DCIP3D manual: 
1.3 General Methodology for Inverting DC and IP Data

The computer programs outlined in this manual solve two inverse problems. In the first we invert the DC potentials (or equivalently the data as illustrated in Figure 3) to recover the electrical conductivity (x,y,z). This is a nonlinear inverse problem that requires linearization of the data equations and subsequent iteration. Next we invert the IP data (such as those in Fig ure 4) to recover the chargeability (x,y,z). Because chargeabilities are usually small quantities ( < 0.3 is usual) it is possible to linearize equations (5) and (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 d = (d₁,d₂,...d_N) denote the data, where N is the number of data. So d_i could be the i^th potential in a DC resistivity data set, or the secondary potential or apparent chargeability in an IP survey. Let the physical property of interest be denoted by the symbol m. The quantity m_i can denote the conductivity or chargeability for the i^th cell. For the inversion we choose m_i = ln_i when inverting for conductivities and m_i = _i when reconstructing the chargeability distribution.

The goal of the inversion is to recover a model vector m = (m₁ ,m₂ ,...,m_M ), where M is the number of model cells that acceptably reproduces the N observations d^obs = (d₁^obs, d₂^obs,...,d_N^obs). Importantly, the data are noise contaminated so we don't want to fit them precisely. To do so would ensure that we do not have the correct earth model because some features observed in the constructed model would assuredly be artifacts of the noise. Alternatively, if we fit the data too poorly then information about the conductivity that is coded in the data will not have been recovered. Our objective therefore is neither to underfit nor overfit the data. Rather, we want to find a model that reproduces the data only to within an amount that is justified by the estimated uncertainty in the data. To accomplish this we introduce a global misfit criterion

     (7)

where W_d is a datum weighting matrix. In this work we shall assume that the noise contaminating the j^th observation is an uncorrelated Gaussian random variable having zero mean and standard deviation _j. As such, an appropriate form for the matrix is W_d = diag{1/₁,...,1/_N}. 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 (and chargeability) distributions are complicated. 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. Therefore 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 possible models. 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 such a reference model, is it better known in some places than others? In other words, should the constructed model be close to the reference model in certain locations while being allowed to 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". One important guiding principle is that the nonuniqueness inherent in the inversion generally means we can generate models which are arbitrarily complicated, yet we cannot 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 DCIP3D our choice for the objective function _m is guided by a desire to find a model which has (i) minimum structure in the vertical and horizontal directions and at the same time is (ii) close to a reference model m₀. To accomplish this we minimize a discretized approximation to

     (8)

In equation (8) the functions are specified by the user and the constant _s controls the importance of closeness of the constructed model to the reference model m₀. The constants _x, _y, _z control the roughness of the model in the three directions. We can de\fn a length scale in each direction as The greater the length scale
is in each direction, the smoother is the model. Varying these scales allows the construction of models that are smoother, thus more elongated, in either x-, y-, or z-direction. To obtain a reasonably smooth model, the length scale should be no less than two cell widths. Given that we always work with a finite model domain, the length scales should be smaller than the respective dimension of the model region.

The discrete form of equation (8) is

     (9)

The matrices W_s, W_x, W_y, W_z, are formed by finite difference approximation of the integrals in eq.(8).

The inverse problem is now properly formulated as an optimization problem:

     (10)

Appropriate techniques can be employed to carry out the minimization and the minimizer _m yields the model we are seeking. For DC resistivity and IP inversions we use different minimization techniques that will be discussed in the following sections. In addition, we also impose positivity in the IP inversion to ensure that the recovered chargeability is positive.