  We consider frequency estimation in a noisy environment with noisy probes.
This builds on previous studies, most of which assume that the initial probe
state is pure, while the encoding process is noisy, or that the initial probe
state is mixed, while the encoding process is noiseless. Our work is more
representative of reality, where noise is unavoidable in both the initial state
of the probe and the estimation process itself. We prepare the probe in a GHZ
diagonal state, starting from $n+1$ qubits in an arbitrary uncorrelated mixed
state, and subject it to parameter encoding under dephasing noise. For this
scheme, we derive a simple formula for the (quantum and classical) Fisher
information, and show that quantum enhancements do not depend on the initial
mixedness of the qubits. That is, we show that the so-called `Zeno' scaling is
attainable when the noise present in the encoding process is time
inhomogeneous. This scaling does not depend on the mixedness of the initial
probe state, and it is retained even for highly mixed states that can never be
entangled. We then show that the sensitivity of the probe in our protocol is
invariant under permutations of qubits, and monotonic in purity of the initial
state of the probe. Finally, we discuss two limiting cases, where purity is
either distributed evenly among the probes or concentrated in a single probe.
