1. Outline Estimation of terrain (Bouguer) density:
Principles
Nettleton, scatterplot, covariance, Parasnis methods
First-difference and Multiscale first-difference methods
Bias in terrain-density estimates:
Subsurface structure correlated with surface topography
Density of composite material
Wet and dry densities
Densities of tills in the area

2. Principles of terrain density estimation
Optimal terrain density leads to the removal of the effect of the topography in the Bouguer-corrected gravity
Bouguer-corrected gravity is due to deeper sources
It is smoother
It may be uncorrelated or correlated with topography
Consequently, optimal density will achieve:
Lower correlation of topographic elevation with Bouguer gravity
Smoothest Bouguer gravity variation

3. Variance reduction and correlation
Note that the “variance” (s2) is the squared mean statistical error:
The variances due to subsurface anomalies (true) and to the surface topography (the one we want to get rid of) are additive in the data :
This is the covariance.
These anomalies should
not be correlated (<…>=0)
So, if we manage to remove the effect of topography in Bouguer gravity gB, we likely reduce its s2

4. Nettleton method
Graphical correlation between the distance variations of the Bouguer-corrected gravity gB(x) and topography h(x)
Criterion for selecting r: absence of longer-range variations in gB(x)

5. Covariance method
Criterion for selecting r : the mean-square detrended Bouguer anomaly gB(x) must be the smallest
“Detrended” means that mean values are subtracted from both free- air gravity gFA(x) and elevation h(x), or from the Bouguer gravity gB(x) = gFA(x) -2prGh(x)

6. Parasnis method
Cross-plot the values of free-air gravity gFA versus elevation h(x)
Criterion for selecting r : the points (h, gFA) should fall on a straight line
The slope of this line equals 2prG

7. Scatter plot method For each line and for some estimate of r:
Calculate gB from gFA and h
Subtract linear trends from gB and h arrays
You can use Matlab’s function polyfit to do this
Make a cross-plot (scatterplot) of these detrended (h,gB)
You will see a cloud of points
Calculate the correlation coefficient (r, next slide)
Criterion for selecting the r: the cloud must be horizontal and the covariance equal zero
If r > 0 (positive slope), the r is too low (under-corrected)
If r < 0 (negative slope), the r is too high (over-corrected)
You can again measure the slope Dr using polyfit
Note that this method only looks at the covariances of small perturbations in (h,gB) but ignores the general trends

8. Correlation coefficient
Correlation coefficient between two series {x} and {y}:
where <x> and <y> are the sample means and sx and sy are the standard deviations

9. The method of first differences
Calculate ratios DgFA /Dh(x) of the differences of free-air gravity DgFA to elevation differences Dh(x) for adjacent stations
Criterion for selecting r : these ratios should equal 2prG

10. Multi-scale first differences
Calculate ratios rapp = DgFA /Dh/2pG (“apparent density”) for many pairs of stations in an area
Draw a histogram p(rapp)
Criterion for selecting r : the histogram will have peaks (most frequently occurring correlation of DgFA and Dh) at the true values of r