Ice mechanical properties, and hence the response of glaciers to climate change, depend strongly on the presence of liquid water at ice-grain boundaries. The propagation velocities of radar and seismic waves are also highly sensitive to this water. Mixing laws, typically the Looyenga and Riznichenko formulae, have traditionally been used to quantify liquid water content within glaciers from such velocity data; however, it has become apparent that these mixing laws are geometrically inconsistent. We present an inclusion-based effective medium approximation in which we model water inclusions within solid ice. Two types of inclusions are used: spherical inclusions to represent water in the grain junction nodes, and high-aspect ratio spheroidal inclusions to represent water in the grain boundary veins. We apply this model to radar and seismic data from a polythermal glacier in Svalbard to quantify both inclusion geometry and the unfrozen water content within the warm ice.
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 The strain rate of glacier ice is sensitively dependent on its water content [Duval, 1977], and so water in ice is an important influence on the dynamics of both temperate and polythermal glaciers. Since the velocities of both radar and seismic waves are also sensitively dependent on ice-water content, both methods have the potential for remotely measuring this parameter [e.g., Murray et al., 2000a; Benjumea et al., 2003; Bradford and Harper, 2005]. Published examples of water content using radar and seismic data vary from ∼0 to 9% [Pettersson et al., 2004], but often without assessment of the associated uncertainty. Recent work has concentrated on improving the accuracy and precision of the velocity models that form the basis of these water content estimates [Barrett et al., 2007; Murray et al., 2007], and has shown the importance of correct data collection and interpretation.
 Simple relationships (known colloquially as “mixing laws”) have been used to calculate water content ϕw from measured velocities. The two most common mixing laws are those of Looyenga  and Riznichenko , derived for radar and seismic data respectively. Both are members of a continuous spectrum of power-mixing laws originally derived by Lichtenecker and Rother , and each makes specific assumptions about the geometry of the water inclusions. The Looyenga formulation,
applied to radar data, uses a power exponent of 1/3 to relate wet ice permittivity ɛ* to the permittivity of water-free ice ɛi and water ɛw. It is often cited as assuming spherical water inclusions, but in fact is general for any isotropic arrangement of two constituents [Landau et al., 1984]. The Riznichenko formulation,
used for seismic interpretation, uses an exponent of -1 to relate the bulk and shear moduli of wet ice (k*, μ*) to its constituent moduli (ki, μi for water-free ice; kw, μw for water). Its basic assumption is of parallel-layered geometry perpendicular to wave propagation. Thus, these two mixing laws are inherently geometrically incompatible, and cannot produce consistent results for ice-water content. This incompatibility has not to date been apparent because there are virtually no co-located measurements of radar and seismic velocities. The only published dataset comparing water content estimates obtained from spatially (but not temporally) coincident radar and seismic velocity data sets used these mixing laws and, as expected, produced contradictory results for the calculated water content [Benjumea et al., 2003; Navarro et al., 2005].
 In this paper we present a new method to interpret water content from radar and seismic velocities that in addition allows differentiation between water inclusion geometries. We apply this method to a dataset collected at a polythermal glacier in Svalbard, and show that simultaneous interpretation of both seismic and radar data provides a powerful technique to infer both water content and inclusion geometry.
2. Effective Medium Approach
 The effective medium approach uses an explicit description of the heterogeneous system microstructure; for warm glacier ice this is the geometry of the water-filled vein and node network within the ice crystal structure. This approach commonly employs an inclusion-based formulation for the system, where one component is treated as a homogeneous matrix while the other components are represented as embedded homogeneous inclusions. Mathematical expressions for the effective properties of this heterogeneous system are derived using a volumetric average of the appropriate fields within each component. Evaluation of the field within a given inclusion requires a description of its interaction with other inclusions in the system. Provided consistent forms of these interactions are specified for different fields (e.g., electric and strain for GPR and seismic respectively), consistent estimates of macroscopic physical properties are obtained [Endres and Knight, 1991].
 Given the water content of glacier ice, we use an ice matrix with water-filled inclusions. The dielectric and elastic properties as well as densities of water and ice used in modeling are given in Table 1. For modeling purposes, these inclusions are divided into two distinct shapes: spheres (representing the nodes) and flattened oblate spheroids (representing veins). The choice of the oblate spheroids is necessary to account for the potential variation in elastic wave velocities due to the high compressibility of the tapered portions of the veins. This node-vein model for glacial ice is equivalent to the sphere-crack model successfully used to model the elastic properties of porous rocks [Endres and Knight, 1997].
Table 1. Physical Properties Used in Effective Media Formulationa
Dielectric permittivity (ɛo = permittivity of free space), elastic moduli and density values used for water and water-free ice components.
 The geometry of the veins and nodes is specified by the inclusion aspect ratio α1 = 1 for the nodes and 0 < α2 < 1 for the veins, where α = a/b, where a is the unique axial length and b is the rotational axial length. The volumetric fraction of nodes and veins in this system are ϕ1 and ϕ2, respectively, and the total water content is ϕw = ϕ1 + ϕ2.
 The macroscopic elastic moduli of the glacier ice system were determined using a version of the equivalent inclusion-average stress approximation [Endres and Knight, 1997]. For seismic wave propagation, the effects of pore geometry are incorporated through the open system bulk and shear moduli (* and *, respectively), which describe the behavior of the medium under drained conditions (i.e., pore fluids can freely enter and leave the medium in response to an applied stress). These moduli are obtained from
the functions P and Q are defined in Berryman . The Gassmann  relationships give the following expressions for the macroscopic elastic moduli k* and μ* during seismic wave propagation:
The P-wave velocity VP predicted by the node-vein model for the wet ice is given by
where the effective density of this heterogeneous system is
An effective medium formulation for dielectric properties that uses an equivalent form of inclusion interactions is given by Fricke [1924, 1953]. When expressed in terms of the node-vein model for glacier ice, we obtained
and A(αn) is the unique axis depolarization coefficient. Assuming that the magnetic permeability is that of free space, the electromagnetic (EM) wave velocity VEM predicted by the effective medium formulation for the wet ice is
where c is the velocity of EM radiation in free space. Both of these inclusion-based formulations give results that satisfy the corresponding Hashin and Shtrikman [1962, 1963] bounds.
3. Field Data and Analysis
 In order to demonstrate the efficacy of the method, we analysed ground-penetrating radar (GPR) and seismic data collected at the polythermal glacier, Bakaninbreen, Svalbard. These data included co-located 100 MHz GPR common midpoint (CMP), seismic refraction surveys, and a 4-fold stacked section of seismic data, all orientated in the cross-glacier direction and collected over a few weeks. Details of the field site and data acquisition, as well as processing of the seismic stacked section are described by Murray et al. [2000b] and Smith et al. .
 The GPR data were processed using a running-mean filter to remove low-frequency source-generated noise (termed wow) and Ormsby bandpass filters (corner frequencies at 20-55-110-180 MHz), first-break static corrections, and an energy-decay gain function; a top mute was also imposed to remove direct wave energy. Processed data were input to semblance analyses, from which a two-layer velocity:depth model was derived; a shallow (2.4 m) layer of snow was included in depth conversions, although a distinct reflection from the snow-ice interface was not observed in the CMP gather. Since true medium velocity can only be expressed by the first-break of a wavelet, output stacking velocities were backshifted to correct for semblance responses to non-zero phase GPR wavelets [Murray et al., 2007; Booth et al., 2008]. Velocity precision was assessed using Monte Carlo simulation [Murray et al., 2007] to consider all model permutations between successive semblance peaks at their 90% threshold.
 Semblance responses to seismic data were difficult to interpret since the CMPs had low fold-of-cover and recorded amplitudes were clipped. However on application of NMO (normal moveout) corrections and horizontal stacking [Smith et al., 2002], the reflection from the glacier bed appears at 67 ms two-way travel time in the stacked section. For the upper 52 m of the glacier, seismic velocity was established from a refraction survey [Smith et al., 2002]. We then calculate seismic velocity through the warm ice using the travel time observed in seismic stacked section and the depth derived from the GPR velocity analysis, which assumes that the GPR and seismic reflect at the same interface. Seismic velocity precision was calculated from that of the GPR depth estimate.
 The velocity-water content relationships for each model were obtained by holding the proportion of water in the veins Φv = ϕ2/ϕw constant while varying ϕw in the model (Figure 1). To obtain a realistic range of P-wave velocity variations, it was necessary to use α2 = 0.001 as the vein aspect ratio. Both VP and VEM decrease monotonically with increasing water content, and the magnitude of the velocity decrease grows as Φv increases (Figure 1). Both VP and VEM are dominated at low water contents by water within veins, and are relatively insensitive to water contained within spherical nodes (Figure 1); this is especially true of seismic velocity. Furthermore, it is clear that with a single measurement of either VP or VEM, the water content is not uniquely determined unless the relative abundance of the veins and nodes are also known (Figure 1).
 The variation of GPR and seismic velocity with depth at Bakaninbreen is presented in Figure 2. At this glacier, radar data show the interface between cold and the underlying warm ice (i.e., ice at its pressure melting point) at 57 ± 0.1 m, with the glacier bed at 117 ± 0.5 m; the radar interval velocity in each layer is 0.1703 ± 0.0003 m/ns and 0.1629 ± 0.0014 m/ns, respectively. The seismic velocity-depth variation shows the propagation velocity in the warm ice to be 3.57 ± 0.04 km/s. If these velocities are interpreted using the standard mixing models (Looyenga and Riznichenko), they give inconsistent results, with GPR suggesting much lower water content than seismic velocities (Table 2). This inconsistency is most likely due to the relatively strong influence of veins on the seismic velocity that is not considered in the Riznichenko model. As noted earlier, previously published results from Johnsons Glacier are also inconsistent if interpreted using the Looyenga and Riznichenko formulae, with the radar velocity again suggesting lower water content [Navarro et al., 2005; Benjumea et al., 2003]. However, it should be noted that the two Johnsons Glacier surveys were collected in different years; furthermore, the radar velocity was derived by fitting hyperbolae to common-offset data, and no correction has been made to either the radar or seismic velocity for non-zero-phase wavelets.
Table 2. Water Contents Derived From the Separate Application of Looyenga Formula to Radar Velocity and Riznichenko to Seismic Velocity Compared to the Joint Inclusion-Based Estimate from the Effective Medium Modelling
 In Figure 3, we cross-plot effective medium model results for P-wave against EM propagation velocities over a range of porosities and vein-node ratios. Superimposed on these plots are results from our Bakaninbreen analysis and from Johnsons Glacier, with their uncertainties. Both lie significantly away from the line that cross-plots the Looyenga and Riznichenko formulations (L-R in Figure 3). Indeed the water contents predicted by effective medium modelling lie between those predicted by Looyenga and Riznichenko models (Table 2). At Bakaninbreen our results are consistent with most water being present in spherical nodes (90%), and suggest that the water content in warm ice is 1.4%. At Johnsons Glacier 96% of water is in nodes, and the predicted water content is 2.0%.
5. Summary and Conclusions
 We have shown that the traditional models for interpreting geophysical properties as ice-water content are inconsistent and, when both VP and VEM have been measured in the field, produce contradictory estimates. We have presented an alternative model that uses consistent effective medium descriptions of the dielectric and elastic velocities for wet ice. This model shows that with a single measurement of either property, the water content is not uniquely determined unless the relative proportion of water in the veins and nodes is also known. Hence, simultaneous measurement of both radar and seismic properties is required to ascertain ice-water contents, and additionally provides the geometry of the included water. We used this model to interpret field data from the polythermal glacier Bakaninbreen, Svalbard, and calculated estimates of the water content simultaneously from the measured seismic and radar velocities. The warm ice at the glacier is interpreted to have a water content of 1.4%, of which 90% is contained within spherical nodes rather than the high aspect ratio veins. No information about the size of the inclusions is available from this modelling, except that the inclusions are assumed to be much smaller than the wavelength (∼2.0 m for radar and 16 m for seismic data).
 Effective medium modelling allows considerable possibilities for the future investigation of inclusions within ice under a coherent physical framework. A wide variety of properties can be incorporated using this method, including the propagation of shear waves and ultrasound. Furthermore, this method provides a means for interpreting attenuation of radar and seismic waves in terms of the ice water-content. A key development will be the modelling of a 3-phase medium, relatively straightforward with the effective medium approach, to represent both air and water, since several studies indicate that a plausible interpretation of EM properties is only obtained if both water and air are present [Bradford and Harper, 2005; Gusmeroli et al., 2009]. A three-phase mixture may actually be required to fully characterise the cold-ice at Bakaninbreen if the high velocity of the GPR wavelet is attributed to some fraction of air. The combined use of radar and seismic velocities, with corresponding attenuations, would provide sufficient parameters to estimate the volumetric content and geometry of both air and water inclusions within glacier ice. More complex physics could also be incorporated, for example to simulate EM wave propagation through electrically conductive water inclusions.
 This research was funded by NERC grants NER/A/S/2002/01000 and GST/02/2192. Seismic data collection was led by A.M. Smith (British Antarctic Survey).