Dictionary:Continuation

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Determining a field over one surface from measurements of the field over another surface (specifically, at another elevation). The field at the elevation z, F(x,y,z), can be found from the field on the surface, F(x ', y ',0). Where the surfaces are horizontal and no sources intervene, the upward-continuation relation (an application of Green's theorem) is

${\displaystyle F(x,y,z)={\frac {\left|z\right|}{2\pi }}\iint {\frac {F(x',y',0}{R^{3}}}dx'dy'}$.

An interchange of the two fields in this equation gives the downward-continuation relation. See downward continuation and Peters (1949), Telford et al. (1990, §2.6.7 and 3.7.5), and Pawlowski (1995).