Upward Continuation

Key Takeaways

• The mathematical foundation of upward continuation rests on the fact that gravity and magnetic anomaly fields satisfy Laplace's equation in regions free of sources (rock masses or magnetic bodies), which means the field is completely determined everywhere in the source-free half-space above the surface if it is known on any horizontal plane within that half-space: the boundary value problem (Dirichlet problem for the half-space) has the solution given by the Poisson integral formula, which is the theoretical basis for the convolution-based implementation of upward continuation; in practice the data are sampled on a regular grid, the 2D discrete Fourier transform is applied, the wavenumber-domain filter is applied, and the inverse transform returns the continued field on the same spatial grid but corresponding to the higher observation level; the filter operator e^(-kz) is a smooth low-pass filter whose cutoff sharpness increases with continuation height z, making upward continuation a self-consistent, stable, and noise-suppressing operation (unlike downward continuation, which amplifies short-wavelength noise and becomes unstable).
• Regional-residual separation is the most important application of upward continuation in petroleum exploration gravity surveys, where the objective is to separate the gravity signal of deep basement structure (the regional) from the signal of shallower sedimentary density contrasts (the residual) that may be associated with structural traps, salt domes, or carbonate buildups: continuing the Bouguer gravity anomaly upward by 5 to 20 kilometers suppresses the sedimentary contributions and produces a smooth regional field dominated by deep crustal density variations and basement topography; subtracting this continued field from the original observed field yields the residual anomaly map that highlights shorter-wavelength density contrasts in the sedimentary section that are of exploration interest; the choice of continuation height controls the wavelength cutoff of the regional: too shallow a continuation removes only the shortest-wavelength noise and leaves many sedimentary features in the regional, while too deep a continuation over-smooths and removes genuine exploration-relevant anomalies into the residual.
• Reduction to equivalent altitude (REA) using upward continuation normalizes airborne magnetic survey data collected at variable flight altitude (due to terrain clearance requirements over rugged topography) to a common smooth drape surface, eliminating altitude-related amplitude variations that would otherwise masquerade as real geological features: in surveys over mountainous terrain, the aircraft may fly at 80 meters above the terrain surface but the terrain itself may vary by 500 meters in relief, so the magnetic data are collected at drastically different distances from the basement sources over different parts of the survey; upward continuing the data from the variable-altitude surface to a flat plane at the maximum terrain elevation (or some higher level) produces a dataset equivalent to what would have been collected on a flat survey at that altitude, enabling consistent interpretation across the full survey area without altitude-correlated artifacts; the upward continuation step is applied as part of all modern airborne magnetic data processing workflows before gridding and enhancement.
• Comparing gravity or magnetic fields continued to multiple heights provides a qualitative depth estimate for anomaly sources because the rate at which an anomaly's amplitude decreases with continuation height is directly related to the source depth: a shallow source produces a narrow, high-amplitude anomaly that attenuates rapidly with increasing continuation height, while a deep source produces a broad, lower-amplitude anomaly that changes little as the continuation height increases through depths shallower than the source; Peters' half-slope method and Werner deconvolution both use this principle to estimate source depth from profile data; for gridded data, computing the rate of change of the field amplitude with upward continuation height (the "pseudo-depth" method) provides a semi-quantitative map of anomaly source depths across the survey area; this depth estimation approach is particularly useful in reconnaissance surveys where no drill holes are available to calibrate the depth to magnetic basement.
• Upward continuation is always stable and noise-attenuating, in fundamental contrast to downward continuation (which continues the field to a lower elevation, closer to or through the sources), which is an ill-posed problem that amplifies short-wavelength noise exponentially and can produce wildly oscillatory results unless regularized with a stabilizing filter: downward continuation is sometimes used to sharpen anomalies and improve resolution when the sources are known to lie between the measurement surface and the continuation level, but it requires careful choice of stabilization parameters (such as a maximum-depth frequency cutoff) to prevent noise blowup; upward continuation requires no such stabilization and is the preferred first-pass enhancement in any potential field interpretation workflow; the asymmetry between the two directions reflects the physical reality that electromagnetic fields diverge with distance from their sources (upward is always safe) but converge toward sources (downward amplifies any errors), and it dictates that upward continuation is the workhorse enhancement while downward continuation is used only in specific, well-constrained situations.