This work proposes a fully automated method for recovering the location of a source and medium parameters in shallow water. The scenario involves an unknown source emitting lowfrequency sound waves in a shallow water environment and a single hydrophone recording the signal. Using the spectrogram of each modal component obtained by a warping method, we investigate how to recover the modal travel times and we provide stability estimates. A penalized minimization algorithm is then presented to estimate the source location and medium parameters. The proposed method is tested on different experimental data, demonstrating its effectiveness in real-world scenarios.