  We reconstruct spectra of secondary X-rays from a tunable 250-350 MeV laser
wakefield electron accelerator from single-shot X-ray depth-energy measurements
in a compact (7.5 $\times$ 7.5 $\times$ 15 cm), modular X-ray calorimeter made
of alternating layers of absorbing materials and imaging plates. X-rays range
from few-keV betatron to few-MeV inverse Compton to >100 MeV bremsstrahlung
emission, and are characterized both individually and in mixtures. Geant4
simulations of energy deposition of single-energy X-rays in the stack generate
an energy-vs-depth response matrix for a given stack configuration. An
iterative reconstruction algorithm based on analytic models of betatron,
inverse Compton and bremsstrahlung photon energy distributions then unfolds
X-ray spectra, typically within a minute. We discuss uncertainties, limitations
and extensions of both measurement and reconstruction methods.
