Thermodynamic profiling using ground-based remote sensing instruments such as differential absorption lidar (DIAL) has the potential to fill observational needs for climate and weather-related research and improve weather forecasting. The DIAL technique uses the return signal resulting from atmospherically scattered light at two closely spaced wavelengths to determine the range-resolved absorption coefficient for a molecule of interest. Temperature profiles can be retrieved using a temperature-sensitive absorption feature of a molecule with a known mixing ratio such as oxygen. In order to obtain accuracies of less than 1 K, the narrowband DIAL equation must be expanded to account for Doppler broadening of molecular backscatter, and its relative contribution to the total signal, the backscatter ratio, must be known. While newly developed low-cost high spectral resolution lidar (HSRL) can measure backscatter ratio with sufficient accuracy, the frequency-resolved DIAL equation, even with this information, remains transcendental, and solving it for temperature can be computationally expensive. In this paper, we present a perturbative solution to the frequency-resolved DIAL equation when we have an HSRL providing the required ancillary measurements. This technique leverages perturbative techniques commonly employed in quantum mechanics and has the ability to obtain accurate temperature profiles (better than 1 K) with low computational cost. The perturbative solution is applied to a modeled atmosphere as an initial demonstration of this retrieval technique. An initial estimate of the error in the temperature retrieval for a diode-laser-based DIAL is presented, indicating that temperature retrievals with an error of less than can be achieved in the lower troposphere. While this paper focuses on temperature measurements, the perturbative solution to the DIAL equation can also be used to improve the accuracy of retrieved number density profiles.
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