Downward Continuation

Key Takeaways

• Instability of downward continuation in the presence of noise is the fundamental mathematical challenge that distinguishes downward continuation from upward continuation in potential field data processing: upward continuation is a stable operation because the harmonic continuation kernel for upward continuation attenuates high-wavenumber components of the data (which correspond to short-wavelength, shallow features) exponentially with the continuation distance, making the operation inherently smoothing and noise-suppressing; downward continuation is an unstable operation because the continuation kernel for downward continuation amplifies high-wavenumber components exponentially with the continuation distance, meaning that noise in the measured data, which appears at the highest spatial frequencies (shortest wavelengths) in the data, is amplified most strongly by downward continuation and can completely overwhelm the signal from the geological source bodies if the continuation depth is large relative to the data station spacing or if the initial noise level in the data is significant; this instability means that downward continuation must always be stabilized (regularized) by applying a low-pass filter that limits the amplification of high-wavenumber components below the noise level, accepting some loss of resolution in exchange for a usable result rather than an amplified noise field; the choice of the stabilization filter cutoff is a trade-off between resolution (sharpness of the continued anomalies, favoring a high cutoff) and stability (freedom from amplified noise, favoring a low cutoff), and must be calibrated to the specific noise level and station spacing of the dataset being processed.
• Wavenumber domain implementation of downward continuation uses the convolution theorem of Fourier analysis to compute the continuation in the spatial frequency domain, where the continuation operation corresponds to multiplication of the Fourier transform of the measured field by the continuation factor exp(+kz) (where k is the horizontal wavenumber and z is the downward continuation distance), followed by inverse Fourier transform to return to the spatial domain: this wavenumber domain approach is computationally efficient (requiring only a 2D FFT, a complex multiplication, and an inverse FFT) and directly reveals the amplification factor as a function of wavenumber, making it straightforward to design and apply the stabilizing low-pass filter in the wavenumber domain (by setting the continuation factor to a maximum value at high wavenumbers rather than allowing it to grow exponentially); modern potential field processing software implements downward continuation in the wavenumber domain with user-controlled stabilization parameters that allow the processor to balance resolution against noise, and the results can be compared at different continuation depths to assess the sensitivity of the interpreted anomaly patterns to the choice of continuation distance and stabilization; the wavenumber domain approach also enables the equivalent-source technique, in which the measured field is modeled as the response of a distribution of point sources at a shallow reference level (the equivalent source layer), and the equivalent source parameters are inverted from the measured data, allowing subsequent continuation, derivative, and transformation operations to be performed on the smoothed equivalent source representation rather than directly on the noisy measured data.
• Application of downward continuation in salt basin exploration helps define the top-of-salt surface and the lateral extent of salt bodies from gravity data collected at the seafloor by ocean bottom gravity measurement systems (gravimeters deployed on the seafloor by ROV in deepwater areas): salt bodies in sedimentary basins have significantly lower density (approximately 2.2 g/cc) than the surrounding sediments (2.3-2.6 g/cc), creating a negative gravity anomaly over salt diapirs and sheets that is a key constraint in salt geometry modeling for seismic depth imaging; when gravity data is measured at the sea surface above a deepwater salt basin, the measured gravity anomaly is a smoothed, attenuated version of the salt gravity signature that has been upward-continued through the water column, blurring the edges and interior structures of the salt body; downward continuation of the sea surface gravity data to the seafloor level (or even to the top-of-sediment level) sharpens the anomaly and improves the lateral resolution of the salt body edges by a factor roughly proportional to the ratio of the continuation distance to the depth of the salt body top; in deepwater areas with water depths of 1,000-3,000 meters, downward continuation of sea surface gravity to the seafloor can provide a 2-4x improvement in the effective resolution of the salt geometry, reducing the ambiguity in the interpreted salt boundary position from several kilometers to hundreds of meters.
• Downward continuation to the source level as an extreme case produces the anomaly that would be measured directly in contact with the causative body (zero distance between measurement and source), which in theory provides perfect resolution of the source geometry but in practice is always a target depth shallower than the actual source to avoid the instability that arises when the continuation factor grows without bound as z approaches the source depth; the practical limit for stable downward continuation is typically 50-75% of the estimated depth to the top of the shallowest source body, beyond which the amplification of noise exceeds the improvement in resolution; the depth to the top of the source can be estimated from the half-width of the gravity or magnetic anomaly (Euler deconvolution and Werner deconvolution provide systematic estimates of source depth from the anomaly shape), and this depth estimate guides the selection of the maximum safe continuation depth; in applications where the primary geological interpretation goal is to separate closely spaced anomalies that overlap at the measurement elevation (two adjacent salt diapirs whose gravity anomalies merge into a single broad anomaly at the sea surface), downward continuation to a depth that is close to the tops of the salt bodies can reveal the two-lobed nature of the merged anomaly and enable the two bodies to be identified and mapped as distinct features.
• Derivative calculations in combination with downward continuation provide the highest-resolution enhancement of potential field anomalies by combining the depth-reduction effect of downward continuation with the edge-enhancement effect of horizontal and vertical derivative computations: the horizontal derivative (computed as the spatial gradient of the continued gravity or magnetic field) is maximum at the edges of source bodies (where the field gradient is steepest) and provides a sharp map of the lateral boundaries of the density or susceptibility contrasts; the vertical derivative (the vertical component of the gradient of the continued field) is particularly useful for identifying shallow, small-magnitude sources because it emphasizes high-wavenumber components of the field (just as downward continuation does) and suppresses long-wavelength regional trends; the analytic signal (the amplitude of the 3D gradient vector of the total magnetic field or gravity field) combines horizontal and vertical derivatives into a single attribute that produces a positive peak over magnetic or density sources regardless of their magnetization direction, simplifying the interpretation of source geometry in areas with complex or variable magnetization; the combination of downward continuation (to reduce the effective source depth and sharpen anomalies) followed by derivative computation (to enhance edges and suppress regional trends) is a standard processing sequence for high-resolution magnetic and gravity mapping of basement structure in petroleum-bearing sedimentary basins.