# Dictionary:Derivative map

2. With potential fields the second vertical derivative [based on Laplace's equation,${\textstyle {\frac {\partial ^{2}\phi }{\partial z^{2}}}=-{\frac {\partial ^{2}\phi }{\partial x^{2}}}+{\frac {\partial ^{2}\phi }{\partial y^{2}}}}$ ] was once used widely, but it has largely been replaced by the total gradient or total horizontal derivative. The horizontal derivatives, ${\textstyle {\frac {\partial ^{2}\phi }{\partial x^{2}}}}$ and ${\textstyle {\frac {\partial ^{2}\phi }{\partial y^{2}}}}$, are usually estimated by finite-difference methods from values measured at gridded points on a map, often using a residualizing template based on polar representation of the Laplacian or by 2D convolution with such a template. See Cordell and Grauch (1985).