# Augmented data for the Backus problem

## TL;DR

This paper investigates an augmented boundary data approach for the Backus problem involving the Laplace equation, using additional boundary information to improve inverse source estimation and numerical recovery of harmonic functions.

## Contribution

It introduces an augmented data framework for the Backus problem, deriving a quasi-linear boundary equation and applying finite element methods for numerical solutions.

## Key findings

- Additional boundary data helps estimate sources more accurately.
- Derived a quasi-linear boundary equation involving augmented data.
- Numerical methods successfully recover harmonic functions from augmented data.

## Abstract

Backus considered a boundary value problem for the Laplace equation with the non-linear data in the form of the magnitude $|Du|$ of the gradient of the solution u. We consider this problem with the data expanded by $(\partial/\partial\nu)|Du|$ given on the boundary of the domain. To justify the requirement for additional data, we use them to estimate the number of sources for the related inverse source problem in the plane. We show that, for an arbitrary dimension, a harmonic function satisfies a quasi-linear equation on the boundary of the domain with the coefficients involving the augmented data. We use the finite element method to recover the harmonic function on the boundary by solving numerically the derived equation.