A new potential method for the $X_{\rm max}$ measurement of extensive air showers based on backtracking radio signals

{Measurements of cosmic-ray composition based on air-shower measurements rely mostly on the determination of the position of the shower maximum ($X_\mathrm{max}$). One efficient technique is to image the development of the air shower using fluorescence telescopes. An alternative technique that has made significant advances in the recent years is to measure the radio emission from air shower. Common methods for $X_\mathrm{max}$ determination in the radio detection technique include fitting a two-dimensional radio intensity footprint at the ground with Monte-Carlo simulated showers which is computationally quite expensive, and others that are based on parameterizations obtained from simulations. In this paper, we present a new method which is computationally extremely efficient and has the potential to reconstruct $X_{\rm max}$ with minimal input from simulations. The method involves geometrical reconstruction of radio emission profile of air showers along the shower axis by backtracking radio signals recorded by an array of antennas at the ground. On implementing the method on simulated cosmic-ray proton and iron showers in the energy range of $\rm 10^{17}-10^{18}\,eV$, we find a strong correlation between the radio emission profile obtained with the method in the $20-80$~MHz frequency range and the shower longitudinal profile, implying a new potential way of measuring $X_\mathrm{max}$ using radio signals.}