The Comparison Model Method - A New Arithmetic Approach to the Discrete Inverse Problem of Groundwater Hydrology .1. One-Dimensional Flow

Let the steady pressure z[.] of a fluid in a one-dimensional domain be governed by a two-point boundary value problem for the equation (a z_x )_x = f. The identification of position-dependent transmissivity, a[.] from knowledge of f[.] and z[.] is addressed. The ``comparison model method'' (CM method), a direct identification method known for some time and applied to two-dimensional inverse problems, is investigated herewith in the one-dimensional setting. Both the infinite- and finite dimensional cases are covered. In the former, CM method relies on the solution, p[.], of an auxiliary flow equation (the CM) subject to the same f[.] and boundary values as the z[.] equation, whereas a tentative, constant transmissivity, c, is assigned. The quotient of the first derivatives p_x and z_x multiplied by c yields a solution b[.] to the inverse problem. The non-uniqueness of b[.] is examined as a function of c ; cases where existence implies uniqueness are presented; the role of positivity constraints is discussed; ``self-identifiability'' is also defined and discussed. All available results are then translated into the finite-dimensional setting: the appropriate unknown are internode transmissivities. Algebraic and numerical examples are provided.