Ablation therapy for supraventricular arrhythmias in modern electrophysiology is guided by voltage and conduction velocity. Low-voltage and low-conduction-velocity areas are associated with remodelled tissue and targeted for ablation to terminate atrial arrhythmias or reduce recurrence. State-of-the-art mapping techniques use multiple bipolar measurements to assess local voltage and conduction velocity, both of which depend on estimation of the electrical field, i.e. the spatial gradient of the extracellular potential. This field is commonly approximated by a first-order Taylor expansion using differences in unipolar voltages and distances between measurement points. However, this approach assumes a constant electrical field between unipolar electrodes, which becomes only approximate as an electrical impulse passes through the mapped domain. To improve electric field estimation, we propose a second-order Taylor expansion of the electrical potential by extending the conventional two-dimensional least-squares approach to include the parameters required to approximate the Hessian matrix. The solutions of the least-squares problems derived from the first- and second-order Taylor expansions are compared with a reference electrical field obtained from a finite-difference approximation of the extracellular potential gradient on a rectangular atrial slab. The resulting electrical fields were normalised, and angular deviation at the point of interest was evaluated. For the least-squares approximation, the number of bipolar measurements was varied, and measurements were sampled pseudorandomly from a circular neighbourhood around the point of interest, with a fixed offset of 1 mm from the surface. In the centre of a planar propagating wave, the mean angular deviation of the electrical field direction over one depolarisation wave was 25.24° and 24.60° for the first- and second-order approaches, respectively, using 10 bipolar measurements. With 15 bipolar measurements, the deviation reached 25.63° for the first-order approach and 24.84° for the second-order approach. In summary, further research is needed to understand the performance of higher-order approximations.