A new numerical algorithm for the analysis of single line to ground faults on short overhead transmission lines is presented in this paper. It is based on synchronised sampling at two line terminals and an accurate fault model including the arcing phenomena and tower footing resistance at the fault point. The core of the algorithm is an efficient non-recursive parameter estimation method. A dynamic arc model is included in the fault model to represent the arc's interaction with the external network. The algorithm accurately estimates the arc voltage amplitude, tower footing resistance, and the fault location, simultaneously. The algorithm is derived in the time domain, and is based on the synchronized acquisition of currents and voltages at both line terminals. Case studies based on simulated data are presented to demonstrate the effectiveness of the new method.