Curves for p 1 = –2, p 2 = 0 and g = 2 are shown in Figure 1.Ī Data-Assimilation Example from Glaciology The function g gives the square of the distance from the origin, so circles centered on the origin represent contours of constant g. The solution to the system of equations f(u, p) = 0 is represented by the point where the line and the cubic curve cross. If we plot solutions of f 2( u 1, u 2, p 1, p 2) = u 3 1 – u 2 + p 2 = 0 in the ( u 1, u 2) plane we obtain a cubic curve that crosses the vertical axis at p 2 with slope 0.
If we plot solutions of f 1( u 1, u 2, p 1, p 2) = u 1 + u 2 + p 1 = 0 in the ( u 1, u 2) plane we obtain a line with slope –1 and vertical intercept – p 1. To gain further appreciation of ∂g/∂ p, we consider a graphical representation of our example problem. Indeed, substituting u 1 = 0 and u 2 = 0 into our expression, confirm this, Since g = u 2 1 + u 2 2 ≥ 0 (always), g( u) = 0 must be a global minimum so we expect ∂g/∂ p = 0. To appreciate the meaning of these derivatives we consider a couple of specific value pairs for the parameters p 1 and p 2.įirst, if p 1 = 0 and p 2 = 0, the equation f( u, p) = 0 has solution u = 0, so g( u) = 0. Reference Brinkerhoff, Meierbachtol, Johnson and HarperBrinkerhoff and others, 2011) no attempt is made to review them here. Although applications using real glaciological data can be found in various published research papers (e.g. Application to an idealized data-assimilation problem from glaciology is described in the final section. In this paper ‘adjoint methods’ are described in the context of optimization problems.
Thus, finding appropriate values for model parameters can be accomplished by solving an optimization problem, namely that of minimizing a measure of the difference between calculated and measured values. a frictional coefficient along the bed) that result in the best match between values calculated by the model (surface velocities, perhaps) and measured data. The idea behind data assimilation is to find the values of the parameter of interest (e.g.
We use the phrase ‘data assimilation’ to refer to a method where more easily measured data, such as surface velocities, are used to estimate the values of such parameters. (Even basal topography can be considered a model ‘parameter’.) Many of these parameters are difficult to measure directly. Numerical models of glaciers and ice sheets include parameters such as viscosity (often expressed using other parameters), density, thermal conductivity, heat capacity, geothermal heat flux and basal traction coefficients.