I. Introduction
Vertical resistivity depth soundings provide an attractive geophysical means for the determination of glacial ice thickness and have been successfully employed in a number of studies (Reference RöthlisbergerRöthlisberger, 1967). The interpretation of field data, which may be performed at the field site, consists of comparing the accumulated measurements with theoretical or master curves from which the ice thickness may be directly inferred. Details of this procedure are found in Reference Van Nostrand and CookVan Nostrand and Cook (1966). The master curves utilized in the interpretation of glacial data are calculated for plane-layered models and do not take into account the trough-like shape of valley glaciers. Reference RöthlisbergerRöthlisberger (1967) recognized the possible significance of the trough-like shape, resulting from the presence of steep valley walls, and attempted to estimate corrections to the plane-layered curves through the results of an analogue model study. The model employed, however, did not account for a layer of high-conductivity ice overlaying much more resistive ice, which is a typical feature of valley glaciers. In addition the effects of varying the ratio of ice thickness to glacier width was not investigated in detail. Thus the correction he obtained with the analogue model was limited to the specific model conditions.
A more general theoretical or experimental treatment of the effect of glacial cross-section does not appear to exist, although Reference CarpenterCarpenter (1955) presents a method which can be used to determine the presence of lateral inhomogeneities.
In view of the utility of vertical resistivity profiles in glacial studies it would appear desirable to obtain master curves more appropriate to typical valley glacier cross-sections. While a finite difference method could be employed to generate such master curves (Madden, unpublished; Jepsen, unpublished) the comparatively simple cross-sectional geometry and relatively homogeneous nature of valley glaciers renders the problem amenable to treatment as a boundary-value problem. Since they are rapidly calculated, the analytical solutions obtained in this manner may then be used to investigate the effect of all involved parameters.
II. Theory
The model of a layered trough embedded in a perfectly conducting half-space (Fig. 1) provides a good approximation of a typical valley glacier. The perfectly conducting halispace is a satisfactory representation of the high conductivity of host rocks in comparison to ice. The two layers within the trough, of resistivity ρ 1 and ρ 2 respectively, allow the effect of resistivity contrasts between the ice layers to be examined.
Fig. 1. Layered trough model.
The potentials ϕ 1( x, y, z) and ϕ 2( x, y, z) obey Laplace’s equation, ∇2 ϕ 1( x, y, z) = 0 and ∇2 ϕ 2( x, y, z) = 0, in the first and second layer, respectively. With a current source I located at the point (x 0, 0, 0) the boundary conditions are:
Solution of Laplace’s equation by separation of variables (Reference Irving and MullineuxIrving and Mullineux, 1959) and evaluation of the arbitrary constants by application of the boundary conditions, yields for the potential in the upper layer:
where
λ l are the solutions of the eigenvalue equation
and
In obtaining this solution it is necessary to note that the y-dependent eigenfunction obeys:
where
For purposes of numerical evaluation it should also be noted that the eigenvalues λ i are independent of m.
The above expression for the potential in the upper layer may be used in any of the definitive expressions for apparent resistivity ρ a . For the Wenner configuration of Figure 1 this yields:
where a is the electrode spacing. Hence
The expression for ρ a converged rather slowly at small a spacings, but provided no serious obstacle to numerical evaluation.
Apparent resistivity and the effect of the valley walls
The effect of the valley walls for a constant conductivity contrast, K = ρ 1/ρ 2, and layer thickness h is to shift the peak of the ρ a curve towards smaller a spacings (Figures 2 and 3). The effect of this shift is to decrease the estimated ice thickness. In both Figures 2 and 3 there exists a value of L x /L y which is the minimum value the ratio may have without significantly affecting the apparent resistivity curves. Denoting the minimum value of the ratio by L min it is evident from Figures 2 and 3 that Lmin is dependent upon the Cagniard parameter α defined as,
Fig. 2. Apparent resistivity versus a spacing.
Fig. 3. Apparent resistivity versus a spacing.
Thus, in Figure 2 with α = 11.1, the valley walls will affect the apparent resistivity curves if L min < 10 while in Figure 3 with α = 0.1 the apparent resistivity will be affected only if L min < 1.5. This is exactly the effect anticipated by Reference RöthlisbergerRöthlisberger (1967). Without attempting to be exhaustive the apparent resistivity curves were generated for several values of α, obtained with various combination of h and K and from these a value of L min selected. Several of these models are listed in Table I. The selected values of L min versus α are shown in Figure 4. A reasonable rule for the use of plane layered master curves in the interpretation of field data is that the ratio L x /L y appropriate for the data be greater than L min of Figure 4.
Fig. 4. Minimum value of Lx/Ly needed for use of plane layer curves; Lx/Ly should lie above the curve.
Table I. Layer Thickness, Contrasts, α and Lmin for Models Considered
III. Application to Field Data
The necessity of accounting for the valley glacier cross-section in the interpretation of field data is evident from the study of Reference RöthlisbergerRöthlisberger and Vögtli (1967) on the Unteraargletscher. The ice thickness was known from a previous seismic study of this glacier. This allowed the known ice thickness to be compared with the results of the resistivity survey. Figure 5 shows the field data obtained with a Schlumberger electrode configuration and converted by Röthlisberger and Vögtli to equivalent Wenner measurements through an approximate transformation, and the theoretical curve for two plane layers over a half space calculated by Röthlisbcrger and Vögtli on the basis of the seismically determined ice thickness. The lack of agreement is evident and, as stated by Röthlisberger and Vögtli, the correct ice thickness could not be determined from the resistivity data.
Fig. 5. Apparent resistivity versus a spacing. For trough model Ly = 500 m, Lx = 1.25 Ly, K = 100, h/Ly = 0.01 For plane layered model α = 0.5, Ly = 350.
The known cross-section of the glacier can be approximated by a trough 2.5 depth units wide (L x = 1.25Ly) with an upper layer thickness of 0.01 depth units and a K of 10. The theoretical curve for this structure is included in Figure 5. The agreement between the field data and theoretical curve is excellent. A comparison of the glacier cross-section determined seismically, taken from Reference RöthlisbergerRöthlisberger and Vögtli (1967), and the glacier model from which the master curve of Figure 5 was calculated, is shown in Figure 6. Röthlisberger and Vögtli concluded that the deviation of field data from theoretical curves was the result of inhomogeneitics within the ice. On the basis of Figure 5 it is evident that a significant part of the deviation resulted from the presence of the valley walls and layers of the glacier.
Fig. 6. Comparison of layer trough model fit to Unteraargletscher field data with seismic cross-section.
IV Summary and conclusions
A formulation has been given for calculating the apparent resistivity of a two-layered glacier in a perfectly conducting valley. The analysis could be modified in a straight-forward manner to allow the glacier to have three or more layers as long as the valley floor and walls are essentially perfect conductors. The same type of eigenfunction analysis could also be used to interpret electrical surveys in valleys where the walls and floor are very resistive compared to layered material filling the valley.
Based on “master curves” computed with this formulation, it has been shown that, in order to neglect the presence of the valley walls, the ratio of L x /L y must increase as the Cagniard parameter α increases. A curve showing the minimum allowable values of L x /L y has been given. It is suggested that plane-layer curves not be used to interpret field data unless the L x /L y ratios are greater than the indicated minimum.
Finally we have reinterpreted an electrical profile of Reference RöthlisbergerRöthlisberger and Vögtli (1967) for the Unteraargletscher. The thickness of the ice as we interpret it, is about 500 m, as compared with the seismic interpretation of 400 m.
