Outline General rules for depth estimation Depth estimation methods Ambiguity of source depth Depth estimation methods Based on characteristic shape of the anomaly Width at half amplitude Width between steepest slopes Gradient-based methods Graphical methods (linear slope, Peters, Sokolov’s) Based on radial Fourier spectra Dominant wavelength Spectral slope (roll-off) Based on downward continuation Roy’s Graphical methods for the top of the source E-line Werner’s (two methods) Logachev’s

Source-depth ambiguity Source depth cannot be unambiguously determined from the recorded field Green’s equivalent layer (stratum) can be placed anywhere below the observation surface This layer represents a distributed source Perfectly reproduces the field recorded on and above the survey surface All methods of depth estimation look for “the simplest” models: Explaining localized and pronounced gravity anomalies Try explaining them by compact sources

Principle Four key ideas for estimating the depth to the source: Depth is proportional to the lateral extent w of the anomaly: The question is how to measure this width conveniently F is the formfactor depending on the shape of the source Depth is inversely proportional to the gradients of log(Dg): Depth is inversely proportional to the dominant wavenumber kD: Depth is proportional to the roll-off of spectral amplitudes: or To see that these quantities should be proportional to the depth, simply check their dimensionalities

Width-based method #1: Width at half amplitude If w is measured at half of the peak amplitude w = w1/2, then, for a spherical monopole source: and therefore F = 0.65. If the line does not pass immediately over the source, h would be closer to the total distance to the source See pdf notes For a line source (rod, pipe) in the direction across its strike: F = 0.5. For this method, the regional trend should be carefully removed The trend distorts the reading of w1/2.

Width-based method #2: Width between steepest points If w is measured between the points of steepest dDg/dx, then for a spherical anomaly: and therefore F = 1. See pdf notes For a line source (rod, pipe) across strike: Steepest-gradient points are more difficult to eyeball, but this w is practically unaffected by the regional trend The trend only adds a constant to the gradients

Gradient-based method #1 h is estimated from the gradient of logDg (scale-invariant gravity anomaly) at the point of largest dDg/dx: For a spherical (point, box) source: See pdf notes For a line source (rod, pipe) across strike:

Gradient-based method #2 Quantities Dg and dDg/dx can also be measured at points of their respective largest magnitudes. Then: For a spherical (point, box) source: See pdf notes For a line source (rod, pipe) across strike:

Gradient-based method #3 Using the second horizontal derivative d2Dg/dx2 (curvature) and Dg at the peak of Dg: For a spherical (point, box) source: For a line source (rod, pipe) across strike, F is the same This suggests a method for any shape in the next slide See pdf notes

Gradient-based method #4 If we use upward continuation to evaluate the second vertical derivative of the field: Poisson’s equation then for any shape of the source, the two second horizontal derivatives can be replaced with the vertical one: and the formfactor becomes:

Graphical shape-based methods for depth Slightly different combinations of the steepest slope dDg/dx with Dg taken at different points See pages 16-20 in pdf notes Linear-slope distance method Peters method Sokolov’s method

Graphical shape-based methods for horizontal position of the dipole source Look for the position of the shallower (south in the northern hemisphere) pole of the effective dipole Remember that the measured positive high over this pole is shifted to the south Green is the observed field, blue and red are its constituents due to the north and south poles

Graphical shape-based methods for horizontal position of the dipole source See pages 21-24 in pdf notes. Two groups of methods: Using only the main lobe of DT (x) Werner’s methods – use bisectors of two or three chords of the main lobe of DT (x) Using both the positive and negative lobes E-line method – connect the positive high and negative low; the intersection with DT (x) gives the x0 of the shallower pole Logachev’s method – measure the positive high DThigh and negative low DTlow, then find point x0 such that:

Spectral method: Spectral roll-off Plot log(amplitude spectrum Dg(k)) vs. the absolute value of radial wavenumber, See pages 13-15 in pdf notes From upward continuation, we know that the spectrum of a potential field is multiplied by “continuation factor” when the observation plane is shifted upward by Dz For a source at depth h: Dg(x) is close to a pulse, Consequently the amplitude spectrum is constant at the source level: Therefore, the spectrum at the surface is: Therefore, log(g) vs. |k| graph makes a straight line with slope (-h):

Downward continuation (Roy’s) method Downward continue the field in small increments in h and plot the amplitude of the anomaly, Dgmax, as a function of h The depth of the anomaly hmax corresponds to the “elbow” after which Dgmax(h) starts quickly increasing When approaching this depth, the amplitude of the anomaly should increase as This has a fairly steep increase as h approaches hmax This method is only “spectral” by the procedure of calculating Dg(h) dependence