82.1 Introduction

Estimating the bed topography and basal slipperiness indirectly through their effect on surface shape and surface velocity is an example of an inverse problem. If we put all available surface measurements into a vector y, denote the basal properties through x, and write the relationship between bed and surface as y = f(x) where f is the forward model, the inverse problem is that of determining the conditional probability distribution function (PDF; P(x | y)) of a system state given the measurement vector. This PDF also can be written as P(b,c | s,u,v,w) where b and c are the bed topography and the basal slipperiness, respectively, and s, u, v and w the surface topography and the three components of the surface velocity vector.

The forward model f may well not have an inverse in the usual sense. We can, for example, expect that for some x, f(x) = 0, indicating that some properties of the bed have no effect on the surface. Using Bayes' theorem, however, this conditional probability also can be written as

where P(y) and P(x) are the prior probability density functions of the measurement vector and the system state, respectively, and P(y|x) is the conditional PDF of y given x. Instead of trying to find the inverse of f, which in general will not exist, we use Bayes' theorem to calculate P(x|y) from P(y|x). Assuming that the PDFs are Gaussian and the forward model can be linearized so that y = Fx + e where F is a matrix and e the measurement errors, taking the minus logarithm of the above expression and maximizing with respect to x leads to a maximum a posteriori solution x of the inverse problem given by x = xa + RaFH(FRaRH + Re)-1(y - Fxa)

where xa is the a priori value of x, Ra and Re are the covariance matrixes of xa and e, respectively, and the superscript H denotes the conjugate transpose (Rodgers, 2000). The above equation gives an estimate for the system state (x) that can be interpreted as a weighted sum of the a priori (xa) and the new information on the system state gained by the measurement (y).

The forward problem consists of determining the surface shape and surface velocities given the bed topography, basal slipperiness and the form of the constitutive equation. The equations to be solved are aijj + fi = 0, and vij = 0 where aij are the components of the symmetrical Cauchy stress tensor, vi are the components of the velocity vector and fi the components of the volume force. A Cartesian coordinate system is used with the x and y axes spanning the horizontal plane, and with the z axis pointing upwards. The constitutive law is Glen's flow law with a stress exponent n. The boundary conditions are the kinematic boundary conditions at the upper (s = s(x,y)) and lower (b = b(x,y)) boundaries, free surface condition for the stresses at the upper boundary and a sliding law, ub = c(x,y)tb, along the lower boundary, where ub is the basal sliding velocity, tb the basal shear stress and c(x,y) the basal slipperiness. This problem is non-linear because the constitutive law is non-linear for n ^ 1 and because the surface fields react in a non-linear fashion to a finite perturbation in either bed topography (b(x,y)) or basal slipperiness (s(x,y)).

The long-wavelength sensitivity of the forward solution to changes in basal properties can be determined analytically for non-linear rheology. For arbitrary length scales, analytical solutions exist for linear rheology and numerical solutions for non-linear rheology (Hutter, 1983; Reeh, 1987; Johannesson, 1992; Gudmundsson, 2003; Raymond et al., 2003, Hindmarsh, 2004). In either case the perturbations in surface topography

(s(x,y)) and the surface velocity (u,v,w), are related in frequency space through transfer functions, for which analytical expressions exist (Gudmundsson, 2003). This relation can be written in the form y = Fx, where y is a vector containing all surface measurements of surface topography and surface velocities, x is a vector containing the to-be-estimated basal slipperiness and bed topography, and F is a matrix with the transfer functions as elements.

The inverse problem consists in determining the maximum of the conditional probability density P(b,c|s,u,v,w) and the covariance of that maximum for b and c, given surface data and the corresponding covariance matrixes. If the measurements of the surface topography and the measurements of each of the surface velocity components are independent, and the bed topography independent of the slipperiness distribution, P(b,c|s,u,v,w) can be written as a product of eight PDFs. Each PDF gives the conditional probability of one surface variable (s,u,v,w) with regard to one unknown basal variable (b,c). Surface and basal quantities (s,u,v,w,b and c) are assumed to be arranged in columns. Assuming all PDFs are Gaussian, maximizing the minus of the logarithm of that product with respect to both b and c leads to b — (Rb)-1(FHR-e's + FHuR-lu + F%R-y + FWR-1w)

where the Fs are the corresponding forward models (for example s = Fsbb), b is the vector giving elevation at each grid location and

An estimate for the slipperiness distribution (c') is of the same form but with c replacing b in all lower indices. To keep the notation simple it has been assumed that the a priori values of bed and the basal slipperiness are equal to zero.

The above expression gives the solution as a function of x and y. To use the transfer function formulation, the surface data must be transformed into frequency space. This usually requires the surface data to be interpolated onto an equidistant grid. This will give rise to interpolation errors and some spatial correlations between interpolation values, both of which must be estimated. Methods of optimal interpolation can be used for this purpose. For the surface topography, for example, we look for an estimate in the form of s' — Fs + e, where F is now an unknown matrix to be determined. The unbiased minimum variance interpolator is given by s' = ssa — Rsa (Rsa + Re)-1 and the covariance matrix of the interpolated values by R' = Rsa - Rsa (Rsa + Re)-1Rsa. The Rsa covari-ance matrix can be determined from an analysis of the experimental variogram (Kitanidis, 1997). Fourier transform of the interpolated surface fields and the covariance matrices leads to an estimate having the same form as the one given above, but with, for example, s substituted by Ws and Rs by WRsWH, where W is the discrete Fourier matrix. If the noise and the a priori fields are uncorrelated, the corresponding covariance matrixes are of diagonal form and the expressions given above can be simplified considerably.

Figure 82.1 Retrieved bed topography and basal slipperiness distribution. The units on the x and y axes are kilometres. The contour interval for the upper figure is 7.5 m and 0.025 m in the lower figure. (See www.blackwellpublishing.com/knight for colour version.)

Figure 82.1 Retrieved bed topography and basal slipperiness distribution. The units on the x and y axes are kilometres. The contour interval for the upper figure is 7.5 m and 0.025 m in the lower figure. (See www.blackwellpublishing.com/knight for colour version.)

As an example for a simultaneous inversion for bed topography and basal slipperiness, synthetic surface data were generated using a Gaussian shaped perturbation in bed topography and slipperi-ness distribution. For the forward problem, 30% uncorrelated noise was added to the solution and the surface data were then inverted. Figure 82.1 shows the retrieved bed topography and basal slipperiness distribution. The (total) basal slipperiness perturbation is CDC, where C is the mean value of the basal slipperiness and DC the (fractional) perturbation. Figure 82.1b shows the contour lines of the retrieved fractional perturbation (DC). In Fig. 82.2 a transverse section along the y axis gives a comparison between the amplitude of the initial (dashed lines) and the retrieved (solid lines) perturbations, with the left-hand peak representing the slipperiness perturbation and the right-hand peak the bed perturbation.

Figure 82.2 Transverse section along the x = 0 line in Fig. 82.1 showing the agreement between retrieved (dashed lines) and original bed properties for bed topography (right-hand peaks) and basal slipperiness (left-hand peaks). (See www.blackwellpublishing.com/knight for colour version.)

Figure 82.2 Transverse section along the x = 0 line in Fig. 82.1 showing the agreement between retrieved (dashed lines) and original bed properties for bed topography (right-hand peaks) and basal slipperiness (left-hand peaks). (See www.blackwellpublishing.com/knight for colour version.)

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