1. 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)
Roy’s
Graphical methods for the top of the source
E-line
Werner’s (two methods)
Logachev’s

2. 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

3. 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

4. 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.
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.

5. 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

6. 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:

7. 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:

8. Gradient-based method #3
Using the second 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

9. 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 a vertical one:
and the formfactor becomes:

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

11. 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

12. Graphical shape-based methods for horizontal position of the dipole source
See 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 the x0 such that:

13. Spectral method #1: Spectral roll-off
Plot log(amplitude spectrum Dg(k)) vs. the absolute value of radial wavenumber,
See 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 at the source level:
Therefore, the spectrum at the surface is:
Therefore, log(g) vs. |k| graph makes a straight line, with slope (-h):

14. Spectral method #2: 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 computation of the Dg(h) dependence