The spatial crosscorrelation method for dispersive surface
waves
Journal: Geophysical Journal International
Manuscript ID: GJI-S-14-0247
Manuscript Type: Research Paper
Date Submitted by the Author: 26-Mar-2014
Complete List of Authors: Lamb, Andrew; USGS, van Wijk, Kasper; University of Auckland, Physics Liberty, Lee; Boise State University, Geosciences Mikesell, Thomas; Massachusetts Institute of Technology, Department of Earth, Atmospheric and Planetary Sciences
Keywords:
Surface waves and free oscillations < SEISMOLOGY, Controlled source
seismology < SEISMOLOGY, Hydrogeophysics < GENERAL SUBJECTS, Hydrothermal systems < GENERAL SUBJECTS
Geophysical Journal International
submitted to Geophys. J. Int.
The spatial crosscorrelation method for dispersive1
surface waves2
Andrew P. Lamb1, Kasper van Wijk2, Lee M. Liberty3, and T. Dylan Mikesell4
1 U.S. Geological Survey, 345 Middlefield Rd, Menlo Park, California 94025, USA
2 Department of Physics, University of Auckland, Auckland, New Zealand
3 Department of Geosciences, Boise State University, Boise, Idaho, USA
4 Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology,
Cambridge, Massachusetts, USA
3
Received 26 March 2014 in original form4
SUMMARY5
The vertical component of dispersive surface waves are routinely inverted to estimate the6
subsurface shear-wave velocity structure at all length scales. In the well-known SPatial7
AutoCorrelation (SPAC) method, dispersion information is gained from the correlation8
of seismic noise signals recorded on the vertical (or radial) components. We demonstrate9
practical advantages of including the crosscorrelation between radial and vertical com-10
ponents of the wavefield in a spatial crosscorrelation (SPaX) method. The addition of11
crosscorrelation information increases the resolution and robustness of the phase veloc-12
ity dispersion information, as demonstrated in numerical simulations and a near-surface13
field study with active seismic sources, where our method confirms the presence of a14
fault-zone conduit in a geothermal field.15
1 INTRODUCTION16
The dispersion of Rayleigh waves is commonly used to characterize the lithosphere (e.g.17
Bensen et al. 2007; Knopoff 1972), as well as the near-surface (e.g. Nazarian et al. 1983;18
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2 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
Park et al. 1999; Socco and Strobbia 2004). In a medium with a vertically heterogeneous19
velocity distribution, frequency-dependent Rayleigh waves will propagate at different phase20
velocities. Retrieving these dispersive properties allows us to invert for (shear) wave velocity21
as a function of depth.22
Dispersion information from the correlation of surface waves in Aki’s SPatial AutoCor-23
relation (SPAC) method (Aki 1957) was more recently exploited by (e.g., Ekstrom et al.24
2009; Stephenson et al. 2009). Tsai and Moschetti (2010) established the equivalence be-25
tween SPAC and the time-domain versions since made popular as “seismic interferometry”26
(e.g., Wapenaar and Fokkema 2006). In related work, the REfraction MIcrotremor method27
(REMI Louie 2001) maps the near-surface velocity distribution using the dispersion observed28
in crosscorrelations of signals from a combination of ambient noise and human activity. Until29
recently, analyses have been limited to the vertical or the radial components of the wavefield.30
However, van Wijk et al. (2011) explored crosscorrelations of the off-diagonal components31
(radial and vertical) of the Green tensor and found these to be more robust in the presence32
of uneven Rayleigh wave illumination on the receivers. This approach was followed by a for-33
mal extension of the SPAC method to the complete Green tensor by Haney et al. (2012). In34
addition to enhancement of the surface wave dispersion information, these additional cross-35
correlations can be used to enhance the body wave arrivals (van Wijk et al. 2010; Takagi36
et al. 2014)37
Here we explore extending SPAC to include SPatial crosscorrelation (SPaX) of radial38
and vertical components. In SPaX, we correlate all combinations of the vertical and radial39
wavefield components to extract more robust estimates of the dispersive Rayleigh wave.40
The additional components increase accuracy and resolution of the surface-wave dispersion41
information. Numerical modeling of the full wavefield illustrates the advantages, and we42
demonstrate this with an example of active-source seismic data from a geothermal field site43
at Mount Princeton Hot Springs, Colorado.44
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The spatial crosscorrelation method for dispersive surface waves 3
2 THEORY45
Consider a Rayleigh plane wave traveling with a phase velocity c along the surface in the
positive x-direction of a homogeneous two-dimensional (2D, x, z) elastic half-space. For a
source at position xs along the surface, the displacement wavefield recorded at surface posi-
tion x is Uz(xs, x, t), where subscript z represents the vertical component and t is time. At
another surface location x′ > x we record Uz(xs, x′, t). These two wavefields are identical to
each other, except for the time delay between the receivers that is (x′− x)/c. It is therefore
quite intuitive that the crosscorrelation of the two wavefields from an impulsive source at xs
is
Uz(xs, x′, t)⊗ Uz(x
s, x, t) = δ (t− r/c) (1)
where ⊗ denotes the crosscorrelation and r = |x′ − x| is the inter-receiver distance. In46
principle, if the two receivers are bounded by a source on each side, the correlation of the47
wavefields results in the causal and acausal component of the Green’s function (δ (t− r/c)48
and δ (t+ r/c)).49
The SPAC method explores this same principle, but in the frequency domain. The real
part of the Fourier transform of the retarded Dirac delta function is
φzz(r, ω) = <[F(δ(t− r/c))] = cos(ωr/c), (2)
where φzz was referred to by Aki (1957) as the SPAC coefficient for the vertical component
correlation. By crosscorrelating all four combinations of vertical Uz(x, t) and radial Ux(x, t)
wavefield recordings, we create the four SPAC coefficients
φ(r, ω) =
φzz φzx
φxz φxx
, (3)
where φzz and φxx are the Spatial AutoCorrelation (SPAC) coefficients, and φzx and φxz are
the spatial crosscorrelation coefficients. Equation (26) of Haney et al. (2012) summarizes the
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4 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
1D derivation of all four components of the frequency domain Green tensor
φ(r, ω) = P (ω)
cos(ωr/c) −R sin(|ω|r/c)
R sin(|ω|r/c) R2 cos(ωr/c)
, (4)
where R is the ratio of the horizontal-to-vertical displacement of the Rayleigh waves and50
P (ω) is the power spectrum of the Rayleigh waves.51
Ekstrom et al. (2009) use the roots of the real part of Aki’s SPAC coefficient φzz(ω) to
estimate the dispersion relationship of Rayleigh waves. For the spatial autocorrelation terms
in Equation 4, this means that ωr/c = nπ/2 and we find the phase velocity relationship
czz(ωn) = cxx(ωn) =ωnr
nπ/2, (5)
where ωn represents the nth root of the cosine function. Here, we add the information about
the phase velocity estimates by correlating all combinations of the wavefields, including the
cross-correlations of vertical and radial components:
czx(ωn) = cxz(ωn) =ωnr
nπ. (6)
In the following, we illustrate some of the strengths and outstanding challenges of the SPaX52
technique with numerical examples, before ending with a near-surface field example from a53
vibroseis survey.54
3 SPATIAL CROSSCORRELATION IN A HOMOGENEOUS SLAB55
The previous section outlined the extraction of the Rayleigh-wave dispersion curve. In reality,56
interference from other wave modes contaminate the Rayleigh-wave information. In a slab57
geometry, body waves reflect from the bottom of the slab. In this section we use elastic-wave58
numerical modeling to investigate degradation to surface-wave dispersion curves caused by59
interfering body waves.60
The parameters of the numerical experiment in a slab are displayed in Figure 1. The slab61
is 50 m thick with P- and S-wave velocities of 900 and 500 m/s, respectively. The Rayleigh-62
wave velocity is 462 m/s. We simulate wave propagation for 11 source positions (black stars)63
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The spatial crosscorrelation method for dispersive surface waves 5
having distances to the first receiver (inverted open triangles) ranging from 10 to 110 m at a64
10 m interval. The Ricker source wavelet has a dominant frequency of 30 Hz. We model the65
radial and vertical components of the wavefield using SPECFEM2D at 60 receiver positions.66
The surface receiver spacing is 2 m and the sample rate is 2000 Hz. SPECFEM2D is a high-67
order variational numerical technique (Priolo et al. 1994; Faccioli et al. 1997) and is widely68
used in application ranging from seismology (Komatitsch and Vilotte 1998; Komatitsch and69
Tromp 1999, 2002; Komatitsch et al. 2002) to ultrasonics (van Wijk et al. 2004).70
[Figure 1 about here.]71
3.1 Vertical component wavefield correlations72
The left panel of Figure 2 is the vertical component of the wavefield on all 60 receivers73
for source 11. The Rayleigh wave with a linear move-out has the largest amplitudes, but74
body-wave reflections intersect the Rayleigh waves. These interfering body waves complicate75
methods based on the isolation of Rayleigh waves in the shot record. In the SPAC method,76
averaging over long recording times ensure a distribution of source locations. Analogous77
to the SPAC method, we illustrate the advantage of using multiple sources with varying78
distances from the receiver array. The middle panel of Figure 2 is the correlation of the79
vertical component of the wavefields at receivers 20 and 40, for sources 1 to 10. Note that80
the correlation between Rayleigh waves is stationary (t ∼ 0.08 s), but the correlated energy81
related to the body-wave reflections varies for each source location. Summing the cross-82
correlated wavefields over the sources results in constructive interference of Rayleigh waves83
and destructive interference of all other wave modes. The top trace (sum) shows the average84
of the 10 lower traces. The crosscorrelation of all receivers with receiver 20, summed over85
all sources leads to the virtual shot record shown in the right panel of Figure 2.86
[Figure 2 about here.]87
In the virtual shot record, the body waves have almost disappeared completely and a88
strong Rayleigh wave is present. To demonstrate how multiple source summation improves89
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6 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
the SPAC coefficients, we show φzz(r, ω) in Figure 3, where r = 40 m as receiver 20 is90
correlated with receiver 40. As observed in the time domain (Figure 2 middle), summing91
wavefield correlations from multiple sources stabilizes the estimate of φzz. The differences92
between the single source correlation and the summed version highlights the influence of93
body-wave interference, which can lead to biased phase velocity estimates. Note that the94
band-limited nature of our numerical modeling decreases the signals below 10 Hz (and above95
70 Hz). We discuss the implications of this in a later section, but first we will consider the96
advantage of adding spatial crosscorrelations to the autocorrelation results.97
[Figure 3 about here.]98
3.2 Multicomponent wavefield correlations99
Analogous to the zz-component of the SPAC method, in Figure 4 we plot the spatial autocor-100
relation (xx and zz components) and crosscorrelation (xz and zx components) coefficients101
for receivers 20 and 40 after summing over shots 1 to 10. These components behave as the102
harmonic functions predicted by equation 4, and we note the π/2 phase shift between di-103
agonal and off-diagonal components, as well as the anti-symmetry between the off-diagonal104
terms.105
[Figure 4 about here.]106
From the roots of the harmonic functions in Figure 4, we estimate the phase-velocity107
curves shown in Figure 5. The top panel is the phase-velocity estimate from a single source,108
whereas the bottom panel contains the estimates after source summation. The difference109
between φzz and the other three SPAC coefficients is due to the root number n in φzz being110
misidentified as a result of the body wave interference. Estimates of the phase velocity from111
the other terms are centered about the correct Rayleigh velocity of 462 m/s with a standard112
deviation of 15 m/s.113
The curves from the summed sources in the bottom panel of Figure 5 show significantly114
less variability. The summing over sources resulted in the correct root n identification, due115
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The spatial crosscorrelation method for dispersive surface waves 7
to the suppression of body-wave correlations. The first root(s) of equations 5 and 6 are most116
susceptible to error. In our slab model, the finite thickness affects the Rayleigh waves at117
long wavelengths, and the source wavelet has limited power at these low frequencies. The118
latter is also the reason why it is common practice that the first root is not used (Ekstrom119
et al. 2009; Tsai and Moschetti 2010).120
[Figure 5 about here.]121
To quantify the improvements of using all four SPaX coefficients, Figure 6 contains the122
mean (solid diamond) and standard deviation of the Rayleigh-wave velocity estimate for123
each source position, and for the sum of all sources on the very right (open diamond). At124
certain individual source positions, the variance due to body-wave interference with the125
Rayleigh-wave signal is large. The variance for the summed result is reduced considerably,126
demonstrating the improved velocity estimation using all possible Rayleigh-wave informa-127
tion.128
[Figure 6 about here.]129
3.3 Missing roots130
When signals lack energy in the low-end of the frequency spectrum, we miss the first root(s)
of equations 5 and 6. To accurately estimate dispersion curves we need an estimate of how
many roots of the harmonic functions are missing. In our estimation, we follow a similar
procedure to Ekstrom et al. (2009) and calculate a series of phase velocity dispersion curves
cm that are based on adaptations of equations 5 and 6. For the diagonal terms we write
cmzz,xx(ωn) =ωnr
(n+m)π/2, (7)
and for the off-diagonal terms we write
cmzx,xz(ωn) =ωnr
(n+m)π, (8)
where m represents the number of missed roots incremented in multiples of 2.131
The SPaX method uses all four correlation coefficients to estimate the missed roots and132
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8 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
thereby improve our estimation of the dispersion curves. In practice, we estimate missed zero133
crossings by knowing that: (1) the roots of the off-diagonal and diagonal terms follow a sine134
and cosine function, respectively; (2) the phase velocities at our minimum and maximum135
frequencies must fall within certain physical limits constrained by prior information (Ekstrom136
et al. 2009); and (3) the dispersion curves for all four SPAC coefficients are jointly interpreted137
to minimize error.138
To demonstrate our approach we again turn to numerical modeling. Instead of a slab139
model, we return to a homogeneous half-space model so that body waves do not exists. The140
model has a Rayleigh-wave phase velocity of c = 500 m/s. We place two receivers at the141
surface, separated by a distance r. In this experiment, we use a single source location, but142
repeat the experiment twice. The first experiment uses a sinusoidal chirp sweep from 30 to143
150 Hz and the second use the same sweep but from 0 to 150 Hz. In this way, we illustrate144
the influence of the missing roots and compare with the correct data.145
For each source, we crosscorrelate all receiver combinations and compute the SPaX coef-146
ficients (Figure 7). We see that for φzz and φxx in the partial-band case (30 - 150 Hz), there147
are three missing zero crossings while for φzx and φxz there are four missing zero crossings.148
The complete-band case (0 - 150 Hz) shows all of the zero crossings – indicated by the149
partially transparent data.150
[Figure 7 about here.]151
We should therefore use m = 3 in equation 7 for the diagonal terms and m = 4 in152
equation 8 for the off-diagonal terms to estimate the correct phase-velocity curves. To see153
the influence of using the incorrect m value, we plot a variety of dispersion curves in Fig-154
ure 8. Figure 7 shows how the slopes of the zero crossings vary between components. In this155
example, the slope is positive for the first crossing of φxz and negative for the other three156
coefficients φzz, φzx, and φxx. When m follows an odd number series (e.g. ±1, ±3, ±5,...),157
this means the first available zero crossing in our data occurs at a slope of φ that is opposite158
in direction to the slope of φ at the first missed root.159
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The spatial crosscorrelation method for dispersive surface waves 9
[Figure 8 about here.]160
We note that the variability between the different dispersion curves for m = ±2 becomes161
less with increasing receiver separation distance r because the larger separation causes the162
signals at each receiver to have more skipped phases. Increasing receiver separation therefore163
makes estimating the number of missed zero crossings more difficult in partial-band data164
because there may be a range of m values that produce realistic dispersion curves. This165
is best overcome by reducing r until two receivers are chosen that give dispersion curves166
with sufficient variability that enables the correct number of missed zero crossings to be167
estimated. The trade-off with selecting a smaller value of r is that we will have less zero168
crossings for the same frequency range thereby giving less resolution.169
Finally, estimating m is relatively straightforward in the absence of dispersion; however,170
it becomes more difficult when the velocity changes with frequency. Furthermore, it is in-171
creasingly difficult to estimate the number of missed roots when the range of the omitted172
frequency band increases, thereby incorporating an increasing number of missed roots of φ.173
To understand the affect of dispersion, we investigate a numerical model with dispersion in174
the next section.175
4 SPATIAL CROSSCORRELATION IN DISPERSIVE MEDIA176
We illustrate the benefits of including the off-diagonal SPaX coefficients in a two-layer nu-177
merical model, where the Rayleigh wave velocity changes from 480 m/s to 600 m/s at a depth178
of 12 m (Figure 9). Again we use SPECFEM2D to simulate wavefields and sum correlations179
from 12 source positions with offsets to the first receiver ranging from 10 to 120 m, at 10 m180
increments. An approximation to the Dirac delta source provides broad-band signals, and181
the recorded wavefields are low-pass (150 Hz) filtered to ensure numerical stability. Sixty182
receivers with 2 m spacing record the wavefield at a sample rate of 2000 Hz.183
[Figure 9 about here.]184
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10 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
The shot records and related correlation function estimates for this example are presented185
in Lamb (2013). The correlation coefficients for a receiver separation of r=40 m are shown186
in Figure 10. The black-solid line is the result of summing all four SPaX coefficients, after187
the crosscorrelation terms φxz and φzx have been phase rotated by 90°. The phase-velocity188
dispersion curves calculated from these correlation coefficients are plotted in Figure 11. The189
dispersion curve of the combination of the four SPaX coefficients matches the correct phase190
velocities best.191
[Figure 10 about here.]192
[Figure 11 about here.]193
The curves in Figure 11 show that the phase velocity decreases with increasing frequency194
from ∼ 600 m/s to 480 m/s. The slower layer of the model is predominantly sampled by195
the higher frequency Rayleigh waves – 48 Hz and above where the maximum Rayleigh196
wavelength is 12 m. Furthermore, these frequencies propagate at a slow enough velocity197
that the body waves do not interfere in time and space. Thus, there is a negligible difference198
in the dispersion curves above 48 Hz. In contrast, below 48 Hz the Rayleigh-wave phase199
velocity increases and body waves interfere. Thus, there is some variability (e.g. at 18 Hz)200
in the dispersion curves. Below 12 Hz the dispersion curves diverge more strongly, because201
the physical dimensions of the numerical model are less than the Rayleigh wavelengths at202
these low frequencies.203
5 A FIELD APPLICATION204
The Upper Arkansas Valley in the Rocky Mountains of central Colorado is the northernmost205
extensional basin of the Rio Grande Rift (Figure 12). The valley is a half-graben bordered to206
the east and west by the Mosquito and Sawatch Ranges, respectively (McCalpin and Shannon207
2005). The Sawatch Range is home to the Collegiate Peaks, which include some of the highest208
summits in the Rocky Mountains. Some of the Collegiate Peaks over 4,250 m (14,000 ft)209
from north to south include Mt. Harvard, Mt. Yale, Mt. Princeton and Mt. Antero. The210
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The spatial crosscorrelation method for dispersive surface waves 11
Sawatch range-front normal fault strikes north-northwest along the eastern margin of the211
Collegiate Peaks and is characterized by a right-lateral offset between the Mount Princeton212
batholith and Mount Antero. This offset in basin bounding faults is accommodated by a213
northeast-southwest dextral strike-slip transfer fault (Richards et al. 2010) and coincides214
with an area of hydrogeothermal activity including Mount Princeton Hot Springs. This215
transfer fault is here termed the ’Chalk Creek fault’ due to its alignment with the Chalk216
Creek valley. A 250 m high erosional scarp, called the Chalk Cliffs, lies along the northern217
boundary of this valley. The cliffs are comprised of geothermally altered quartz monzonite218
(Miller 1999). These cliffs coincide with the Chalk Creek fault, whose intersection with the219
Sawatch range-front normal fault results in a significant pathway for upwelling geothermal220
waters.221
The Deadhorse lake (DHL) site in Chalk Creek Valley (Figure 12) coincides with a north-222
south trending boundary (dashed-dotted line in Figure 13) between hot- and cold-water wells223
to the west and east, respectively. The site is characterized by 10 to 50 m deep glacial, fluvial224
and alluvial deposits overlying a quartz monzonite and granite basement. Deadhorse lake is225
empty for most of the year. We conducted a gravity survey using a Scintrex CG-5 gravimeter226
on a 50 m grid; the data have been corrected for drift, latitude, free-air, Bouguer, and terrain.227
The reduced gravity data are shown in the color overlay in Figure 13. The gravity data show228
an eastward dip in the underlying bedrock with a difference of approximately 4 mGal across229
the survey area (∼600 m) in the west-east direction. We also conducted a vertical seismic230
profile (VSP) survey in the 168 m deep MG-1 borehole (Figure 13) to constrain the seismic231
interval velocities and to determine the depth to competent granite (Lamb 2013).232
[Figure 12 about here.]233
[Figure 13 about here.]234
The location of the active seismic experiment is indicated in Figure 13 – coincident dotted235
and solid black lines. We acquired multicomponent seismic data using a 2720 kg Industrial236
Vehicles T-15000 vibroseis source along a dirt track in an east-northeasterly direction. The237
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12 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
seismic line extends from station 1 to 236, and the station interval is 2 m, giving a total line238
length of 472 m. The vibroseis data were recorded using a 120 channel Geometrics recording239
system and 10-Hz three component geophones. The vibroseis source was a 14 second linear240
sweep from 30-300 Hz.241
Nine component data were acquired between stations 115 and 236. We apply the SPaX242
analysis to the stations between 115 and 220 (solid black line in Figure 13) where the243
acquisition geometry is more or less linear. In the case that the sweep trace is not perfectly244
known (or recorded), correlating the data can introduce errors. Fortunately, one of the245
advantages of the SPaX method is that we can use the uncorrelated vibroseis data. Lamb246
(2013) provide examples of the benefits of using uncorrelated sweep data by comparing247
the SPaX results from correlated and uncorrelated seismic data when the sweep trace is248
incorrect. The examples demonstrate that using correlated versus uncorrelated sweeps can249
cause up to a 10% variation in phase-velocity dispersion curve estimates for zero crossings250
near 30 Hz. The field data results presented in the remainder of this paper are all based251
upon uncorrelated vibroseis data.252
5.1 Dispersion curves from spatial crosscorrelation253
The shot records and correlation coefficients for this field example are presented in Lamb254
(2013), which also describes a 32-layer numerical model that we use to estimate the correct255
number of missed zero crossings (m) in the DHL seismic data. The 32-layer numerical model256
is parametrized and refined through iterative forward modeling to find a satisfactory fit257
between the numerical and field data. The data considered include shot gathers, SPaX-258
derived dispersion curves, a refraction-tomography velocity model and the 32-layer numerical259
velocity model. We simplify the iteration process by selecting a receiver separation that yields260
a single dispersion curve that represents the average field-dispersion curve. We ensure that261
this separation does not cause unrealistic dispersion curves for those values of m that over262
or under estimate the missed zero crossings by a factor of 2.263
Figure 14 shows the phase-velocity dispersion curves at an offset of 30 m in the virtual264
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The spatial crosscorrelation method for dispersive surface waves 13
shot gathers. Curves A through D represent dispersion curves in which we have added265
missed zero crossings (m). The zero crossings are selected such that the direction of the first266
zero crossing matches the theoretical slope of the cosine and sine functions. The numerical267
dispersion curve (gray line) from the 32-layer model is shown to help determine the number268
of missed zero crossings in the field data; the numerical data are full-band (i.e., 0-300 Hz).269
Because the vibroseis sweep did not cover the frequency range 0-30 Hz, we see that the270
number of missed zero crossings should be mzz,zx,xx = 5 and mxz = 6. With the number of271
missed zero crossings estimated, we can consider the variability in dispersion curves related272
to the lateral geologic heterogeneity along the DHL seismic line.273
[Figure 14 about here.]274
5.2 Lateral variations275
To develop an understanding of the lateral variability along the DHL seismic line, we plot the276
SPaX coefficient gathers in Figure 15. We show two cases – (lower left) (φxx) and (lower right)277
(φzz+φxz+φzx+φxx). The lower left panel in Figure 15 shows φxx(r = 30, ω) for receivers 168278
to 213. The traces are plotted at the midpoint between the two receivers and the red dashed279
line follows the 7th zero crossing for each curve. This line provides a qualitative measure280
of lateral variability because it does not represent velocity at a specific depth. Equations 5281
and 6 help to intuitively understand these variations, where the frequency (ωn) and phase282
velocity (cn) are linearly proportional to each other. A trend in the red line toward lower283
frequency near midpoint 187 represents a lower seismic wavespeed locally. The blue dashed284
line in the upper panel shows the corresponding phase velocity at the 7thzero crossing. The285
phase velocity profile generally decreases from midpoint 187 to 212 by approximately 60 m/s.286
[Figure 15 about here.]287
The bottom right panel in Figure 15 shows the combined SPaX coefficients for the288
same receivers as in bottom left plot. Comparison of the SPAC and SPaX gathers in the289
bottom panels demonstrate the improvement in signal coherency from combining all four290
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14 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
components. In particular, the stations from 168 to 194 show less variability in the SPaX291
coefficients. Furthermore, the averaging process fills in some of the missing channels in the292
SPAC gather such as midpoints 170, 184, 191, and 205. There is also an improvement in293
the signal-to-noise ratio for the zero crossings in the frequency range of 41 to 47 Hz and294
midpoints 168 to 184.295
Refraction tomography analysis along the entire seismic line was performed on the ver-296
tical component of the seismic data. The geologic model interpreted from tomography and297
the SPaX method is show in Figure 16. The phase velocity slowdown at midpoint 200 is also298
observed in the refraction tomography results (Lamb 2013) and supported by the decrease299
in gravity to the east (Figure 13). Therefore, the SPaX coefficients as presented in Figure 15300
demonstrate the ability to identify potential lateral lithology variations. The loss in coher-301
ence of signal beyond midpoint 190 may be a function of geometric spreading of the seismic302
energy, but given prior information from the other geophysical data discussed, we attribute303
the coherency loss to deformation associated with the N-S ramp fault shown in Figure 13.304
[Figure 16 about here.]305
The midpoint gather of SPaX coefficient curves is a useful tool to quickly assess spa-306
tial changes in lateral velocity. Different receiver separations r can be used to control the307
number of SPaX-coefficient gather zero crossings in a specific frequency range. For instance,308
increasing r will cause more phase changes between two receiver pairs and therefore a cor-309
responding increase in the zero crossings. Even though a larger r adds a higher density of310
zero crossings for a given frequency range, this can mask lateral heterogeneity because the311
wavefield is averaged over more of the structure with increasing r. Therefore, increases in312
r make it more difficult to determine the correct number of zeros to be added because the313
sensitivity of the dispersion curves to changes in m is reduced.314
The joint geologic interpretation of all of our geophysical techniques for the Deadhorse315
lake field site fit a four-layer model that, with increasing depth, is composed of unsaturated316
sediments, saturated sediments, altered quartz monzonite and competent granite. The in-317
terpretation of these layers, along with further detail about all the analyses performed, are318
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The spatial crosscorrelation method for dispersive surface waves 15
described in Lamb (2013). The seismic and gravity data show evidence for the N-S ramp319
fault (Miller 1999) between Mount Antero and Mount Princeton near station 200 on the320
seismic line. The SPaX results provide further evidence for the N-S ramp fault commencing321
from midpoint 200 eastwards. This fault may well be a pathway for hot geothermal waters,322
but requires further geophysical investigations along its strike through Chalk Creek valley.323
6 CONCLUSION324
The extension of the SPatial AutoCorrelation (SPAC) method to include crosscorrelations325
of the wavefield (SPaX) results in dispersion curves that are more robust and higher in326
resolution. The increased robustness is a function of the added data, and the resolution327
improvements stem from estimating phase velocities at additional frequencies in SPaX. As328
in the SPAC method, prior information is needed for the SPaX method when the lowest329
frequencies are not present in the data. In the field, there are practical advantages to include330
crosscorrelations of the radial and vertical components of the wavefield to characterize near-331
surface velocities. In the case of vibroseis sources, there is no need for (potentially damaging)332
correlations with the sweep, and SPaX coefficients can be used directly to identify lateral333
variations in the subsurface.334
ACKNOWLEDGMENTS335
All data used in this study is freely available upon request from the authors. Funding for336
this research was provided by the Department of Energy award G018195 and the National337
Science Foundation award EAR-1142154. Additional funding for the vibroseis was provided338
by the NSF. We would like to thank the local community and land owners for giving us339
access to their properties and particularly Bill Moore and Stephen Long. We also thank,340
Chaffee County, the SEG Foundation, ION Geophysical, field camp students and other local341
stakeholders for their support. The seismic and gravity data were acquired from 2008 to342
2010 as part of the spring geophysical field camp attended by students from the Colorado343
School of Mines, Boise State University and Imperial College London. Other individuals344
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16 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
who contributed to this work include Mike Batzle, Thomas Blum, Justin Ball, Seth Haines,345
Andre Revil, Kyle Richards, and Fred Henderson.346
REFERENCES347
Aki, K. (1957), Space and time spectra of stationary stochastic waves, with special reference to348
microtremors, Bull. Earthq. Res. Inst., 35, 415–456.349
Bensen, G. D., M. H. Ritzwoller, M. P. Barmin, A. L. Levshin, F. Lin, M. P. Moschetti, N. M.350
Shapiro, and Y. Yang (2007), Processing seismic ambient noise data to obtain reliable broad-band351
surface wave dispersion measurements, Geophysical Journal International, 169 (3), 1239–1260.352
Colman, S. M., J. P. McCalpin, J. P. Ostenaa, and D. A. Kirkham (1985), Map showing upper353
cenozoic rocks and deposits and quaternary faults, rio grande rift, south-central colorado, U.S.354
Geological Survey Miscellaneous Investigations Map I-1594, u.S. Geological Survey, Denver, CO.355
Ekstrom, G., G. A. Abers, and S. C. Webb (2009), Determination of surface-wave phase veloc-356
ities across USArray from noise and Aki’s spectral formulation, Geophysical Research Letters,357
36 (L18301), 1–5.358
Faccioli, E., F. Maggio, R. Paolucci, and A. Quarteroni (1997), 2D and 3D elastic wave propagation359
by a pseudo-spectral domain decomposition method, Journal of Seismology, 1, 237–251.360
Haney, M. M., T. D. Mikesell, K. van Wijk, and H. Nakahara (2012), Extension of the spatial361
autocorrelation (SPAC) method to mixed-component correlations of surface waves, Geophysical362
Journal International, 191 (1), 189–206.363
Knopoff, L. (1972), Observation and inversion of surface-wave dispersion, Tectonophysics, 13 (1),364
497–519.365
Komatitsch, D., and J. Tromp (1999), Introduction to the spectral-element method for 3-D seismic366
wave propagation, Geophysical J, International, 139, 806–822.367
Komatitsch, D., and J. Tromp (2002), Spectral-element simulations of global seismic wave368
propagation-I. Validation, Geophysical Journal International, 150, 390–412.369
Komatitsch, D., and J. P. Vilotte (1998), The spectral-element method: an efficient tool to simulate370
the seismic response of 2D and 3D geological structures, Bulletin of the Seismological Society of371
America, 88, 368–392.372
Komatitsch, D., J. Ritsema, and J. Tromp (2002), The spectral-element method, Beowulf com-373
puting, and global seismology, Science, 298, 1737–1742.374
Lamb, A. P. (2013), Geophysical investigations of the seattle fault zone in western washington375
and a geothermal system at mount princeton, colorado, Ph.D. thesis, Boise State University,376
Geosciences Department, 1910 University Dr., Boise, ID 83725.377
Page 16 of 36Geophysical Journal International
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The spatial crosscorrelation method for dispersive surface waves 17
Louie, J. N. (2001), Faster, better: shear-wave velocity to 100 meters depth from refraction mi-378
crotremor arrays, Bulletin of the Seismological society of America, 91 (2), 347–364.379
McCalpin, J. P., and J. R. Shannon (2005), Geologic map of the buena vista west quadrangle,380
chaffee county, colorado, Open-File Report 05-8, Colorado Geological Survey, Department of381
Natural Resources, Denver, CO.382
Miller, M. G. (1999), Active breaching of a geometric segment boundary in the sawatch range383
normal fault, colorado, usa, Journal Of Structural Geology, 21 (7), 769–776.384
Nazarian, S., K. Stokoe II, H, and W. R. Hudson (1983), Use of spectral-analysis-of-surface-385
waves method for determination of moduli and thickness of pavement systems, Tech. Rep. 930,386
Transportation Research Record.387
Park, C. B., R. D. Miller, and J. Xia (1999), Multichannel analysis of surface waves, Geophysics,388
64 (3), 800–808, doi:10.1190/1.1444590.389
Priolo, E., J. M. Carcione, and G. Seriani (1994), Numerical simulations of interface waves by390
high-order spectral modeling techniques, Journal of the Acoustical Society of America, 95, 681–391
693.392
Richards, K., A. Revil, A. Jardani, F. Henderson, M. Batzle, and A. Haas (2010), Pat-393
tern of shallow ground water flow at mount princeton hot springs, colorado, using geoelec-394
trical methods, Journal Of Volcanology And Geothermal Research, 198 (1-2), 217–232, doi:395
10.1016/j.jvolgeores.2010.09.001.396
Scott, G. R., R. E. Van Alstine, and W. N. Sharp (1975), Geologic map of the poncha springs397
quadrangle, chaffee county, colorado, U.S. Geological Survey Miscellaneous Field Studies Map398
MF-658, u.S. Geological Survey, Denver, CO.399
Socco, L. V., and C. Strobbia (2004), Surface-wave method for near-surface characterization: a400
tutorial, Near Surface Geophysics, 2 (4), 165–185, doi:10.3997/1873-0604.2004015.401
Stephenson, W., S. Hartzell, A. Frankel, M. Asten, D. Carver, and W. Kim (2009), Site charac-402
terization for urban seismic hazards in lower manhattan, new york city, from microtremor array403
analysis, Geophysical Research Letters, 36 (L03301), 1–5.404
Takagi, R., H. Nakahara, T. Kono, and T. Okada (2014), Separating body and rayleigh waves with405
cross terms of the cross-correlation tensor of ambient noise, Journal of Geophysical Research:406
Solid Earth, pp. n/a–n/a, doi:10.1002/2013JB010824.407
Tsai, V. C., and M. P. Moschetti (2010), An explicit relationship between time-domain noise cor-408
relation and spatial autocorrelation (SPAC) results, Geophysical Journal International, 182 (1),409
454–460.410
van Wijk, K., D. Komatitsch, J. A. Scales, and J. Tromp (2004), Analysis of strong scattering at411
the micro-scale, Journal of the Acoustical Society of America, 115.412
Page 17 of 36 Geophysical Journal International
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18 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
van Wijk, K., D. Mikesell, T. Blum, M. Haney, and A. Calvert (2010), Surface wave isolation with413
the interferometric Green tensor, in SEG Technical Program Expanded Abstracts, pp. 3996–4000,414
doi:10.1190/1.3513692.415
van Wijk, K., T. D. Mikesell, V. Schulte-Pelkum, and J. Stachnik (2011), Estimating the Rayleigh-416
wave impulse response between seismic stations with the cross terms of the Green tensor, Geo-417
physical Research Letters, 38 (L16301), 1–4.418
Wapenaar, K., and J. Fokkema (2006), Green’s function representations for seismic interferometry,419
Geophysics, 71 (4), SI33–SI46, doi:10.1190/1.2213955.420
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The spatial crosscorrelation method for dispersive surface waves 19
LIST OF FIGURES421
1 A schematic of the 50-m thick slab model with sources and receivers at the422
surface.423
2 Left: the vertical component of the wavefield in a slab from single shot, Uz424
at source position 6. Middle: the Green’s functions between receivers 20 and 40425
for 10 separate shots, Gzz. The top trace shows the results of summing these 10426
shots. Right: the virtual shot record for all 60 receivers, with the virtual source at427
receiver 20.428
3 The real part of the Fourier transform of the cross-correlation of the vertical429
components for a single shot at source position 6 and for the sum of 10 sources.430
4 SPaX coefficients φzz, φzx, φxz, and φxx from the 10 summed shots.431
5 The roots of the real part of the Fourier transform of the cross-correlations432
between receivers 20 and 40 for all four components for a single shot at source433
position 6 (top) and for 10 summed shots (bottom). Both the top and bottom434
panels have the same vertical scale and color relationship between φ terms.435
6 The mean phase velocity from all four SPaX coefficients with the correspond-436
ing standard deviation. Each point has been averaged over all frequencies before437
averaging the individual SPaX components. Solid diamonds represent single source438
locations, while the open diamond shows the results after summing over 10 sources.439
7 The semi-transparent (0-150 Hz) and full-color (30-150 Hz) SPaX coefficients440
from a homogeneous half-space model.441
8 The semi-transparent and full color dispersion curves cxx and cxz are for full-442
band (0 to 150 Hz) and partial-band (30 to 150 Hz) data respectively. The receiver443
separation is 25 m. The effect of adding up to 6 missing zero crossings (m=6) on444
the partial-band data is shown with mxx = 3 and mxz = 4 resulting in a match445
between the partial-band and full-band dispersion curves.446
9 A schematic of the two-layer numerical model. Because of the two layers, the447
Rayleigh wave velocity is dispersive.448
10 The individual SPaX coefficients and the combined SPaX coefficient after449
summing the 12 shots for a receiver separation of 40 m.450
11 Phase-velocity dispersion curves from all four individual SPaX coefficients451
between receivers separated by 40 m. The solid-black line shows the result of av-452
eraging all four SPaX coefficients to calculate phase velocity c.453
12 Top Right: Colorado state map showing the approximate locations of Chaffee454
County, Mount Princeton, and Denver. Bottom Right: A topographic map of the455
Upper Arkansas basin in Chaffee County superimposed with the study ’DHL’ site.456
Left: Location of faults mapped along the Sawatch Range according to work by457
Scott et al. (1975), Colman et al. (1985), and Miller (1999).458
13 The Dead Horse Lake (DHL) study site showing water wells and their temper-459
atures, the location of the seismic line, and overlain with the results of the gravity460
survey. The well labeled MG-1 was used to conduct a 168 m deep vertical seismic461
profile for which the results are presented in Lamb (2013).462
14 Phase-velocity dispersion curves for field data where r = 30 m. The semi-463
transparent and full-color dispersion curves are for full-band (numerical) and partial-464
band (field) data, respectively. The full-band curve E is based upon a 32-layer465
numerical model. Because the vibroseis sweep did not cover the frequency range466
0-30 Hz, we see that the number of missed zero crossings should be mzz,zx,xx = 5467
and mxz = 6.468
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20 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
15 Lower left: A gather of SPAC coefficient traces for φxx(r = 30, ω) for the field469
data from station 168 to 213. The dashed red line shows the 7th zero crossings. The470
phase velocity corresponding to the 7th zero crossings is shown in the top panel471
and indicates laterally changing velocity. Bottom right: A gather of the average472
correlation coefficients. Top right: The phase velocity corresponding to 7th zero473
crossing of the combined SPaX coefficients.474
16 A geologic interpretation of the SPaX result and refraction tomography ve-475
locities constrained by the MG-1 well (Lamb 2013).476
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The spatial crosscorrelation method for dispersive surface waves 21
Receiver60
Rayleigh wave velocity (c) = 462 m/sP-wave velocity (Vp) = 900 m/sS-wave velocity (Vs) = 500 m/s
50 m
120 m
Receiver1
100 m 10 m
¬¬ ¬ ¬ ¬ ¬ ¬ ¬ ¬ ¬ ¬
Source1 11 20 40
Figure 1. A schematic of the 50-m thick slab model with sources and receivers at the surface.
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22 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
evaw P detcelfeR
Rayleigh wave
Multiples
Tim
e (s
)
Gzz between receivers 20 and 40 Correlations at receiver 20
Source-receiver offset (m)0 20 40 60 80 100 120
Source-receiver offset (m)0 20 40 60 80 100 120
Receiver number1 10 20 30 40 50 60 1 10 20 30 40 50 60
Figure 2. Left: the vertical component of the wavefield in a slab from single shot, Uz at sourceposition 6. Middle: the Green’s functions between receivers 20 and 40 for 10 separate shots, Gzz.The top trace shows the results of summing these 10 shots. Right: the virtual shot record for all60 receivers, with the virtual source at receiver 20.
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The spatial crosscorrelation method for dispersive surface waves 23
0 10 20 30 40 50 60 70 80 90
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x 10
Frequency (Hz)
Sp
ec
tra
l a
mp
litu
de
(a
.u.)
zz
0 10 20 30 40 50 60 70 80 90
−1.5
−1
−0.5
0
0.5
1
1.5
Frequency (Hz)
Sp
ec
tra
l a
mp
litu
de
(a
.u.)
φzz single source
φzz summed sources
Figure 3. The real part of the Fourier transform of the cross-correlation of the vertical componentsfor a single shot at source position 6 and for the sum of 10 sources.
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24 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
0 10 20 30 40 50 60 70 80 90
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Frequency (Hz)
Sp
ec
tra
l a
mp
litu
de
(a
.u.)
φzz
φzx
φxz
φxx
Figure 4. SPaX coefficients φzz, φzx, φxz, and φxx from the 10 summed shots.
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The spatial crosscorrelation method for dispersive surface waves 25
0 10 20 30 40 50 60 70 80 90300
350
400
450
500
550
Frequency (Hz)
Ph
as
e v
elo
cit
y (
m/s
)
φzz
φzx
φxz
φxx
462
0 10 20 30 40 50 60 70 80 90
450
500
525
Frequency (Hz)
Ph
as
e v
elo
cit
y (
m/s
)
475
425
462
Figure 5. The roots of the real part of the Fourier transform of the cross-correlations betweenreceivers 20 and 40 for all four components for a single shot at source position 6 (top) and for 10summed shots (bottom). Both the top and bottom panels have the same vertical scale and colorrelationship between φ terms.
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26 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
200
300
400
462
500
600
700
800
1 2 3 4 5 6 7 8 9 10
MuSPAC phase velocity using a single shot
True slab phase velocity (462 m/s)
Phas
e Ve
loci
ty (m
/s)
Shot Number
MuSPAC phase velocity using 10 summed shots
10
Shot=1Σ
Figure 6. The mean phase velocity from all four SPaX coefficients with the corresponding standarddeviation. Each point has been averaged over all frequencies before averaging the individual SPaXcomponents. Solid diamonds represent single source locations, while the open diamond shows theresults after summing over 10 sources.
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The spatial crosscorrelation method for dispersive surface waves 27
0 10 20 30 40 50 60 70 80
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Frequency (Hz)
Spec
tral
am
plitu
de (a
.u.)
MuspacV02a11aCHRP0DISF0to150m0
zz zx xz xx
Figure 7. The semi-transparent (0-150 Hz) and full-color (30-150 Hz) SPaX coefficients from ahomogeneous half-space model.
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28 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
0 10 20 30 40 50 60 70 800
500
1000
1500
2000
2500
3000
3500
Frequency (Hz)
Phas
e ve
loci
ty (m
/s)
Direction of zero crossings are shown by triangles. MuspacV02a11aCHRP0DISF0to150m
φxzφxx
m(xx)=3, m(xz)=4
m(xx)=5, m(xz)=6
m(xx)=0 m(xz)=0
m(xx)=1, m(xz)=2
Figure 8. The semi-transparent and full color dispersion curves cxx and cxz are for full-band (0to 150 Hz) and partial-band (30 to 150 Hz) data respectively. The receiver separation is 25 m.The effect of adding up to 6 missing zero crossings (m=6) on the partial-band data is shown withmxx = 3 and mxz = 4 resulting in a match between the partial-band and full-band dispersioncurves.
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The spatial crosscorrelation method for dispersive surface waves 29
Receiver (virtual source)
Receiver
Rayleigh wave velocity (C1) = 480 m/s
Rayleigh wave velocity (C2) =600 m/s
12 m
414 m
Layer 1
40 m
Vp
= 900 m/s
Vs
= 522 m/s
ρ = 2000 kg/m3
r12
= 0
Layer 2V
p = 1000 m/s
Vs
= 674 m/s
ρ = 1800 kg/m3
NOT TO SCALE
Figure 9. A schematic of the two-layer numerical model. Because of the two layers, the Rayleighwave velocity is dispersive.
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30 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
0 10 20 30 40 50 60
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Frequency (Hz)
Spec
tral
am
plitu
de (a
.u.)
MuspacV01f40bLAY21DISZ426ShotAllF999Abs1Stk
φzzφzxφxzφxx
φzz+zx’+xz’+xxφzz+xxφzx+xz
Summation of all four coefficients
Figure 10. The individual SPaX coefficients and the combined SPaX coefficient after summingthe 12 shots for a receiver separation of 40 m.
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The spatial crosscorrelation method for dispersive surface waves 31
0 10 20 30 40 50 60450
500
550
600
650
Frequency (Hz)
Phas
e ve
loci
ty (m
/s)
Direction of zero crossings are shown by triangles. MuspacV01f40b2LAY21DISZ426ShotAllF999Abs1
φzz
φzx
φxz
φxx
(φzz+φzx+φxz+φxx)/4
Figure 11. Phase-velocity dispersion curves from all four individual SPaX coefficients betweenreceivers separated by 40 m. The solid-black line shows the result of averaging all four SPaXcoefficients to calculate phase velocity c.
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32 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
Sawatch Fault
Chalk Creek Fault
Sawatch Fault
Chalk Creek
Valley UpperArkansas
Basin
Mt. Princeton
Mt. Antero
Mt. HarvardChalk CreekValley Buena Vista
FOP Study Site
Mt. PrincetonHot Springs
100
km
Mount Antero
Mount Yale
Buena Vista
Mount Princeton
Chalk Cli�s
UpperArkansas
Basin
2 km
N
State of Colorado
Mt. Princeton
200 km
Cha�eeCounty
Denver
FOP Study Area
Sawatch Fault
Collegiate Range
DHL Site
DHL Study Site
Modi�ed from Richards et al., 2010
Northern Segment
Southern Segment
FOP Site
Figure 12. Top Right: Colorado state map showing the approximate locations of Chaffee County,Mount Princeton, and Denver. Bottom Right: A topographic map of the Upper Arkansas basin inChaffee County superimposed with the study ’DHL’ site. Left: Location of faults mapped along theSawatch Range according to work by Scott et al. (1975), Colman et al. (1985), and Miller (1999).
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The spatial crosscorrelation method for dispersive surface waves 33
397400 397500 397600 397700 397800 397900 398000 398100
4286300
4286400
4286500
4286600
4286700
4286800
4286900
UTM x (m)
UTM
y (m
)
Well Water Temperature (0C)0 - 10
10 - 1515 - 2020 - 2525 - 30
30 - 3535 - 4040 - 4545 - 5050 - 70
0 50 100Meters
MuSPAC Line 115-220 Gravity (mGal)3806
3802 3804
�N
115
161
220236
611
Originally came from proposal �gure �le DHL_Proposal_AllMag200dpi_bingV01.ai
Refraction Line 1-236
MG-1
200
N-S Ramp Fault
Figure 13. The Dead Horse Lake (DHL) study site showing water wells and their temperatures,the location of the seismic line, and overlain with the results of the gravity survey. The well labeledMG-1 was used to conduct a 168 m deep vertical seismic profile for which the results are presentedin Lamb (2013).
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34 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
0 10 20 30 40 50 60 70 800
200
400
600
800
1000
1200
1400
1600
Frequency (Hz)
Phas
e ve
loci
ty (m
/s)
UNCV04c02c
φzz: m=1φzx: m=1φxz: m=2φxx: m=1
φzz: m=3φzx: m=3φxz: m=4φxx: m=3
φzz: m=5φzx: m=5φxz: m=6φxx: m=5
φzz: m=7φzx: m=7φxz: m=8φxx: m=7
A
B
C
D
φzz: m=0E
A
B
C
D
E (Full Band)
Figure 14. Phase-velocity dispersion curves for field data where r = 30 m. The semi-transparentand full-color dispersion curves are for full-band (numerical) and partial-band (field) data, respec-tively. The full-band curve E is based upon a 32-layer numerical model. Because the vibroseis sweepdid not cover the frequency range 0-30 Hz, we see that the number of missed zero crossings shouldbe mzz,zx,xx = 5 and mxz = 6.
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The spatial crosscorrelation method for dispersive surface waves 35
20
25
30
35
40
45
50
55
60
Freq
uenc
y (H
z)
170 180 190 200 210Common midpoint (station no.)
Velo
city
(m/s
)
170 180 190 200 210 170 180 190 200 210
SPAC: 1C MuSPAC: 4C
N-S Ramp Fault N-S Ramp Fault320
240260280300
170 180 190 200 210Common midpoint (station no.)
Common midpoint (station no.) Common midpoint (station no.)
Figure 15. Lower left: A gather of SPAC coefficient traces for φxx(r = 30, ω) for the field datafrom station 168 to 213. The dashed red line shows the 7th zero crossings. The phase velocitycorresponding to the 7th zero crossings is shown in the top panel and indicates laterally changingvelocity. Bottom right: A gather of the average correlation coefficients. Top right: The phase velocitycorresponding to 7th zero crossing of the combined SPaX coefficients.
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36 A. P. Lamb, K. van Wijk, L. M. Liberty and T. D. Mikesell
0 50 100 150 200 250 300 350 400 450
2490
2500
2510
2520
Vp ~ 3200 m/s
Vp ~ 4400 m/s
Distance (m)
Elev
atio
n ab
ove
seal
evel
(m)
VSP
Saturated sediments
Altered quartz monzonite
Vp ~ 2200-2700 m/sVp~500 m/s Vp~900 m/s
West EastMG-1 well
Unsaturated sediments Vp~700 m/s Vs~550 m/s
2490
2500
2510
2520
Granite
V.E. = 4
500
1000
1500
2000
2500
3000
3500
4000
4500
Tomography velocity (m/s)
N-S RampFault
1
61 121 141 161181 201 22181 101
41
21
**********
MuSPAC AnalysisSources Receivers
**********
Figure 16. A geologic interpretation of the SPaX result and refraction tomography velocitiesconstrained by the MG-1 well (Lamb 2013).
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