We consider the problem of determining a small elliptical conductivity anomaly in a unit disc from boundary measurements. The conductivity of the anomaly is assumed to be a small perturbation from the constant background. A measurement of voltage across two point-electrodes on the boundary through which a constant current is passed. We further assume the limiting case when the distance between two electrodes go to zero, creating a dipole field. We show that three such measurements suffice to locate the anomaly size and location inside the disc. Two further measurements are needed to obtain the aspect ratio and the orientation of the ellipse. The investigation includes the studies of the stability of the inverse problem and optimal experiment design.

翻译：我们研究从边界测量确定单位圆盘内微小椭圆导电异常体的问题。假设异常体的电导率相对于恒定背景场为微小扰动。通过边界上两个点电极施加恒定电流来测量电压。进一步假设两个电极间距趋近于零的极限情形，从而产生偶极场。我们证明仅需三次此类测量即可确定圆盘内异常体的尺寸和位置。还需额外两次测量以获取椭圆的纵横比和方向。本研究包括该反问题的稳定性分析及最优实验设计。