## 1. Introduction

[2] 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.

[3] Simple relationships (known colloquially as “mixing laws”) have been used to calculate water content ϕ_{w} from measured velocities. The two most common mixing laws are those of *Looyenga* [1965] and *Riznichenko* [1949], derived for radar and seismic data respectively. Both are members of a continuous spectrum of power-mixing laws originally derived by *Lichtenecker and Rother* [1931], 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 ɛ_{i} and water ɛ_{w}. 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 (*k*_{i}, *μ*_{i} for water-free ice; *k*_{w}, *μ*_{w} 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].

[4] 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.