# Inverse problem with uncertain forward operator

Suppose I want to solve a linear inverse problem. In this example we take a convolution with the kernel: $$\frac{1}{(y^2+z^2)^{3/2}}$$ We only take a fixed $$z$$ for the computation and convolve with respect to $$y$$. The associated matrix $$A$$ is dependent on $$z$$, denoted as $$A_z$$.

The output vector $$b$$ is (spatially) located at the same $$y$$ values, i.e. measurements done along $$y$$. Furthermore, we impose a nonnegativity constraint on the sources $$x$$.

We want to estimate the sources $$x$$ via minimizing the cost function $$||A_zx-b||^2+\lambda\mathcal{R}(x) \,\,\,\,\,\,s.t.\,\,\,x\geq0$$ With $$\mathcal{R}(x)$$ being the regularizer.

My question: If I do not know $$z$$ exactly, am I somehow able to both correctly estimate $$x$$ and $$z$$ simultaneously? I could impose that the solution $$x$$ lies on some ball via the regularizer, such that $$z$$ could (maybe) be estimated correctly. But is this necessary, or can I somehow use less information than knowing the norm of $$x$$ beforehand?

I also think this may be dependent on the spatial frequencies present in the vector $$b$$. That is, if $$b$$ has high spatial frequencies with strong amplitude present, then we must (I think) have a minimum value for $$z$$, because too large $$z$$ can not result in $$b$$ having these high frequencies amplitudes.