Ground-penetrating radar (GPR) signals are attenuated by both absorption and scattering. We performed low-frequency (<100 MHz) GPR surveys at the Volcanic Tableland of the Bishop (California) Tuff to evaluate the factors that control GPR depth of investigation and to develop insight into the capabilities of such radars for Mars. The subsurface reflection character was very different for two different commercial systems used; together, they revealed both internal welding contacts in the tuff and an abundance of discrete scatterers. Attenuation coefficients were computed from profiles that showed distributed scattering: the semilogarithmic signal decay is directly analogous to seismic coda. The absorption (intrinsic loss) was determined to be ∼1 dB/m from low-frequency vertical-electric soundings. The residual attenuation (that is, the attenuation in the absence of absorption) is attributed to scattering. Scattering attenuation of ∼1 dB/m at 25–50 MHz corresponds to mean-free paths as short as 4 m, a fraction of the two-way propagation distances of 20–40 m. Therefore the Bishop Tuff is formally a strong scatterer to GPR. The mean-free path is also comparable to the subsurface radar wavelength in this case, maximizing scattering loss. The scatterers themselves likely originate as welding heterogeneities; contrasts in dielectric constant due to density differences may be supplemented by moisture variations. On Mars, scattering is likely to contribute significant losses to GPR signals in all but the most uniform materials, and unfrozen thin films of water in the lower cryosphere could influence both absorption and scattering.
 Ground-penetrating radar (GPR) has become firmly established in the last decade or so as an important tool for high-resolution shallow subsurface investigation, principally in environmental geology, archeology and forensics, and engineering, infrastructure, and nondestructive testing (see Annan  for a review). Commercial GPRs operate from ∼10 MHz to ∼1 GHz. Planetary radiofrequency subsurface sounding was pioneered by the Apollo 17 Lunar Sounder Experiment (ALSE) [Phillips et al., 1974] and Surface Electrical Properties (SEP) [Simmons et al., 1972] experiment. These systems operated at 5–150 MHz and 1–32 MHz, respectively, and using the lowest available frequencies, returned information about the lunar interior to depths of ∼1 km. The MARSIS and SHARAD Mars orbital radar sounders, jointly spanning 2–20 MHz center frequency, are intended to map portions of that planet's interior from tens of meters to a few kilometers depth [Picardi et al., 2004; Seu et al., 2004].
 Although numerous studies have examined Mars geological analogs [e.g., Farr et al., 2001], Mars geophysical analogs remain poorly defined. We have undertaken a multiinstitutional, multiyear campaign to better define and investigate Mars geophysical analogs, particularly with GPR in the 10–100 MHz range. This report, and two companion papers [Heggy et al., 2006a, 2006b] describe initial investigations in the arid western United States.
2. Site Description
 The Bishop Tuff is located at the northern end of the Owens Valley, California (Figure 1), at the western margin of the North American Basin and Range physiographic province. The Bishop Tuff was deposited circa 0.76 Ma [e.g., Sarna-Wojcicki et al., 2000] as a series of pyroclastic flows and falls (e.g., Figure 2) from the Long Valley caldera, ∼40 km to the northwest [e.g., Wilson and Hildreth, 1997; Evans and Bradbury, 2004]. The entire sequence was originally 80–150 m thick but has been variably eroded. Our GPR tests were performed at three sites on the Volcanic Tableland to the northwest of the town of Bishop (Figure 3). Quaternary extension has cut the surface of the Volcanic Tableland with a series of north-trending normal faults [Pinter, 1995; Ferrill et al., 1999] (see Figure 3). The southern margin of the Volcanic Tableland is defined by an erosional escarpment cut by the Owens River (Figure 3), exposing massive ignimbrite, bedded tephra-fall deposits, and several faults in cross section, with typical displacements of several meters [Evans and Bradbury, 2004].
 The Volcanic Tableland was chosen as a potential Mars analog on the basis of its arid climate, likely deep water table, and extensive prior geological investigation. The specific sites were selected on the basis of unimproved-road access and progressive distance from the edge of the Tableland (Figure 3). Rainfall for Bishop averages 14 cm yearly and 1.4 cm during November, the month of our survey. Hydrogeological data are sparse but water table depths can be constrained by the Owens River to the west and south, the Fish Slough to the east, and a lone domestic well on the Volcanic Tableland. Depth to water in this well, ∼1 km from Site 2, is ∼76 m (R. Harrington, Inyo County Water Department, personal communication, 2004). Sites 1, 2, and 3 are ∼67 m, ∼95 m, and ∼180 m above the nearest surface waters, respectively, providing upper limits to the depth of the water table. Transient electromagnetic (TEM) reconnaissance soundings [Dinwiddie et al., 2005] performed several months before our GPR survey showed strong conductors at depths of ∼80 m for Site 2 and ∼180 m for Site 3. Regional groundwater is relatively fresh (100–325 mg/l TDS [Hollett et al., 1991; R. Harrington, personal communication, 2004]), but even such modest ionic concentrations can significantly enhance electrical conductivity in fresh, unweathered rocks (see McNeill  for a review). The TEM conductor is in broad agreement with known depths to the water table, and the unsaturated zone below the test sites is many tens of meters or more in thickness. These conditions prompted our expectation of good opportunities for deeply penetrating GPR surveys at lower frequencies. Finally, the National Research Council Decadal Study on Terrestrial Analogs to Mars recommended the Bishop Tuff as an analog site because some Martian paterae are likely composed of pyroclastic deposits, as is the large “Stealth” region of southwestern Tharsis [Edgett et al., 1997].
3. GPR Systems and Methods
 We performed GPR surveys at the Volcanic Tableland using two commercial systems: the pulseEKKO 100 (PE) manufactured by Sensors and Software, Inc., Mississagua, Ontario, and the 3200 Multiple Low Frequency (MLF) manufactured by Geophysical Survey Systems, Inc., Salem, New Hampshire. The PE uses antennas with center frequencies of 12.5, 25, 50, 100, and 200 MHz; the first three antennas were used here. The MLF uses antennas with center frequencies of 16, 20, 32, 40, and 80 MHz; 20, 40, and 80 MHz were used here. The antennas for both systems are unshielded, loaded dipoles. Both systems accommodate a bandwidth comparable to the center frequency, although we noticed that the dominant frequency coupled into the ground for at least the PE is ∼80% of the nominal center frequency. PE productivity was inhibited by intermittent data-transmission failure across an 80-m fiber-optic cable.
 Data acquired from the two systems were obtained using different techniques. PE data were collected statically with the antennas stationary while 128–512 pulses were averaged (stacked) at each station. Station spacings of 0.5–2 m (about subsurface wavelength) were compiled into lines 50–80 m long. The PE data then typically have a total of ∼256 scans/m. Antenna orientation was broadside (i.e., perpendicular to the survey line). Antenna separations were 10, 4, and 2 m for 12.5, 25, and 50-MHz surveys, respectively. Common-midpoint (CMP) soundings were also performed.
 MLF data were collected dynamically by continuously pulling the antennas parallel to the 40–100 m survey lines. These antennas were separated by a constant offset of 6 m and elevated by ∼40 cm to alleviate surface drag and antenna deformation. Data were recorded at 4–16 scans/sec, so at an average speed of 0.5 m/s, the single-pulse station spacing was 0.031–0.125 m. Therefore MLF data density is ∼16 scans/m, a difference of 12 dB from the density acquired with the PE. All together, our field experience indicates that the effective dynamic range available for all ground losses averages ∼105 dB for the PE and ∼85 dB for the MLF [see also Heggy et al., 2006b]. Each MLF profile was repeated for each site, however, and the better selected for interpretation.
 Nominal data processing for the two systems was broadly similar but differed in the details. Topography was relatively flat over the length of the survey lines, so no elevation corrections were applied. PE data were corrected for low-frequency induction (“dewow”) and then were subjected to a 3-station median spatial filter to alleviate individual bad traces, followed by FIR band-pass filtering at 9–30, 15–60, and 20–120 MHz for 12.5, 25, and 50 MHz data, respectively. These ranges were nominally selected to span f0 to 3f0, where f0 is the effective center frequency, but were adjusted to try to filter out low-frequency coherent noise (ringing) at 5–15 MHz. The MLF data were similarly corrected for low-frequency induction, then frequency filtered at 10–30, 30–50, and 60–100 MHz for 20, 40, and 80 MHz center frequencies, respectively. A 122–200 trace (6–15 m horizontal) moving average was subtracted from the MLF data to attenuate ringing.
4. GPR Survey Results
4.1. Site 1
 This site is 200 m from the SE escarpment of the Volcanic Tableland (Figure 3). The survey profile azimuth was 95° true, nearly perpendicular to a large fault scarp ∼200 m to the west. This was the only profile acquired off-road (here, on desert pavement) and with the transect specifically oriented with respect to a local tectonic feature. Radargrams from Site 1 are shown in Figures 4a–4c.
 The 50-MHz PE radargram (Figure 4a) is dominated by numerous hyperbolic reflections, indicating an abundance of discrete scatterers in the form of compact velocity anomalies or linear (cross-profile) discontinuities. Such scatterers are characteristic of most of the Volcanic Tableland GPR data acquired with PE. Comparable signatures were observed in the Tumalo Tuff (Oregon) by Rust and Russell . Note that the flatter set of reverberations from 200–350 ns near x = 35 m is an artifact of multiple reflections from within an ice chest that was left too close to the survey line.
 Subsurface velocities were determined by CMP sounding. CMPs were expanded around the center of each profile line. Several prominent horizontal reflectors are evident as hyperbolic patterns in the 50-MHz display (Figure 4b), with zero-offset times of 70–300 ns. Semblance analysis [e.g., Sheriff and Geldart, 1995; Annan, 2003] and direct travel time fitting indicate velocities of 0.11 m/ns for the direct wave and a maximum of 0.12 m/ns in the subsurface. At 25 MHz (not shown), CMP velocities were also 0.11–0.12 m/ns. For a 0.12 m/ns average wave speed, then, the reflectors lie in the depth range 4–18 m. The principal welding contacts at approximately 7, 10, 17, and 22 m (Figure 2) would have zero-offset times of approximately 120, 170, 280, and 370 ns, respectively. Of these, only the contact between the uppermost welded tuff and an underlying partially welded tuff at 7-m depth (Figure 2) has an obvious correspondence in Figure 4b. A strong signature is expected for the large density contrast of this contact (Figure 2). The lack of lateral offsets in these patterns demonstrates that the interfaces are close to horizontal at this location.
 The horizontal contacts observed in the PE CMP data are not obvious, however, in the PE profiles. The tops of the scatterers are dispersed over times of 50–200 ns (3–12 m depth) in the PE profiles and do not obviously group at known contact depths or within separate welding units. Kirchhoff migration collapsed the scattering patterns as expected, leaving broadly subhorizontal patterns, but did not obviously enhance expected contacts. Individual scatterers are still evident at lower resolution in the 25 MHz profile (not shown), with a slightly greater (∼10%) depth of investigation. A right-dipping linear feature in the upper left of the 50-MHz radargram (Figure 4a) is likely a fault: converting time to depth and hand-migrating [e.g., Sheriff and Geldart, 1995], the true dip is ∼54°.
 The MLF radargrams at all frequencies are qualitatively different from the PE data: scattering patterns are weak to absent but subhorizontal reflections are evident. At both 40 MHz (Figure 4c) and 20 MHz (not shown), a major reflection is evident beginning at 110–140 ns (6–8 m), again corresponding to the depth of the first documented welding contrast (Figure 2). The reflector has some apparent vertical relief of <20 ns (<1 m) over horizontal distances of 2–10 m; however, given timing mismatches that are evident between adjacent traces, we cannot reliably determine whether these offsets are real. A second reflector is suggested beginning at ∼250 ns (15 m), particularly in the 20-MHz profile, and may correspond to another welding contact. A weak, right-dipping event in the 40 MHz data (x = 25–55 m, t = 200–300 ns) appears to be an airwave. The remaining portions of the MLF radargrams at this site are characterized by coherent noise and multiple reflections.
4.2. Site 2
 This site is 300 m from the southern escarpment of the Volcanic Tableland, ∼2 km WSW from Site 1. The survey profile azimuth was 43° true. Large scarps at distances of 0.9–1.7 km trend 13–18°, so the survey profiles may cross regional structure at an angle of ∼30°. Radargrams from Site 2 are shown in Figures 5a–5f.
 The 50-MHz (Figure 5a) and 25-MHz (Figure 5b) PE profiles are similar in character to those from Site 1 in terms of the density of scatterers resolved and the depth of investigation. Another likely fault is visible in Figure 5a, projecting from the near-surface at x = 25 m to 200 ns at x = 15 m. Following hand migration and additional correction for profile azimuth, the true dip is ∼64°.
 This is the only site for which useful 12.5-MHz data were recovered (Figure 5c). Little detailed structure is visible, but the character from 200–400 ns can be distinguished as subsurface reflections compared to earlier and later reverberations.
 The CMPs for this site (not shown) yielded velocities of 0.09–0.11 m/ns at 50 MHz and 0.10–0.12 m/ns at 25 MHz. Both frequencies resolved the 7-m reflector in addition to the direct wave.
 Data at three MLF frequencies were successfully obtained at this site. At 80 MHz (Figure 5d) the reflector at 120 ns (6 m) appears crisp and flat. Careful inspection reveals numerous small scattering patterns mostly below the contact.
 More structure on the reflector is apparent at 40 MHz (Figure 5e). A large, concave-down feature in the center of the radargram could correspond to a diffraction hyperbola in the center of the 50 and 25 MHz PE lines, but has no analog in the 80 MHz MLF profile. The 20-MHz radargram (Figure 5f) again shows laterally varying structure that is difficult to assess in the presence of timing offsets and reverberation.
4.3. Site 3
 This final site was located in the interior of Volcanic Table 1 and, 8 km from Sites 1 and 2 (Figure 3). The survey azimuth was 325° true. Fault scarps in 3 directions at distances of 1.3–1.8 km trend 338–348° true, so the survey line is within ∼20° of major fault strikes. Radargrams from Site 3 are shown in Figures 6a and 6b.
 The 50-MHz profile (Figure 6a) shows just four major diffractors, all with their tops at <40 ns. Greater reverberation and attenuation limited the depth of investigation at this site. The 25-MHz profile (not shown) exhibits reflection character to slightly greater depth but with poor resolution. The 50-MHz CMP for this site (not shown) resolved only the direct wave at an anomalously high velocity of 0.14 m/ns; no velocity was recovered from the 25-MHz CMP due to air-wave reverberation. The depth to the top of the scatterers therefore is ∼2.2–2.6 m at 0.12–0.14 m/ns.
 Depth of investigation using the MLF at this site also low. The 80 MHz profile (Figure 6b) shows a sharp reflection at ∼80 ns (3.7–4.8 m using velocities above). The reflector is also resolved at 40 MHz (not shown) but the 20 MHz profile (not shown) is highly reverberatory, and the backscattered signal is generally weak.
5. Attenuation Measurement
 We seek to develop an experimental methodology for measurement of attenuation in GPR and its separation into intrinsic attenuation (absorption) and scattering attenuation. After a brief review of attenuation nomenclature, we describe how the PE Bishop data are suited to such investigations, and we derive coefficients of total attenuation.
 In addition to geometric spreading, the amplitude of an electromagnetic signal following a two-way path 2R attenuates as e−2αR, where α is the spatial attenuation coefficient. The latter can be expressed as η in units of dB/m by the conversion η = 20log10(eα) = 8.686α. The general form of the spatial attenuation coefficient [e.g., Ward and Hohmann, 1988] is
where ω is the angular frequency, μ is the magnetic permeability, ɛ is the dielectric permittivity, and tanδ is the loss tangent. For the wave-propagation regime tanδ ≪ 1 and permeability equal to that of free space, this may be simplified to α = πf tanδ√ɛ′/c0, where f is the frequency, ɛ′ = Re(ɛ/ɛ0) is the real part of the relative permittivity (dielectric constant), and c0 is the speed of light in vacuum. This in turn can be written as
 The loss tangent is the ratio of the total energy loss to the total energy storage. For purely ohmic dissipation, tanδ = σ/ɛω, where σ is the electrical conductivity. If dielectric dispersion dominates, tanδ = ɛ″/ɛ′, where ɛ″ = −Im(ɛ/ɛ0). Because ultimately only the ratio of total loss to total storage determines the loss tangent, all such relationships, including scattering, can be expressed as ɛ = ɛ′–iɛ″, where ɛ is now taken to be an effective complex dielectric constant for the remainder of this paper. In a linear system, the real and imaginary parts of the dielectric constant (less DC conduction) must be a Hilbert transform pair, satisfying the Kramers-Kronig relations for causality [e.g., Sihvola, 1999].
 In a medium where the distance separating reflectors is greater than a wavelength, the radargram is characterized by discrete reflections separated by noise. Reflections overlap where reflectors are more closely spaced. Ultimately, the radargram can show a continuous decrease in amplitude with distance, without obvious discrete reflectors. For uniform layer thicknesses z and constant reflection coefficients, loss is directly proportional to distance and so amplitude will decrease as e−2αz. A similar falloff occurs for a random distribution of scatterers in an isotropic medium, although now it must be recognized that returns can come from any direction and be projected onto the radargram time series, e−2αR = e−αvt, where R is range, t is two-way time, and v = c0/√ɛ′ is the mean wave speed.
 A continuous terminal wave train has long been recognized in earthquake seismology as the coda, and its interpretation has been a major branch of investigation for the past several decades [e.g., Aki, 1969; Aki and Chouet, 1975; Dainty and Toksöz, 1981; Jin et al., 1994; Shapiro et al., 2000]. In these studies, the spatial attenuation is usually couched in terms of the quality factor Q or its reciprocal Q−1 = tanδ = αλ/π, where λ is the wavelength [see also Sheriff and Geldart, 1995]. A major theme of “coda-Q” has been the separation of intrinsic attenuation (absorption) from scattering attenuation. This has been accomplished by modeling the joint scattering and absorption using both spatial and temporal information on the coda [Wu, 1985; Zeng et al., 1991]. Here we directly measure the low-frequency absorption in the field and, after extrapolating to radiofrequencies, attribute the remainder of the loss to scattering. Almost all of the PE Bishop Tuff data showed a clearly identifiable segment of continuous semilogarithmic amplitude decay with time (e.g., Figure 7). The strong reverberation present in the MLF radargrams and uncertainty in gain functions prohibited comparable amplitude analysis.
 In order to calculate the attenuation, the relative amplitudes must be corrected for other ground losses, i.e., geometric spreading and backscatter cross section. For such purposes, the radar range equation [e.g., Skolnik, 2001] may be written simply as
where PT is the transmitted power, PR is the received power, G represents all of the gains and losses associated with the system and its antennas, and ξ is the backscatter cross section. We consider three models for the reflecting target. For a smooth, planar reflector, ξ = πR2Γ, where Γ is the power reflection coefficient [Annan and Davis, 1977]. Therefore geometric spreading and the backscatter cross section jointly result in PR/PT ∝ 1/R2. The other end-member is to consider reflections to be due to small (subwavelength) spheres, i.e., as Rayleigh scatterers, ξ = π5D6Γ/λ4, where D is the radius of the sphere [Stratton, 1941; Annan and Davis, 1977]. Here PR/PT ∝ 1/R4. An intermediate case assumes the GPR return is integrated only over the diameter √2λR of the first Fresnel zone, so ξ = πλRΓ/2 [Annan and Davis, 1977], resulting in PR/PT ∝ 1/R3. The Fresnel zone is also used to represent the response of a rough, planar reflector [Cook, 1975].
 We compress the radargram to a single time-dependent function by taking the mean square over all traces at each time; this is the energy-average trace (Figure 7a). This operation implicitly assumes a medium that is composed of horizontal layers or isotropic, random scatterers. The latter follows by considering that energy on the limb of a scattering hyperbola is by definition at some slant range to the scatterer, and therefore should show attenuation commensurate with that range in an isotropic medium. Depth is converted to time assuming a constant velocity and correcting for the antenna separation (Figure 7b). A velocity of 0.11 m/ns was used for all calculations: a minimum velocity will result in a conservative, minimum estimate of the overall attenuation. A model-dependent gain function (R2, R3, or R4) is then applied and the trace renormalized (Figure 7c).
 The attenuation η (dB/m) then follows simply from a least squares fit of the two-way distance versus dB power (Table 1). Because the selected segments are quite linear, formal errors on the attenuation are small; instead, we report the mean and standard deviation of the results obtained using the three different gain functions (Table 1). Measured total attenuations are ∼1–3 dB/m (Table 1) over a median depth interval 5–14 m. Corresponding Q = 2–9 and tanδ = 0.1–0.5. Site 3 has the largest overall attenuation and Site 2 the least. Attenuation increases slightly with frequency for all three sites.
Table 1. Measured Total Attenuations in PulseEkko Data
Two-Way Time, ns
Semilog Fit r2
One-Way Attenuation, dB/m
Spreading/Backscatter Range Exponent
Site 1, 25 MHz
1.56 ± 0.20
Site 1, 50 MHz
1.73 ± 0.22
Site 2, 12.5 MHz
0.95 ± 0.13
Site 2, 25 MHz
1.17 ± 0.24
Site 2, 50 MHz
1.40 ± 0.25
Site 3, 25 MHz
2.85 ± 0.21
Site 3, 50 MHz
3.15 ± 0.33
 The contribution of absorption to the GPR attenuation was derived from ground conductivity, measured in situ by DC resistivity. Schlumberger vertical electric soundings (VES) were performed at each site using a SuperSting R8/IP 8-channel automatic resistivity/IP system. The electrode array was aligned with and centered at the middle of the GPR transects. Wire-mesh electrodes were necessary to decrease contact impedance. Separations between current electrodes varied from 3–200 m, whereas potential-electrode separations of 1 and 10 m were used. All three soundings (Figure 8) show the same overall character, with resistivity decreasing to a minimum at 5–10 m depth before reversing. Note that a contact is evident at 7-m depth for Sites 1 and 2, consistent with the GPR results. While the large resistivities (>1000 Ω-m) observed at the surface and at depth are representative of relatively dry rocks, measured resistivities as low as 300 Ω-m are widely representative of rocks and soil containing some water [e.g., Telford et al., 1990; McNeill, 1990]. Indeed, the three curves show a regular progression suggestive of variations in infiltration efficiency: the increasing vertical variance from Site 1 to Site 2 to Site 3 may indicate development of perched water and a corresponding moisture shadow-zone at depth. The higher CMP velocity for Site 3 implies less water here, however: if this velocity is correct, higher salinity at lower saturation may be required to jointly explain both the resistivity and radar data.
 The low-resistivity zones occupy much of the region imaged by GPR; averaged resistivities within these zones are 1930 ± 350, 910 ± 165, and 475 ± 90 Ω-m for Sites 1, 2, and 3, respectively. The contributions to absorption η (using equation (2)) are η ≈ 0.3, 0.7, and 1.3 dB/m for Sites 1–3, respectively. Absorption due to ground conductivity is important to GPR attenuation at the Volcanic Tableland, in contrast with the original hypothesis.
 Laboratory measurements of Bishop Tuff samples (Figure 9) show a regular negative dispersion (dielectric constant decreasing with increasing frequency) without significant specific relaxations. These measurements were obtained using an Agilent E4991A impedance analyzer connected to a guarded capacitive cell and calibrated using a reference Teflon sheet. The samples, both collected from near the midpoint of Site 2, were vacuum dried at 78°C for 48 hr and subsequently measured at room temperature. The losses implied by the measured imaginary dielectric constants are small (<0.3 dB/m), confirming that field properties (ions in groundwater) control GPR absorption.
 The excess attenuation is the measured attenuation at the GPR center frequencies minus the absorption determined from ground conductivity (Table 2). The excess attenuation is positive in every case, i.e., there is truly a residual attenuation exceeding that predicted from absorption alone. In the absence of other absorptive mechanisms, we attribute the excess attenuation to scattering. Absorption and scattering appear uncorrelated except for Site 3, which has both the largest absorption and the largest scattering. The mean scattering attenuation over all sites and frequencies is 1.1 dB/m. There is a slight trend toward greater scattering loss at higher frequency.
Table 2. Absorption Computed From Ground Conductivity and Inferred Scattering Attenuationa
Excess Attenuation, dB/m
12.5-MHz results for Site 2 only: scattering albedo 0.31 ± 0.19.
1930 ± 350
0.31 ± 0.06
1.25 ± 0.21
1.42 ± 0.23
0.80 ± 0.17
0.82 ± 0.17
910 ± 165
0.66 ± 0.12
0.51 ± 0.27
0.74 ± 0.28
0.44 ± 0.25
0.53 ± 0.22
475 ± 90
1.26 ± 0.24
1.59 ± 0.32
1.89 ± 0.41
0.56 ± 0.12
0.60 ± 0.14
 The scattering albedo B0 is defined as the ratio of scattering attenuation to total attenuation [Jin et al., 1994] (the original term “seismic albedo” is descriptive neither of GPR nor seismic scattering, but “albedo” is used because higher scattering attenuation also means more energy directed into the coda). Computed B0 at 25–50 MHz varies from ∼0.4 to 0.8, with a mean value of ∼0.6. Scattering and absorption both significantly attenuate GPR signals at these sites, each contributing comparably.
 Modern modeling of the seismic coda jointly treats absorption and arbitrary multiple scattering [Wu, 1985; Zeng et al., 1991; Jin et al., 1994]. Simpler approaches previously considered the medium as either weak (single) scattering or strong (diffusion) scattering [Aki and Chouet, 1975; Dainty and Toksöz, 1981]. The transition between the two can be taken to be where the GPR propagation distance 2R is equal to the mean-free path L between scatterers. Because L = 1/2α = 4.34/η by definition, η = 1.1 dB/m and R = 15 m implies L ∼ 4 m and 2R/L ∼ 8, confirming formal strong scattering.
 The mean-free path can be compared to the separation of scattering hyperbolas observed in the 50-MHz PE radargrams. We identified eleven obvious hyperbolas in Figure 4a, ten in Figure 5a, and three in Figure 6a. Mean horizontal separations (a lower bound to the true 3-D spacings) are 5–18 m. The hyperbolic scatterers therefore likely contribute some of the scattering loss, with the remainder at subwavelength scales such that discrete signatures are not observed. Rayleigh scattering would be maximized where the diameter of the anomaly πD = λ, or D ∼ 1 m at 25–50 MHz and 0.12 m/ns for this survey. Contributing heterogeneity must be meter-scale or greater, because backscatter amplitudes in the Rayleigh region at smaller diameters fall off rapidly as D3.
 Dielectric heterogeneity necessary to generate the observed scattering hyperbolas and the overall scattering attenuation could stem from some combination of (1) vertical variations in welding that generate lateral heterogeneity by fault displacement, (2) lateral and vertical variations in welding, or (3) variations in moisture.
 Vertical variations in dielectric constant will be associated with the density differences characteristic of vertical changes in welding [Rust and Russell, 2000; Russell and Quane, 2005; Sheridan and Wang, 2005]. Where horizontal contrasts vary slowly compared to the GPR wavelength, vertical displacements due to faulting could create the lateral contrasts necessary to generate scattering hyperbolas. Extensive experience in seismic reflection indicates that layers as small as λ/30–λ/20 can be detected; furthermore, faults with throws as small as λ/16, while not resolving the offset, still generate diffraction patterns [Sheriff and Geldart, 1995; Yilmaz, 2001]. Vertical offsets of tens of centimeters could in principle therefore act as discrete GPR scatterers. This case is strengthened by a correlation (not shown) between the apparent velocities of the scatterers at the three field sites and regional fault strikes: if the scattering hyperbolas are from linear features, apparent velocities increase as the fault trend more closely parallels the survey line. Although the Volcanic Tableland is cut by numerous faults, there are no reports of the ubiquitous (few-meter spacing) small-scale faulting that would be called for. Furthermore, Rust and Russell  detected similar scattering hyperbolas in 50-MHz PE profiling of the welded Tumalo Tuff. Rust and Russell also detected a reflector below the scattering zone that they interpreted to be a major welding discontinuity. Because this horizon is very continuous, faulting cannot be invoked for the overlying scattering. Therefore we reject faulting as the dominant source of scattering heterogeneity in the Bishop Tuff, and the trend in apparent velocities of the scatterers remains unexplained.
Rust and Russell  interpreted the Tumalo Tuff hyperbolic scatterering patterns to be due to “relatively large lithic or pumice clasts of unusual porosity compared to the surrounding, welded matrix.” They modeled the overall background response of this zone as a layered series of dielectric steps: discrete reflections appear, as expected, when the layer thickness exceeds ∼λ/4 [see Sheriff and Geldart, 1995; Yilmaz, 2001]. These calculations illustrate only the style and not the strength of reflectivity, however, as they do not include geometrical spreading or attenuation.
 We can constrain the magnitude of material contrasts necessary to generate these scattering patterns using a simple radar-range calculation, solving equation (3) for R. On the basis of work presented herein, we take the dielectric constant and total attenuation of the background to be ɛ′ = 6.25 and η = 1.6 dB/m, median values for the three test sites. Note that η now includes the effect of all absorption and scattering between the radar and the target. We assume the prestacking effective dynamic range of the PE available for spreading, absorption, and reflection losses is 113 dB at 50 MHz, on the basis of the manufacturer's performance specifications (150 dB full dynamic range, 20 dB each transmitter and receiver losses, 3 dB antenna gains, −3 dB effective antenna area). This upper limit will minimize the dielectric contrast necessary to detect a target at a specified depth. We use 180 stacks, the geometric average of values at Sites 1 and 2 at this frequency. Backscatter cross sections ξ for an infinite smooth plane, a rough plane, and a sphere are taken from formulas summarized above. We test dielectric contrasts Δɛ′ above and below the background value and average the results for depth of investigation (Table 3). The observed depth of investigation on Lines 1 and 2 is ∼15 m. A background of subplanar reflectors would therefore require modest dielectric contrasts ∼0.1–0.4. The discrete scatterers, however, must have ∣Δɛ′∣ > 1 from Table 3. For Site 3, the lower dielectric contrast that might be associated with ɛ′ = 4.6 and a smaller depth of investigation is more than offset by the need for a larger contrast to compensate for the higher attenuation there.
Table 3. Maximum Depth of Detection and Material Contrasts for Specified Dielectric Contrasta
Zmax computed using ɛ′ = 6.25, 113 dB dynamic range in ground, 180 stacks, and 1.6 dB/m attenuation.
Rust et al.  found that the dielectric constant of dry tuffs measured in the laboratory could be related to bulk density using the simple relationship ɛ′ = 2.26ρb + 1. Changes in dielectric constant in dry tuff exceeding unity would therefore require density contrasts >0.4 g/cm3. Such density changes would be characteristic of the difference between sintered tuff and moderately welded tuffs exposed nearby in vertical section [Wilson and Hildreth, 2003] (Figure 1). Comparable lateral variations on meter scales that would explain GPR diffractions have not been documented, however. Although the dielectric contrast is relatively large between two samples collected at Site 2 (ɛ′ = 4.25 and 6.0 at 25–50 MHz), the apparently nonwelded sample occurred as alluvial float, so actual lateral variations in bedrock are unknown.
 Moisture variations controlled by even minor lateral heterogeneity in welding can generate strong contrasts in dielectric constant. A parallel-plate mixing model weights net dielectric constant proportional to the volume of each component [e.g., Sihvola, 1999] and was reported by Rust et al.  to yield the best fit to dielectric constant versus porosity in laboratory measurements of dry tuffs. The change in dielectric constant Δɛ due to a change in saturation ΔS is then Δɛ = ϕ[1 + ΔS(ɛw − 1)], where ϕ is the porosity, ɛw ≈ 80 is the dielectric constant of water, and the dielectric constant of air in the pore space near unity is implicit. For porosities of 10–15% in the uppermost moderately welded tuff, a unit increase in dielectric contrast can be caused by an increase in saturation of only 7–11% (expressed as a fraction of total porosity). We conclude that GPR scattering in the Bishop tuff is likely dominated by lateral welding heterogeneity, either directly from bulk density contrasts or alternatively from associated moisture variations.
8. Concluding Discussion
 The Volcanic Tableland that caps the Bishop Tuff was selected for Mars-analog geophysical investigation on the basis of its arid environment, young, relatively unaltered rocks, and comparatively deep water table. Vertical electrical soundings, however, revealed zones of low resistivity in the upper 10–15 m that suggest a variably saturated zone or perhaps perched groundwater. Even the moderate salinity of this groundwater is sufficient to strongly impact GPR penetration. Numerous hyperbolic reflection patterns in the GPR profiles attest to significant scattering due to lateral heterogeneity. The total GPR attenuation, measured in direct analogy with seismic “coda Q,” is composed of comparable parts absorption and scattering. Radar-range calculations indicate that contrasts in dielectric constant of order unity are necessary to generate the scattering hyperbolas; such contrasts are likely due to lateral heterogeneity in welding, either as a direct result of density differences or because of moisture variations that track minor lithologic differences.
 Sampling of near-surface bedrock on submeter centers is necessary to determine whether or not significant lateral density variations due to welding actually occur. Additional geophysical surveys may elucidate the nature of scattering heterogeneity (density versus moisture) in the subsurface without the need for extensive drilling. High-frequency, high-resolution seismic reflection will be less sensitive to water and more sensitive to bulk material properties. Very small loop TEM or dipole-dipole electrical profiling could reveal horizontal changes in ground conductivity that could be associated with GPR scatterers. Because the frequency dependence of attenuation here is weak (but is indeed typical of the “GPR Plateau” [Annan, 2003]), depth of GPR investigation varied weakly in the frequency band investigated (∼10–100 MHz). Higher-frequencies (>100 MHz) could still penetrate many meters but provide much sharper images. The geometry of the scatterers could be better defined using multiple survey lines and/or multiple polarizations.
 The Bishop Tuff survey provided valuable insights into GPR scattering and absorption. The derived scattering attenuation is, of course, specific to this site, but such values of ∼1 dB/m would obviously strongly affect depth of investigation in heterogeneous materials elsewhere on either Earth or Mars. The inferred role of water, however, is probably very unlike the upper cryosphere of Mars. Where geothermal heat raises temperatures to within several tens of kelvins of the freezing point in the lower cryosphere, unfrozen films of water coat mineral surfaces [see Anderson and Tice, 1973; Clifford, 1993; Grimm, 2002, 2003]. Unfrozen water could contribute to both absorption and scattering, in analogy with this study. Mars will present both new challenges and new opportunities for GPR.
 We thank Donald Bannon and Brandi Winfrey for field assistance, David Ferrill for helpful discussions, and Saurav Biswas, Gary Walter, and an anonymous referee for their reviews. Kelly Russell provided a particularly thorough and insightful review. This work was partly supported by the Southwest Initiative for Mars (SwIM) internal research program under contracts R9499 (R.E.G.) and R9470 (C.L.D. and D.A.F.). S.M.C. and E.H. were supported by a cooperative agreement between NASA and the Lunar and Planetary Institute (CAN-NCC5-679) and by NASA grants NNG05-GL39G and JPL-1249514.