DCIP2D:
General Methodology for Inverting 2D DC and IP Data

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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 d = (d₁,d₂,...,d_N) denote the data. So d_i could be the i^th potential in a dc resistivity data set or the i^th 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 section.

The goal of the inversion is to recover a model vector m = (m₁,m₂,...,m_M) 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. 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 which 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 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

  (8)

In equation (8) the functions w_s ,w_x ,w_z 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

    (9)

If w_s ,w_x ,w_z are set equal to unity then W_s is a diagonal matrix with elements where x is the length of the cell and z is its thickness, W_x has elements where dx is the distance between the centers of horizontally adjacent cells, and W_x 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:

    (10)

In equation (10) m₀ is a base model and W_m 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 W_m 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 W_d can be set to and the target misfit should be _d^* = N.