  Source conditions of the type $x^\dag \in\mathcal{R}((A^\ast A)^\mu)$ are an
important tool in the theory of inverse problems to show convergence rates of
regularized solutions as the noise in the data goes to zero. Unfortunately, it
is rarely possible to verify these conditions in practice, rendering
data-independent parameter choice rules unfeasible. In this paper we show that
such a source condition implies a Kurdyka-\L{}ojasiewicz inequality with
certain parameters depending on $\mu$. While the converse implication is
unclear from a theoretical point of view, we demonstrate how the Landweber
method in combination with the Kurdyka-\L{}ojasiewicz inequality can be used to
approximate $\mu$ and conduct several numerical experiments. We also show that
the source condition implies a lower bound on the convergence rate which is of
optimal order and observable without the knowledge of $\mu$.
