DCIP2D:
General Background for DCIP2D

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Introduction

This manual presents theoretical background, numerical examples, and explanation for implementing the program library DCIP2D. This suite of algorithms, developed at the UBC-Geophysical Inversion Facility, are needed to invert DC potentials and IP responses over a 2-D earth structure. The manual is designed so that a geophysicist who has an understanding about DC resistivity and Induced Polarization field experiments, but who is not necessarily versed in the details of inverse theory, can use the codes and invert his or her data.

A typical DC/IP experiment involves inputting a current I to the ground and measuring the potential away from the source. In a time-domain system the current has a duty cycle which alternates the direction of the current and has off-times between the current pulses at which the IP voltages are measured. A typical time-domain signature is shown in Fig. 1. In that Figure, is the potential that is measured in the absence of chargeability effects. This is the "instantaneous" value of the potential measured when the current is turned on. In mathematical terms this potential is related to the electrical conductivity by

    (1)

where the forward mapping operator is defined by the equation

    (2)

and appropriate boundary conditions. In equation (2) is the electrical conductivity in Siemen/metre (S/m), is the gradient operator, I is the strength of the input current in Amperes, and r_s is the location of the current source. For typical earth structures , while positive, can vary over many orders of magnitude. The potential in equation (2) is the potential due to a single current. This is the value that would be measured in a pole-pole experiment. If potentials from pole-dipole or dipole-dipole surveys are to be generated then they can be obtained by using equation (2) and the principle of superposition.

When the earth material is chargeable the measured voltage will change with time and reach a limit value which is denoted by in Fig. 1. There are a multitude of microscopic polarization phenomena which collaborate so that this final value is achieved but all of these effects can be consolidated into a single macroscopic parameter called "chargeability". We denote chargeability by the symbol . Chargeability is dimensionless, positive, and confined to the region [0,1).

To carry out forward modelling to compute we adopt Siegel's (1959) formulation which says that the effect of a chargeable ground is modelled by using the dc resistivity forward mapping but with the conductivity replaced by . Thus

    (3

    (4)

The IP datum, which we refer to as apparent chargeability, is defined by

    (5

    (6

Equation (6) shows that the apparent chargeability can be computed by carrying out two DC resistivity forward modellings with conductivities and (1- ) . Note that in this definition apparent chargeability is dimensionless and, in the case of data acquired over an earth having constant chargeability ₀ , we have _a = ₀ .

The field data from a DC/IP survey are a set of N potentials (ideally but usually ) and a set of N secondary potentials or a quantity that is related to . The goal of the inversionist is to use these data to acquire quantitative information about the distribution of the two physical parameters of interest: conductivity (x,y,z) and chargeability (x,y,z) .

The distribution of conductivity and chargeability in the earth can be extremely complicated. Assuredly earth structure is 3D but for the DC/IP codes developed here we restrict ourselves to 2D structures and assume that the survey has been carried out along a traverse which is perpendicular to strike. The cross-section of the earth is divided into rectangular prisms each having a constant value of conductivity and chargeability.