Geophysical inversion

Definition

Let $\mathcal{M}$ be a space of Earth models, such as functions giving a physical property at each point of a region. Let $\mathcal{D} = \R^N$ be the space of data. Let $G \colon \mathcal{M} \to \mathcal{D}$ be a forward operator given by the governing physics, and let $d \in \mathcal{D}$ be observed data, with $d = G(m_{\mathrm{true}}) + e$ for an unknown error $e$. Geophysical inversion is the inverse problem of finding models $m \in \mathcal{M}$ with $G(m)$ close to $d$ within the size of the errors. The usual formulation is the regularised least squares problem $$\min_{m \in \mathcal{M}} \norm{W_d (G(m) - d)}^2 + \lambda \norm{W_m (m - m_{\mathrm{ref}})}^2.$$ Here $W_d$ weights the data by their errors, and $W_m$ measures the roughness or size of the model. The model $m_{\mathrm{ref}}$ is a reference model, and $\lambda > 0$ is a trade-off parameter.