The computing programs outlined in this manual solve two inverse problems. In the first we invert the DC potentials (or equivalently the data in Fig 3b) to recover the electrical conductivity (x,z). This is a nonlinear inverse problem that requires linearization of the data equations and subsequent iteration. Next we invert the IP data in Fig 4b to recover the chargeability (x,z). Because chargeabilities are usually small quantities ( 0.3) it is possible to linearize equation (6) and derive a linear system of equations to be solved. Irrespective of which data set is being inverted however, we basically use the same methodology to carry out the inversions.
To outline our methodology it is convenient to introduce a single
notation for the "data" and for the "model". We let
The goal of the inversion is to recover a model vector
where Wd is a datum weighting matrix. In this work we shall assume that the noise contaminating the jth observation is an uncorrelated Gaussian random variable having zero mean and standard deviation j . As such, an appropriate form for the N×N matrix is . With this choice, d is the random variable distributed as chi-squared with N degrees of freedom. Its expected value is approximately equal to N and accordingly, d*, the target misfit for the inversion, should be about this value.
Earth conductivity distributions are complex. To allow maximum flexibility to produce a model of arbitrary shape it is important that M, the number of cells representing the model, is large. In our inversions M will almost always be greater than N, the number of data. The inverse problem therefore reduces to finding a set of M parameters using only N data constraints under the condition that M,N. Clearly the solution is nonunique and this nonuniqueness represents the principle obstacle for obtaining unambiguous information about earth structure from the observations.
Any inversion algorithm (if it works) will produce a model which reproduces the data. But there are infinitely many models possible. So which one does the algorithm produce? It is not good practise to let the program make a random selection. Rather, a responsible approach is to direct the inversion algorithm to produce a model that is geologically reasonable and is constrained by additional information if that information is available. This can be implemented by formulating a "model objective function" which, when minimized, produces a model with desirable characteristics. The critical aspect of the inversion is therefore to form the model objective function which we characterize by m . To do this, the inversionist must ask the question "what type of model is desired?". Should the model be smooth, should it be blocky? Is there a reference or background model that the constructed model should emulate? If there is a reference model, is it better known in some places than others so that the constructed model should be close to the reference model in certain locations but can depart from our preconceived ideas in other areas? Whatever the answer to these questions, a guiding philosophy should always be to find a model which (in some sense) is "as simple as possible". The nonuniqueness inherent in the inversion generally means that we can generate models which are arbitrarily complicated. We cannot however, make models that are arbitrarily simple. For example a halfspace will generally not reproduce data acquired from a geophysical survey.
In the inversion algorithms in DCIP2D our choice for the objective function m is guided by a desire to find a model which has minimum structure in the vertical and horizontal directions and at the same time is close to a base model m0 . To accomplish this we minimize a discretized approximation to
In equation (8) the functions ws ,wx ,wz are specified by the user and the constant s controls the importance of closeness of the constructed model to the base model m0 and x , z control the roughness of the model in the two directions. Varying the ratio x / z allows the construction of models that are smoother, thus more elongated, in either x- or z-direction. The discrete form of equation (8) is
If ws ,wx ,wz are set equal to unity then Ws is a diagonal matrix with elements where x is the length of the cell and z is its thickness, Wx has elements where dx is the distance between the centers of horizontally adjacent cells, and Wx has elements where dz is the distance between the centers of vertically adjacent cells.
The inverse problem is now properly formulated as an optimization problem:
In equation (10) m0 is a base model and Wm is a general weighting matrix which is designed so that a model with specific characteristics is produced. The minimization of m yields a model that is close to m0 with the metric defined by Wm and so the characteristics of the recovered model are directly controlled by these two quantities. If the data errors are Gaussian and their standard deviations have been adequately estimated then Wd can be set to and the target misfit should be d* = N.