A THEOREM FOR DIRECT CURRENT REGIMES AND SOME OF ITS CONSEQUENCES *

For any direct current regime, the theorem holds, where φ_p is the total measured or calculated potential at any point P, φ is the potential distribution known a priori, r is the distance between P and any volume element dV, the gradients are evaluated at the element, and the current sources and sinks have finite dimensions. Thus, each space element behaves as a dipole of moment (1/4π) ?φdV and contributes its share of signal or potential accordingly. By suitable summation or integration, the contribution from any assigned portion of space to the total measured signal can be determined. Except for the chargeability factor m, the formula also establishes Seigel's initial postulate for the time domain induced polarisation theory. The contribution depends on the potential gradient, not the current density, and the integration extends over the entire space. Although an insulating target carries no current, it contributes a signal that is in general larger than normal by virtue of its higher potential gradient, and thus helps in creating an overall positive anomaly or resistivity high. On the other hand, an infinitely conducting target—even though it supports a larger amount of current than normal—contributes nothing to the measured signal as the potential gradient is zero everywhere inside. Thus, by contributing less than normal, a conducting target promotes the creation of what is usually a resistivity low. In all cases, the contributions from the space elements add up exactly to the measured or total calculated value. Some other consequences of the theorem are also discussed, especially in relation to a simple two-layer earth. For instance, the contribution from the upper half-space (air) turns out to be equal to that from the lower (real ground), for all observation points on the ground surface and for any ground configuration.