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.
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