Herglotz pairing with an obstacle far field

Pairing a far-field pattern with the conjugate density of a Herglotz wave function converts the angular integral into boundary data:This follows by substituting the surface representation of an obstacle far-field pattern and interchanging the integrals.

If measured far-field patterns determine a density $g$ and its Herglotz wave function equals a prescribed spherical wave on the obstacle boundary, that trace identity can locate the boundary. A star-shaped axisymmetric obstacle can be represented by one radial function $h(\theta )$. Numerically, expand $g$ and $h$ in suitable bases, match the far-field constraints, and minimize the complex boundary residual at angular collocation points, using regularization of an inverse problem for noise and small singular values. A single scalar pairing constraint alone does not determine an arbitrary density.