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

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

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

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)

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)

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

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

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

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

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