Signal model and event reconstruction for the radio detection of inclined air showers

The detection of inclined air showers (zenith angles $\theta \gtrsim 65^\circ$) with kilometer-spaced radio-antenna arrays allows measuring cosmic rays at ultra-high energies ($E \lesssim 10^{20}\,\mathrm{eV}$). Radio and particle detector arrays provide independent measurements of the electromagnetic and muonic shower components of inclined air showers, respectively. Combined, these measurements have a large sensitivity to discriminate between air showers initiated by lighter and heavier cosmic rays. We have developed a precise model of the two-dimensional, highly complex and asymmetric lateral radio-signal distributions of inclined air shower at ground -- the ``radio-emission footprints''. Our model explicitly describes the dominant geomagnetic emission with a rotationally symmetric lateral distribution function, on top of which additional effects disturb the symmetry. The asymmetries are associated with the interference between the geomagnetic and sub-dominant charge-excess emission as well as with geometrical projection effects, so-called ``early-late'' effects. Our fully analytic model describes the entire footprint with only two observables: the geometrical distance between the shower impact point at the ground and the shower maximum $d_\mathrm{max}$, and the geomagnetic radiation energy $E_\mathrm{geo}$. We demonstrate that with this model, the electromagnetic shower energy can be reconstructed by kilometer-spaced antenna arrays with an intrinsic resolution of 5\% and a negligible bias.