# The geomagnetic/magneto-telluric induction problem with spatial conductivity fluctuations

## TL;DR

This paper generalizes the electromagnetic induction equation to include spatial conductivity fluctuations, providing a framework to recover the fluctuation spectrum from inverse geomagnetic data.

## Contribution

It introduces a new formulation of the Helmholtz equation incorporating arbitrary spatial conductivity fluctuations and discusses how to retrieve the fluctuation spectrum from inverse problems.

## Key findings

- Analytical relation between spectral density and conductivity profile.
- Method to recover conductivity fluctuation spectrum from inverse data.
- Extension of the induction problem to complex, fluctuating Earth models.

## Abstract

The electromagnetic induction equation (Helmholtz equation) for the electrically conducting Earth is generalised to the inclusion of a spatially fluctuating internal conductivity spectrum that is superimposed on a one-dimensional large-scale conductivity reference-profile which depends solely on the vertical coordinate z>0. This large-scale profile is assumed to be known. The distribution of the fluctuations is allowed to be arbitrary. The conductivity-fluctuation generalised Helmholtz equation depends on the spectral density of the fluctuations as function of the vertical and horizontal coordinates. Application to the inverse geomagnetic/magneto-telluric problem is discussed. We show analytically that the inverse problem yields the conductivity fluctuation spectrum. The relation between the spectral density and the obtained conductivity profile is given.