The Jeffreys-Bullard theory of the topographic correction to geothermal gradients cannot be applied with confidence if the height of the relief is large relative to the horizontal distance and depth of the measurement points. It cannot be generally applied to shallow probe measurements in the ocean bottom if bold relief occurs on a scale exceeding a few meters, or on continents to observation in shallow boreholes in extremely rugged terrain.
In an important special case, where the measurement depth is small relative to the distance to the relief, the ‘superficial’ gradient anomaly may be approximated by the value applicable at zero depth. A fairly general two-dimensional steady-state theory for this case can be based on the solution for heat flux through an inclined plane of arbitrary height and slope angle. These two parameters are easily visualized and represented graphically so that models which approximate or bracket real topography can be identified quickly. The results can be applied to stations on planes, valleys, ridges, or benches bounded by irregular slopes. They are valid for points arbitrarily close to slopes of any height or inclination. Finite slope and curvature of the surface at the station can be accommodated if they are not too great. Even if other theories of the topographic correction are applicable, the present method can be useful, as it leads to rapid estimates by graphical means and to useful limits even if the superficial condition is not satisfied.
Curvature in an ocean-bottom temperature profile justifies suspicion of a topographic disturbance from undetected relief. The temperature probe's length should be 2 or 3 times the uncertainty in local elevation difference, and measured curvature should be negligible for reasonable assurance that undetected relief is not causing gradient errors greater than ±10%. Relief not detectable with conventional echo sounders, but of the type observed with deeply towed sounding equipment, can cause heat-flow anomalies of 50–100%, and relatively little curvature will be indicated by probes a few meters long. The very high oceanic heat flows are difficult to explain by undetected relief, but the very low ones are not.
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