# Variance estimate of solution of inverse problem by nonlinear least squares with multi-dimensional model function?

The inverse problem I solve has the following basic outline: a number $$n_g$$ of Gaussian kernels ($$\textbf{x}$$) of some width $$\sigma$$ and each with an amplitude $$s_j$$ is propagated in n-d space to $$\textbf{x}+\textbf{v}(\textbf{x})*dt$$ by some n-d polynomial function $$\textbf{v}(\textbf{x})$$ with coefficients $$\boldsymbol{\beta}$$, where $$R^{n_g}\rightarrow R^{n_p}$$ function $$f$$ then generates an image consisting of $$n_p$$ pixels, which is compared to a measurement $$\textbf{y}$$. The coefficients $$\boldsymbol{\beta}$$ are then found as that which minimizes $$S=\sum_i (y_i-f_i(\textbf{x},\boldsymbol{\beta}))^2$$

The usual least squares formulation is to minimize $$S=\sum_i (y_i-f(x_i,\boldsymbol{\beta}))^2$$, where $$f:R\rightarrow R$$. This has an associated covariance matrix $$F_{ij}=\frac{\partial f(x_i,\boldsymbol{\beta})}{\beta_j}$$ and variance $$\text{var}(\beta_j)\approx\frac{S}{n-m}([F^TF]^{-1})_{jj}$$, where $$n$$ is the dimension of $$\textbf{y}$$ and $$m$$ is the dimension of $$\boldsymbol{\beta}$$.

When attempting to estimate the variance in the $$\beta_j$$ obtained by the solution in my inverse problem, I was wondering if it would be valid to define the covariance as $$F_{ij}=\frac{\partial f_i(\textbf{x}, \boldsymbol{\beta})}{\beta_j}$$, and to keep $$n$$ as the dimension of $$\textbf{y}$$. I was also wondering if there were any useful reading references for inverse problems of that form?