  We constrain the halo profiles outside the halo boundaries by solving for the
matching profiles required by the halo model. In the halo model framework, the
matter distribution in the universe can be decomposed into the spatial
distribution of halos convolved with their internal structures. This leads to a
set of linear equations in Fourier space which uniquely determines the optimal
halo profiles for any given halo catalog. In this work, we construct three halo
catalogs with different boundary definitions, and solve for the optimal
profiles in each case using measurements of halo-matter and halo-halo power
spectra. Our results show that for a given halo field, there is always a set of
matching profiles to accurately reconstruct the input statistics of the matter
field, even though it might be complex to model the profiles analytically.
Comparing the solutions from different halo catalogs, we find their mass
distributions inside the inner depletion radii are nearly identical, while they
deviate from each other on larger scales, with a larger boundary resulting in a
more extended profile. For the depletion radius based catalog, the numerical
solution agrees well with the Einasto profile. Coupling the Einasto profile
with the depletion catalog, the resulting halo model can simultaneously predict
the halo-matter power spectra to $10\%$ and matter-matter power spectrum to
$5\%$, improving over conventional models in both the interpretability and
versatility.
