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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.

The boundary identity gives a numerical route to shape recovery from a Herglotz boundary identity. First recover an admissible density $g$ from measured far-field patterns, enforcing the supplied Herglotz pairing with an obstacle far field for available incident directions:Then evaluate its Herglotz wave function throughout a search region and find a smooth positive radial profile satisfyingThe same $h(\theta )$ must work for every azimuth $\varphi$, expressing the assumed axial symmetry. In practice expand $g$ and $h$ in basis functions, enforce these equations at collocation points, and minimize the complex residual together with the far-field-data residual. The exclusion of an interior Dirichlet eigenvalue ensures uniqueness of the interior Dirichlet problem for the trial boundary data. Use regularization of an inverse problem to control measurement error and unstable density components; neither uniqueness nor numerical stability follows merely from writing the residual equations.

For rich incident-direction data, the conjugated pairing constraints form an adjoint far-field integral equation for $g$. For only one fixed incidence, the displayed pairing is a single complex scalar constraint and cannot by itself determine an arbitrary density. Additional data or a joint constrained shape fit is then needed; the trace condition supplies the actual geometric information.

For large $r=|\mathbf{r}|$, the outgoing kernel hasThus the surface representation of an obstacle far-field pattern, with ${\psi }_s=e^{ikr}{f}_{\infty }/r+O(r^{-2})$, givesConjugating the Herglotz wave function gives $\overline{v_g}({\mathbf{r}}^{\prime })={\int }_{S^2}{e}^{-ik\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime }}\overline{g(\hat{\mathbf{r}})}dS$. Interchanging the angular and obstacle-surface integrals yields the Herglotz pairing with an obstacle far field:This contains only the scattered-field boundary traces and the Herglotz function, with the outward-obstacle normal convention fixing the sign.

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.