# The Amplitude-Phase Decomposition for the Magnetotelluric Impedance Tensor

## TL;DR

This paper introduces a novel impedance tensor decomposition into a Phase Tensor (PT) and a new Amplitude Tensor (AT), enabling comprehensive analysis of subsurface resistivity and distortion effects in magnetotelluric data.

## Contribution

The paper presents a complete impedance tensor decomposition into PT and AT, with the AT capturing amplitude information and being independent of PT, applicable to 3D subsurface models.

## Key findings

- AT contains galvanic and inductive amplitudes.
- Geometric AT parameters converge to PT in 1D/2D models.
- AT can recover galvanic electric anisotropic distortion.

## Abstract

The Phase Tensor (PT) marked a breakthrough in understanding and analysis of electric galvanic distortion but does not contain any impedance amplitude information and therefore cannot quantify resistivity without complementary data. We formulate a complete impedance tensor decomposition into the PT and a new Amplitude Tensor (AT) that is shown to be complementary and mathematically independent to the PT. We show that for the special cases of 1D and 2D models, the geometric AT parameters (strike and skew angles) converge to PT parameters and the singular values of the AT correspond to the impedance amplitudes of the transverse electric and transverse magnetic modes. In all cases, we show that the AT contains both galvanic and inductive amplitudes, the latter of which is argued to be physically related to the inductive information of the PT. The geometric parameters of the inductive AT and the PT represent the same geometry of the subsurface conductivity distribution that is affected by induction processes, and therefore we hypothesise that geometric PT parameters can be used to approximate the inductive AT. Then, this hypothesis leads to the estimation of the galvanic AT which is equal to the galvanic electric distortion tensor at the lowest measured period. This estimation of the galvanic distortion departs from the common assumption to consider 1D or 2D regional structures and can be applied for general 3D subsurfaces. We demonstrate exemplarily with an explicit formulation how our hypothesis can be used to recover the galvanic electric anisotropic distortion for 2D subsurfaces, which was, until now, believed to be indeterminable for 2D data. Moreover, we illustrate the AT as a mapping tool and we compare it to the PT with both synthetic and real data examples. Lastly, we argue that the AT can provide important non-redundant amplitude information to PT inversions.