The spatial crosscorrelation method for dispersive surface...

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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

Transcript of The spatial crosscorrelation method for dispersive surface...

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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

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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

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propagation-I. Validation, Geophysical Journal International, 150, 390–412.369

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Louie, J. N. (2001), Faster, better: shear-wave velocity to 100 meters depth from refraction mi-378

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normal fault, colorado, usa, Journal Of Structural Geology, 21 (7), 769–776.384

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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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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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