simulated, it is inverted to obtain a single, “best” fit model, as shown
in Figure 6d. Note that while all previous procedures to account for Vs
uncertainty required the use of a global-search algorithm to ultimately
develop suites of Vs profiles, in contrast, this new procedure is inversion-
algorithm agnostic, thereby permitting the analyst to use any algorithm of
their choice (global or local). Furthermore, by selecting only the single,
“best” model the issue of dependence on the number of trial models is avoided
altogether, provided of course that a reasonable/sufficient number of models
has been attempted to produce a good fit to the simulated dispersion curve,
which is easily verifiable. Returning to Figure 6d, it is important to note
that the fit may not be perfect, however, the proposed procedure does not
require it. The issue of primary importance is that the simulated experimental
dispersion data and the theoretical fit are in good agreement over the
experimental frequency range, though not necessarily strictly identical. The
procedure of simulating a realization of the dispersion data and fitting it
through inversion is repeated for some number of trials (N), which is the only
input parameter to be defined by the analyst. The best inversion-derived fit
to each realization and the corresponding ground model is saved as a potential
solution to the experimental data considering its uncertainties, refer to
Figure 6c. The N fits (i.e., N theoretical dispersion curves) can then be used
to calculate implied dispersion data statistics which can be compared directly
against the experimental dispersion data statistics to assess the
successfulness of the procedure. Note that the value of N is expected to be
problem dependent, however, a minimum acceptable value for N can be checked
prior to performing any inversions by comparing the statistics of the
simulated dispersion realizations and the experimental dispersion data. If the
statistics of the simulated dispersion data (i.e., the targets used for
inversion) are unable to reproduce the experimental statistics, it can then be
inferred that the inversion-derived fits to those simulated curves will also
not reproduce the experimental statistics. Thus, care must be taken to select
N to be sufficiently large to ensure agreement between the simulated and
measured experimental data, although not too large to avoid excessive
computational cost when performing N different inversions. An N=250 was shown
to perform well over the course of this study, however the choice of N is left
to the discretion of the analyst. Importantly, following the inversion of the
N simulated dispersion curves to obtain N theoretical dispersion curves, the
statistics implied by the N theoretical dispersion curves (i.e., the implied
dispersion data, refer to Figure 6c), must be compared quantitatively with the
measured experimental dispersion data in a manner similar to that shown in
Figure 5. If the agreement between the implied and measured experimental
dispersion data is found to be unsatisfactory, additional simulations and/or
increasing the value of N can be used to improve the results. The remainder of
this paper will address the application of this new approach to two synthetic
tests and a real-world example.

##  4 Synthetic Tests

###  4.1 For a Single Layering Parameterization

To demonstrate the effectiveness of the newly proposed procedure, we will test
it using the same synthetic dataset as that was used previously to evaluate
the literature-based approaches for accounting for Vs uncertainty. Recall, the
experimental dispersion data (i.e., mean and standard deviation as a function
of frequency) for this dataset is presented in Figure 1c. As the new approach
requires the correlations between frequencies, and the synthetic data, as
developed, does not include such information, the correlations had to be
synthesized. The procedure for synthesizing the correlations involved first
simulating ground models whose theoretical dispersion curves were consistent
with the experimental dispersion’s uncertainty. The statistics used to inform
the simulation of these ground models were developed based on the mean values
from an M0-type inversion and the uncertainty from an M1-type inversion, and
while these types of inversions have been shown to be lacking in their ability
to propagate dispersion uncertainty (refer to Figures 4 and 5), they provided
a reasonable correlation structure. The synthesized correlations were combined
with the known/assumed statistics (refer to Figure 1c) to define the
experimental dispersion data. With this information, the procedure outlined in
Figure 6 was performed using N=250 realizations. The choice of N=250 was made
prior to performing the inversions by checking that the statistics of the
realizations consistently reproduced the statistics of the experimental
dispersion data. Each realization was inverted using the same 3-layer
parameterization and 10,000 random plus 50,000 neighborhood algorithm trial
models, as was used in the previous section. Inversions were performed on the
Texas Advanced Computing Center’s (TACCs) cluster Stampede2 using a single
Skylake (SKX) node. The entire analysis (i.e., the inversion of all N=250
simulated dispersion curves) took less than 2 hours to complete.

The results from the inversion analyses are provided in Figures 7, 8, and 9.
Figure 7a compares the theoretical dispersion curves fit to the N=250
realizations of the experimental dispersion data with the original
experimental dispersion data. Note the theoretical dispersion curves have been
colored in terms of their dispersion misfit relative to the original
experimental data and not their respective realizations. In contrast to the
previous methods (refer to Figure 3) we observed a reasonable mixture of
theoretical dispersion curves with low misfits (i.e., <0.2) near the mean
trend of the experimental data and higher misfits extending outward to capture
the dispersion data’s uncertainty. This indicates qualitatively that the
theoretical curves from the new procedure are being distributed according to
the uncertainty in the experimental dispersion data rather than directly as a
consequence of the inversion process. To better illustrate the agreement
between the measured and inverted dispersion uncertainty, the vertical dashed
lines at approximately 3, 6, 13, and 30 Hz in Figure 7a indicate the locations
of “slices” shown in Figures 7b, c, d, and e, respectively. These “slices”
indicate excellent agreement between the measured dispersion data’s
probability density functions (black lines) and the inverted distributions of
Vr (histograms), especially in comparison to the previous methods considered
(refer to Figure 4). Figure 8 presents the results in a quantitative manner
across the entire frequency range. Figures 8a and b, present the normalized
dispersion residual mean and dispersion residual COV, respectively. While
minor differences are observed, both metrics illustrate excellent agreement
between the experimental and inverted dispersion data’s mean and uncertainty
(i.e., residuals approximately equal to zero at all frequencies), especially
when compared to previous methods (refer to Figure 5).

Figure 7: Qualitative assessment of the newly proposed procedure for
propagating experimental dispersion uncertainty into the inverted Vs profiles
considering a single inversion parameterization for a synthetic example. Panel
(a) shows the experimental dispersion data and 250 inverted theoretical
dispersion curves fit to the 250 realizations of the experimental dispersion
data. The theoretical dispersion curves have been colored according to their
dispersion misfit values relative to the original experimental dispersion
data, not their respective realizations. The vertical dashed lines in panel
(a) at approximately 3, 6, 13, and 30 Hz denote the location of the “slices”
shown in panels (b), (c), (d), and (e), respectively. These “slices” compare
the measured experimental dispersion data’s distribution (solid black line)
with the distribution of Rayleigh wave velocity (Vr) derived from inversion
(histogram). Figure 8: Quantitative assessment of the newly proposed procedure
for propagating experimental dispersion uncertainty into the inverted Vs
profiles considering a single inversion parameterization for a synthetic
example. The comparison is made on the basis of: (a) the dispersion residual
mean ($\mu_{res}$) [i.e., the difference between the mean of the inverted
theoretical dispersion curves ($\mu_{inv}$) and the mean of the measured
experimental dispersion data ($\mu_{meas}$)] normalized by $\mu_{meas}$ and
expressed in percent, (b) the dispersion residual coefficient of variation
[i.e., the difference between the coefficient of variation of the inverted
theoretical dispersion curves ($\delta_{inv}$) and the coefficient of
variation of the measured experimental dispersion data ($\delta_{meas}$)]. The
vertical dashed lines in panels (a) and (b) denote the location of the
“slices” shown in Figure 7. Figure 9: Uncertainty-consistent Vs profiles
considering a single inversion parameterization for a synthetic example using
the newly proposed procedure. In order to better illustrate their
concentration, the profiles have been discretized in terms of depth and Vs,
binned into cells, and color mapped in terms of the number of profiles in each
cell. Panel (a) summarizes the Vs profiles resulting from the inversion of the
N=250 realizations of the experimental dispersion data alongside the true
solution, the discretized log-normal median, and the 95% confidence interval
(CI). Panel (b) illustrates the lognormal standard deviation of Vs
($\sigma_{ln,Vs}$) of the 250 profiles shown in panel (a). The sharp spikes in
$\sigma_{ln,Vs}$ are due to the uncertainty in the profile’s layer boundaries,
reflecting a shortcoming in how $\sigma_{ln,Vs}$ has been calculated
historically, and are not the result of actual uncertainty in Vs directly.

With the theoretical dispersion curves from the inverted ground models having
been shown to be consistent with the uncertainty of the experimental
dispersion data, we can now examine the effects on the Vs profiles. Figure 9a
presents the single-lowest misfit Vs profile from the inversion of each of the
N=250 dispersion realizations. To better illustrate their concentration about
the true solution the Vs profiles have been discretized in terms of depth and
Vs, binned into cells, and color mapped in terms of the number of profiles in
each cell. The discretized lognormal median Vs profile and the 95% lognormal
confidence interval (CI) profiles are also shown for reference. Figure 9a
shows that the discretized median profile reasonably captures the true
solution and that the 95% confidence interval qualitatively captures the
variance in the Vs profiles, with the notable exception of the sharp spikes at
layer boundaries. When we compare $\sigma_{ln,Vs}$ from the new approach
(i.e., Figure 9b) directly with those from the literature (i.e., Figure 2b, d,
f, and h) we observe that the new approach shows increased uncertainty from
the (b100) alternatives, as should be expected, but surprisingly slightly less
uncertainty than the (n100) alternatives. This observation indicates that (at
least for this example) the n100 models tend to over-estimate Vs uncertainty
even when under-predicting the dispersion data uncertainty (refer to Figure
5). Further examination of Figure 9b shows that $\sigma_{ln,Vs}$ for this
example is approximately 0.06, which is quite low compared to values commonly
assumed in practice (EPRI, 2012; Stewart et al., 2014; Toro, 1995). This is
certainly due in part to the use of only a single inversion layering
parameterization, as others have shown that the variability within a
parameterization is generally much less than that between parameterizations.
Hence, if the true subsurface layering is unknown a priori, it is not simply
enough to accurately represent the experimental dispersion data’s uncertainty
using a single, assumed layering parameterization. Rather, one must also
incorporate the epistemic uncertainty in the layering parameterization itself
(Cox and Teague, 2016; Di Giulio et al., 2012; Vantassel and Cox, 2021). The
incorporation of multiple inversion parameterizations into the procedure and
their effects on the Vs profiles are presented in the following section.

###  4.2 Extension to Multiple Inversion Layering Parameterizations

Uncertainty within surface wave inversion is generally split into intra- and
inter-parameterization variability, which address the uncertainty inside a
single inversion parameterization (i.e., the previous example) and between
various parameterizations (i.e., current example), respectively. Previous
studies have shown that the inter-parameterization variability is generally
more significant than the inter-parametrization variability. However, as the
methods used in previous studies to account for intra-parameterization
variability tended to underestimate the experimental uncertainty (refer to
Figures 4 and 5), it is of interest to reexamine this conjecture using the new
procedure.

The new procedure was repeated using the same synthetic experimental
dispersion data as the previous example (refer to Figure 1c), except rather
than using only a single layering parameterization consisting of 3 layers,
five different layering parameterizations were used to account for epistemic
uncertainty in the parameterization selection. The five parameterizations were
based on the Layering by Number (LN) schema and included LNs of 3, 5, 7, 9,
and 14. These parameterizations were selected because they were all deemed
acceptable when this same set of experimental dispersion data was inverted by
Vantassel and Cox (2021). In practice, the analyst must decide carefully which
parameterizations to pursue, as this can strongly impact the resulting
inversion-derived Vs profiles. We recommend investigating a number of trial
parameterizations, and the reader is referred to the previous study by
Vantassel and Cox (2021) for a more thorough discussion on the number and type
of parameterizations to use in their inversions. The five parameterizations
noted above were used to invert different sets of N=250 dispersion
realizations (i.e., 1250 realizations in total). N=250 realizations per
parameterization was selected to ensure that each parameterization would have
a sufficient number of realizations to each individually capture the
uncertainty of the experimental dispersion data, as it is important to not
discount the intra-parameterization variability when investigating the inter-
parameterization variability. This, of course, is not the only approach
available to consider the inter-parameterization variability. An alternative
method examined during the course of this study is to invert a single set of N
realizations with various parameterizations. This alternate approach was found
to produce similar results to those using the presented method, however, since
the presented method is believed to be more robust, as it ensures each
parameterization remains separate, it is the one selected here.

The results from the newly proposed procedure when accounting for multiple
layering parameterizations are shown in Figures 10, 11, and 12. Figure 10a
compares the theoretical dispersion curves fit to the 1250 realizations of the
experimental dispersion data with the original experimental dispersion data.
The theoretical dispersion curves are colored in terms of their dispersion
misfit relative to the original experimental dispersion data (not their
respective realizations) and are again shown to follow a reasonable
distribution of low misfit (<0.2) curves near the mean trend of the
experimental data and higher misfit curves capturing its uncertainty. The
vertical colored dashed lines at approximately 3, 6, 13, and 30 Hz in Figure
10a indicate the locations of “slices” presented in Figures 10b, c, d, and e,
respectively. These slices reveal excellent agreement between the measured and
inverted distributions of Vr. To quantitatively assess performance over the
entire frequency range, Figures 11a and b present the normalized dispersion
residual mean and dispersion residual COV, respectively. While again some
minor fluctuations are observed (e.g., at 6 Hz), both metrics illustrate
excellent agreement between the experimental dispersion data’s mean and
uncertainty (i.e., residuals approximately equal to zero at all frequencies).

With the theoretical dispersion curves from the inverted ground models having
been shown to be consistent with the uncertainty of the experimental
dispersion data, we can now examine the effects on the Vs profiles. Figure 12a
presents the single-lowest misfit Vs profile from the inversion of each of the
1250 dispersion realizations (i.e., N=250 profiles per parameterization * 5
parameterizations = 1250 profiles). To better illustrate their concentration,
the Vs profiles have been discretized in terms of depth and Vs, binned into
cells, and color mapped in terms of the number of profiles in each cell. The
inverted Vs profiles are shown alongside the true solution, discretized
lognormal median Vs profile, and lognormal 95% CI profiles. As expected, the
inverted profiles qualitatively show more uncertainty than when a single
parameterization is considered (refer to Figure 9a for comparison). This
distinction is most apparent at the layer boundaries, as the locations of
these boundaries are controlled predominantly by the assumed layering
parameterization. Parameterizations with many layers will tend to result in
smoother profiles with gradual changes in Vs, while parameterizations with
only a few layers will tend to result in profiles with sharper contrasts (Cox
and Teague, 2016). As the true site layering may not be known prior to surface
wave inversion, there is often a need to consider multiple parameterizations
with different numbers of layers to properly address one of surface wave
inversion’s main sources of epistemic uncertainty. Figure 12b presents
$\sigma_{ln,Vs}$ of the 1250 profiles. The uncertainty across multiple
parameterizations is shown to increase in regards to what was observed
previously for a single parameterization (refer to Figure 9b for comparison).
Specifically, $\sigma_{ln,Vs}$ is now closer to 0.1 within any given layer,
thereby confirming quantitatively what was already observed qualitatively in
Figure 12a. The increase in $\sigma_{ln,Vs}$ lends confidence to the
conjecture that the uncertainty resulting from multiple parameterizations will
tend to exceed that for a single parameterization, although perhaps not to the
same extent as may have been inferred previously due the tendencies to
underestimate intra-parameterization variability in previous studies.
Regardless, at least for this synthetic example, $\sigma_{ln,Vs}$ remains
quite low compared to values commonly assumed in practice (EPRI, 2012; Stewart
et al., 2014; Toro, 1995).

Figure 10: Qualitative assessment of the newly proposed procedure for
propagating experimental dispersion uncertainty into the inverted Vs profiles
considering multiple inversion parameterizations for a synthetic example.
Panel (a) shows the experimental dispersion data and the inverted theoretical
dispersion curves fit to the 1250 realizations (5 parameterizations with N=250
realizations each) of the experimental dispersion data. The theoretical
dispersion curves have been colored according to their dispersion misfit
values relative to the original experimental dispersion data, not their
respective realizations. The vertical dashed lines in panel (a) at
approximately 3, 6, 13, and 30 Hz denote the location of the “slices” shown in
panels (b), (c), (d), and (e), respectively. These “slices” compare the
measured experimental dispersion data’s distribution (solid black line) with
the distribution of Rayleigh wave velocity (Vr) derived from inversion
(histogram). Figure 11: Quantitative assessment of the newly proposed
procedure for propagating experimental dispersion uncertainty into the
inverted Vs profiles considering multiple inversion parameterization for a
synthetic example. The comparison is made on the basis of: (a) the dispersion
residual mean ($\mu_{res}$) [i.e., the difference between the mean of the
inverted theoretical dispersion curves ($\mu_{inv}$) and the mean of the
measured experimental dispersion data ($\mu_{meas}$)] normalized by
$\mu_{meas}$ and expressed in percent, (b) the dispersion residual coefficient
of variation [i.e., the difference between the coefficient of variation of the
inverted theoretical dispersion curves ($\delta_{inv}$) and the coefficient of
variation of the measured experimental dispersion data ($\delta_{meas}$)]. The
vertical dashed lines in panels (a) and (b) denote the location of the
“slices” shown in Figure 10. Figure 12: Uncertainty-consistent Vs profiles
considering multiple inversion parameterization for a synthetic example using
the newly proposed procedure. In order to better illustrate their
concentration, the profiles have been discretized in terms of depth and Vs,
binned into cells, and color mapped in terms of the number of profiles in each
cell. Panel (a) summarizes the Vs profiles resulting from the inversion of
1250 realizations (5 parameterizations with N=250 realizations each) of the
experimental dispersion data alongside the true solution, the discretized log-
normal median, and the 95% confidence interval (CI). Panel (b) illustrates the
lognormal standard deviation of Vs ($\sigma_{ln,Vs}$) of the 250 profiles
shown in panel (a). The sharp spikes in $\sigma_{ln,Vs}$ are due to the
uncertainty in the profile’s layer boundaries, reflecting a shortcoming in how
$\sigma_{ln,Vs}$ has been calculated historically, and are not the result of
actual uncertainty in Vs directly.

##  5 Application at a Real Site

To illustrate the application of the newly proposed method at a real site, we
present the following case study. For reference, the site is located in the
southern California and has a known geology consisting of alluvial soils
(sands and clays) approximately 15 – 25 m thick, overlaying highly variable
weathered bedrock, over competent bedrock. While an extensive set of active-
source and passive-wavefield surface wave measurements are available at this
site, to keep the example simple, we will only describe how dispersion data
was extracted from some of the passive-wavefield microtremor array
measurements (MAM). To incorporate aleatory variability across the 70 m by 70
m site into the estimates of dispersion uncertainty, four 35 m diameter
circular arrays each composed of nine sensors will be considered. In essence,
each of the four MAM arrays were responsible for characterizing one quarter of
the overall site. While using only one size of MAM array will result in
somewhat bandlimited dispersion data, we do so herein to maintain clarity in
how the combined aleatory and epistemic uncertainties are estimated. MAM array
sensors were buried and left undisturbed to record ambient noise, seven hours
of which were used to extract dispersion data from each array. Each of the
four MAM arrays were processed separately to obtain experimental dispersion
data using the Rayleigh three-component beamforming method developed by
Wathelet et al. (2018) and implemented in the open-source software Geopsy
(Wathelet et al., 2020). Constant length time blocks were preferred over
frequency-dependent time blocks to facilitate statistical calculations on the
experimental dispersion data. Time blocks were selected to be 60 seconds long
to ensure at least 100 cycles at the lowest processing frequency. The
recommended 4 time blocks per sensor per block set were used to compute the
average cross-correlation matrices. As sufficiently long noise records were
available, no overlap was permitted between blocks or block sets. No higher
modes were apparent in preliminary dispersion processing, so only the spectral
peaks with the single highest power at each frequency were selected from each
block set to represent the experimental dispersion data. This resulted in 11
estimates of Vr at each frequency for each array. Or, in other words, 44
estimates of Vr per frequency across the entire site. The experimental data
was binned and resampled in terms of log-wavelength following the
recommendations of Vantassel and Cox (2021). The raw experimental dispersion
data and the resulting statistical representation is shown in Figure 13a. The
frequency-dependent COVs tend to increase from approximately 0.02 at short
wavelengths to approximately 0.08 at long wavelengths. The minimum and maximum
wavelength of the experimental dispersion data are approximately 10 m and 250
m, respectively. The reader will note that the maximum wavelength (250 m) is
substantially longer than what one would generally expect to be able to
resolve based on the array resolution limits (typically 2 or 3 times the
maximum array aperture). However, this was done only after confirming that the
dispersion data from these arrays were consistent with data from larger
aperture arrays also performed at the site (not shown). The minimum depth of
profiling can be approximated by dividing the minimum wavelength by a depth
factor (df) of 3 or 2 (Foti et al., 2018) to obtain an estimate of the minimum
resolvable thickness of the near-surface layer between 3 m and 5 m. These
relatively large near-surface layers are the direct result of using only
passive arrays of medium size for this example. The maximum depth of profiling
can be approximated by dividing the maximum wavelength by a df of 3 or 2.
However, the use of even a df of 3 may be optimistic when the dispersion curve
has an “L”-shape which does not flatten at long wavelengths (Vantassel and
Cox, 2021). Nonetheless, we adopt a df of 3 for parameterizing the inversion’s
maximum depth based on the experimental dispersion data, and then limit the
depth to which the inverted Vs profiles are presented based on the quality of
the resulting Vs profiles (i.e., depth limited based on high $\sigma_{ln,Vs}$
values when resolution is poor).

The frequency-dependent uncertainty in the experimental dispersion data was
propagated through the inversion process using the newly proposed procedure
outlined in Figure 6. To incorporate the additional epistemic uncertainty from
the inversion’s parameterization, five LN-type parameterizations using LN = 3,
5, 7, 9, and 14 were considered following the same procedure as presented in
the previous synthetic example. Inversions took 2 hours to complete using 5
SKX nodes on the Stampede2 cluster. After inversion, the parameterization-
quality criteria proposed by Vantassel and Cox (2021) was used to assess the
performance of the five parameterizations. The criteria showed the LN=3
parameterization underperformed its counterparts and was therefore removed
from further consideration to avoid biasing the results. The resulting 1000
theoretical dispersion curves (i.e., 250 curves/parameterization * 4
acceptable parameterizations = 1000 curves) that were fit to the 1000
experimental dispersion data realizations are shown alongside the experimental
dispersion data in Figure 13b. Note that the theoretical curves in Figure13b
have been colored in terms of their dispersion misfit relative to the original
experimental data, and not their respective realizations. Again, we observe
that the majority of the curves have a misfit less than 1.0 (i.e., are on
average within 1 standard deviation of the mean), with quite a few below 0.2
that closely follow the mean experimental dispersion trend. Importantly,
however, a number of curves with misfits greater than 1.0 are also present, as
should be expected if the dispersion uncertainty is accurately being
represented in the inversion results. This leads to a qualitative comparison
that the experimental dispersion data’s uncertainty and the uncertainty
implied by the suites of theoretical dispersion curves derived from the
inversion are in excellent agreement. Figure 14 presents a quantitative
comparison between the uncertainty of the experimental dispersion data and
inverted theoretical dispersion curves, highlighting four different “slices”
using vertical colored dashed lines at approximately 3, 6, 13, and 30 Hz. The
quantitative comparison confirms that the inversion-derived models are
accurately capturing the experimental dispersion uncertainty in terms of their
mean (Figure 14a), standard deviation (Figure 14d), and distribution shapes
(Figures 14b, c, e, and f). It is important to note that the minor
discrepancies between the measured and inverted data’s mean value and
uncertainty (refer to Figure 14a and 14d) are to be expected, because unlike
synthetic data, real experimental data may have some error in its mean and is
likely to have variable frequency-dependent uncertainty, making the exact
replication of some realizations difficult. For example, the dispersion COVs
at this site varied between 0.02 – 0.08, unlike the synthetic examples which
had constant dispersion COVs of 0.05. However, despite these complicating
factors the new procedure is able to produce results that fit the experimental
dispersion data remarkably well, with good agreement being observed between
the measured and inverted dispersion uncertainty across the entire bandwidth
of interest.

Figure 13: Dispersion data used to study the application of the newly proposed
procedure for propagating experimental dispersion uncertainty into the
inverted Vs profiles considering multiple parameterizations at a real site:
(a) raw experimental dispersion data with its binned and resampled statistical
representation, and (b) agreement between the experimental dispersion data and
1000 inverted theoretical dispersion curves fit to the 1000 realizations (4
acceptable parameterizations with N=250 realizations each) of the experimental
dispersion data. The theoretical dispersion curves have been colored according
to their dispersion misfit values relative to the original experimental
dispersion data, not their respective realizations. The vertical dashed lines
in panel (b) indicate the location of the “slices” shown in Figure 14. Figure
14: Quantitative assessment of the newly proposed procedure for propagating
experimental dispersion uncertainty into the inverted Vs profiles considering
multiple parameterizations at a real site. The comparison is made on the basis
of: (a) the residual mean ($\mu_{res}$) [i.e., the difference between the mean
of the inverted theoretical dispersion curves ($\mu_{inv}$) and the mean of
the measured experimental dispersion data ($\mu_{meas}$)] normalized by
$\mu_{meas}$ in percent, (d) the residual coefficient of variation [i.e., the
difference between the coefficient of variation of the inverted theoretical
dispersion curves ($\delta_{inv}$) and the coefficient of variation of the
measured experimental dispersion data ($\delta_{meas}$)]. The vertical dashed
lines in panels (a) and (d) at approximately 3, 6, 9, and 14 Hz denote the
locations of the “slices” shown in panels (b), (c), (e), and (f),
respectively. These “slices” compare the measured experimental dispersion
data’s distribution (solid black line) with the distribution of Rayleigh wave
velocity (Vr) derived from inversion (histogram).

Figure 15 shows the inversion-derived, uncertainty-consistent Vs profiles at
two depth scales; Figure 15a to a depth of 60 m and Figure 15b to a depth of
30 m. The Vs profiles reveal a three-layered system consistent with the
anticipated geologic conditions and comprised of approximately 15 m of soft
soil (Vs of approximately 200 m/s), overlying at least 35 m of weathered rock
(Vs of approximately 600 m/s), overlying stiffer material. The uncertainty in
the Vs profiles is quantified using $\sigma_{ln,Vs}$ and presented in Figures
15b and 15d at the same depth scales as those for Vs (i.e., Figures 15a and
15c). $\sigma_{ln,Vs}$ increases from close to zero for the surface layer,
where the high frequency/short wavelength surface-wave dispersion data had low
COVs, to a $\sigma_{ln,Vs}$ of 0.6 at depth, where due to the lack of very low
frequency/long wavelength data coupled with relatively high dispersion COVs
the velocity of the material cannot be accurately resolved. We consider the
depth where the Vs uncertainty becomes so large as to make meaningful
inferences about the site’s stiffness intractable as the maximum depth
limitation of the presented Vs profiles (i.e., a depth of approximately 45 m,
where $\sigma_{ln,Vs}$ starts to exceed about 0.2). To illustrate the accuracy
of the surface wave measurements, a P-S-suspension log from the site is shown
in comparison to the Vs profiles in Figures 15a and 15c. The P-S-suspension
log is seen to reside solely within the 95% CI of the inverted Vs profiles,
despite two complicating factors. First, the inversion-derived Vs profiles are
the result of measurements made over a 70 m by 70 m area with known spatial
variability, whereas the P-S-suspension log is more-or-less a point
measurement. And second, the P-S-suspension log was performed at the western
edge of the 70 m by 70 m site rather than at its center due to site
constraints. When these two factors are combined, one would expect some
differences between the invasive and non-invasive Vs profiles. Nonetheless,
the agreement is quite good and the Vs uncertainty associated with the surface
wave profiles has been robustly quantified for use in subsequent engineering
analyses, while that of the single P-S log is unknown and would have to be
assumed. This case study illustrates that the proposed procedure can
rigorously propagate experimental dispersion uncertainty through the surface
wave inversion process and produce suites of Vs profiles that, while
uncertain, can produce results that are consistent with invasive
characterization methods.

Figure 15: Uncertainty-consistent Vs profiles considering multiple inversion
parameterizations at a real site using the newly proposed procedure. Panels
(a) and (c), summarize the Vs profiles resulting from the inversion of the
1000 realized experimental dispersion curves (4 parameterizations with N=250
realizations each) at two separate depth scales. In order to better illustrate
their concentration, the profiles have been discretized in terms of depth and
Vs, binned into cells, and color mapped in terms of the number of profiles in
each cell. The profiles are shown alongside their discretized log-normal
median and 95% confidence interval (CI) and a P-S-suspension log from nearby.
Panels (b) and (c), show the lognormal standard deviation of Vs
($\sigma_{ln,Vs}$) of the 1000 profiles shown in panel (a) and (c),
respectively. The sharp spike in $\sigma_{ln,Vs}$ at approximately 15 m is due
to the uncertainty in the profile’s layer boundaries, reflecting a shortcoming
in how $\sigma_{ln,Vs}$ has been calculated historically, and is not the
result of actual uncertainty in Vs directly.

##  6 Conclusion

Surface wave methods are increasingly being preferred over traditional site
characterization methods for measuring a site’s shear-stiffness due to their
non-invasive nature. The quantification and propagation of uncertainty from
non-invasive measurements into Vs profiles for use in subsequent engineering
analyses has been the focus of much work in recent years, and while
significant progress has been made, no study has yet been able to show
quantitative evidence of the successful propagation of experimental dispersion
uncertainty into the resulting Vs profiles. Therefore, this study began by
examining methods currently available from the literature and showing their
deficiencies in three specific categories. First, current methods are shown to
be highly sensitive to their many user-defined inversion input parameters
(e.g., number of global-inversion trial models), making it
difficult/impossible for them to be performed repeatedly by different
analysts. Second, the suites of inverted Vs profiles derived from these
methods, when viewed in terms of their implied theoretical dispersion data,
are shown to systematically under-estimate the uncertainty present in the
experimental dispersion data, though they may appear satisfactory when viewed
purely qualitatively. And third, if the uncertainties in the implied
theoretical dispersion data were to be examined quantitatively, which has not
been done previously, there is no obvious remedy available to the analyst to
resolve any inconsistency between the measured and inverted dispersion
uncertainty. Therefore, a new approach has been proposed that seeks to remedy
these shortcomings. First, in addition to the necessary considerations for any
inversion, the method is governed by only one additional user-defined input
parameter (N), which controls the number of dispersion curve realizations. A
value of N=250 has been shown to perform well throughout this study, however,
as N is expected to be problem dependent the selection of an appropriate value
of N is left to the analyst. Second, the new method requires the quantitative
comparison between the measured and inverted dispersion uncertainties to
ensure suites of Vs profiles which quantitatively reproduce the uncertainties
in the experimental dispersion data. And third, should the measured and
inverted dispersion uncertainty be in disagreement, specific guidance is
provided on the actions necessary (e.g., increasing the value of N) to produce
Vs profiles that better account for the experimental uncertainty. The
application of the new procedure has been demonstrated using two synthetic
tests and a real-world example. Results show the procedure’s ability to
produce suites of Vs profiles which accurately capture the site’s Vs
structure, while rigorously propagating the measured, site-specific dispersion
data’s uncertainty through the inversion process, yielding uncertainty-
consistent Vs profiles that can be used in subsequent engineering analyses.

##  7 Acknowledgements

The first author would like to thank Professor Bob Gilbert for his course
Decision, Risk, and Reliability taught at The University of Texas at Austin
which inspired the basis of this study. The second author would like to thank
numerous colleagues, like Brendon Bradley, Cecile Cornou, Sebastiano Foti,
Albert Kottke, Tony Martin, Ellen Rathje, Adrian Rodriguez-Marek, Marc
Wathelet, and Liam Wotherspoon and former graduate students Trent Ellis, Shawn
Griffiths, David Teague, and Clint Wood who have exchanged thoughts on this
topic over the years. All of the inversion results presented in this paper
were produced using the Dinver module of the open-source software Geopsy
(Wathelet et al., 2020) which is freely available for download at
www.geopsy.org. All inversion pre- and post-processing was performed with the
assistance of the open-source Python package SWprepost (Vantassel, 2020).
Inversions were performed using the Texas Advanced Computing Center’s (TACCs)
cluster Stampede2 using an allocation provided through DesignSafe-CI (Rathje
et al., 2017). The figures in this paper were created using Matplotlib 3.1.2
(Hunter, 2007) and Inkscape 0.92.4.

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