Introduction. Matlab output for Problems 1 2. SOLUTIONS (Carl Tape) Ge111, Assignment #3: Gravity April 25, 2006

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1 SOLUTIONS (Carl Tape) Ge111, Assignment #3: Gravity April 25, 26 Introduction The point of this solution set is merely to plot the gravity data and show the basic computations. This document contains a Matlab source code that generated Figures 1 and 2, as well as the output below. In practice, various interpolations are much easier using a computer, which also allows for more complex interpolating schemes (e.g., using a cubic function instead of a line). Note that 1 5 mgal = 1 Gal = 1 m s 2. The standard free-air gravity correction is g fa / h =.386 mgal/m =.386 Gal/m = s 2. Matlab output for Problems 1 2 We have measurements of relative gravity, from which we can compute predicted heights. And we have measurements of heights, from which we can computed predicted relative gravity. Using both the gravity and the heights measurements, we can compute the local free-air correction. Plots related to this problem are shown in Figure 1, and the Matlab output is here (see source code): Predicted and measured heights (m): h_pred h_meas Predicted and measured gravity (mgal): g_pred g_meas Predicted and known free-air correction (mgal/m): Note that the measured gravity has been corrected for drift using a linear interpolation (Figure 1), and it has also been referenced to the base station, which is taken to be the ground floor. Similarly, the height measurements are relative to the base station as well. 1

2 Matlab output for Problem 3 The earth is approximated by an ellipsoid. The gravitational field for the reference ellipsoid is that of the World Geodetic System 1984 (WGS84), which is given by (Blakely, 1995, p. 136) g (λ) = sin2 λ sin 2 λ. (1) Another option is from Reynolds (1997): g (λ) = ( sin 2 λ sin 4 λ ). (2) This function is plotted in Figure 2, along with the interpolated points for Pasadena and Chalfant Valley. Note that this is not the geoid: the reference ellipsoid is the equipotential surface of a uniform earth, whereas the geoid is the actual equipotential surface at mean sea level, which takes into account heterogeneous distributions of mass within the earth s interior (Blakely, 1995, p. 136). The observed gravity is modeled by the reference ellipsoid gravity, plus various corrections. Considering the free-air correction, g fa = h, and the Bouguer correction due to an infinite mass slab, g sb = 2πGρh, we can write the estimated gravity as g est (λ, h, ρ) = g (λ) + g fa (h) + g sb (ρ, h), (3) where we have emphasized the key parameters by showing the functional dependence. Here is the Matlab output (see source code), assuming no Bouguer correction 1 : Reference gravities, considering latitudes: PAS : m s^-2 CHV : m s^-2 PAS - CHV : -3.91e-3 m s^-2 = mgal Reference gravities, considering latitudes and elevations: PAS : m s^-2 CHV : m s^-2 PAS - CHV : 1.75e-3 m s^-2 = 17.5 mgal --> So the gravity at PAS should be higher, since the elevation effect outweighs the latitude effect. Reference gravities, considering latitudes, elevations, and Bouguer (rho =. kg m^-3): PAS : m s^-2 CHV : m s^-2 PAS - CHV : 1.75e-3 m s^-2 = 17.5 mgal --> So gravity at either place might be higher, depending on rho. In this case (ρ = ), the gravity at Pasadena is greater than that at Chalfant Valley. 1 Mathematically, we assume the density of the infinite slab to be zero, i.e., ρ = kg m 3. 2

3 Now we consider the case where the infinite slab has a density of ρ = 267 kg m 3, something closer to typical values for rock: Reference gravities, considering latitudes: PAS : m s^-2 CHV : m s^-2 PAS - CHV : -3.91e-3 m s^-2 = mgal Reference gravities, considering latitudes and elevations: PAS : m s^-2 CHV : m s^-2 PAS - CHV : 1.75e-3 m s^-2 = 17.5 mgal --> So the gravity at PAS should be higher, since the elevation effect outweighs the latitude effect. Reference gravities, considering latitudes, elevations, and Bouguer (rho = 267. kg m^-3): PAS : m s^-2 CHV : m s^-2 PAS - CHV : -4.36e-4 m s^-2 = mgal --> So gravity at either place might be higher, depending on rho. In this case (ρ = 267), the gravity at Chalfant Valley is greater than that at Pasadena. Thus, we see that if we use the Bouguer correction, the choice of ρ is crucial. References Blakely, R., Potential Theory in Gravity & Magnetic Applications, Cambridge U. Press, Reynolds, J., An Introduction to Applied and Environmental Geophysics, John Wiley, Chichester, New York,

4 Matlab code close all clear earthr = 6371e3; earthm = 5.97e24; G = 6.673e-11; g = earthm*g/earthr^2; a = earthr; deg = 18/pi; % radius of earth (m) % mass of the earth (kg) % gravitational constant (N m^2 kg^-2) % gravitational acceleration at the surface (m s^-2) % % Problems 1 and 2 fa_known = -.386; % standard free-air correction (mgal/m) % data for S Mudd % N = DATENUM(Y,MO,D,H,MI,S) mins = [ ]; t_meas = datenum(26,4,11,1,mins,); g_meas = [ ]; t_base = t_meas([1 5]); g_base = g_meas([1 5]); % assume linear function for interpolation tbase_smooth = linspace(t_base(1),t_base(2),1); gbase_smooth = linspace(g_base(1),g_base(2),1); % gravity relative to base station measurement g_corr = g_meas - interp1(tbase_smooth,gbase_smooth,t_meas); % height computed from free air anomaly (see notes) h_pred = g_corr / fa_known; % measured heights d_meas = [ ]; h_meas = cumsum(d_meas); disp( Predicted and measured heights (m): ); disp( h_pred h_meas ); disp([h_pred(1:4) h_meas(1:4) ]); % predicted gravity from measured heights using known free-air correction g_pred = fa_known * h_meas; disp( Predicted and measured gravity (mgal): ); disp( g_pred g_meas ); disp([g_pred(1:4) g_corr(1:4) ]); % predicted free air correction from measured heights and measured gravity fa_meas = diff(g_corr(1:4))./ diff(h_meas(1:4)); 4

5 fa_pred = mean(fa_meas); disp( Predicted and known free-air correction (mgal/m): ); disp([fa_pred fa_known]); % figure; nr=3; nc=2; msize = 2; ticks = datenum(26,4,11,1,[:1:3],); subplot(nr,nc,1); hold on; grid on; plot(tbase_smooth, gbase_smooth, b ); plot(t_base, g_base, b., markersize,msize); xlabel( Time ); ylabel( Relative gravity at base station (mgal) ); title({ Correction for drift at base station,, datestr(t_meas(1),)}); set(gca, xtick,ticks, xticklabel,datestr(ticks,15)); subplot(nr,nc,2); hold on; grid on; plot(t_meas, g_meas, b., markersize,msize); xlabel( Time ); ylabel( Relative gravity (mgal) ); title( Gravity data for S. Mudd ); set(gca, xtick,ticks, xticklabel,datestr(ticks,15)); subplot(nr,nc,3); hold on; grid on; plot(t_meas, g_corr, b., markersize,msize); xlabel( Time ); ylabel( Gravity relative to base station (mgal) ); title( Gravity data for S. Mudd ); set(gca, xtick,ticks, xticklabel,datestr(ticks,15)); subplot(nr,nc,4); hold on; grid on; plot(t_meas, h_pred, b., markersize,msize); xlabel( Time ); ylabel( Height relative to base station (m) ); set(gca, xtick,ticks, xticklabel,datestr(ticks,15)); subplot(nr,nc,5); hold on; grid on; ax1 = [ ]; plot(ax1(1:2),ax1(1:2), r-- ); plot(h_pred(1:4), h_meas(1:4), b., markersize,msize); xlabel( Predicted height (m) ); ylabel( Measured height (m) ); title( Gravity --> heights ); axis equal, axis(ax1); subplot(nr,nc,6); hold on; grid on; ax1 = [ ]; plot(ax1(1:2),ax1(1:2), r-- ); plot(g_pred(1:4),g_corr(1:4), b., markersize,msize); xlabel( Predicted free-air gravity (mgal) ); ylabel( Measured free-air gravity (mgal) ); title( Heights --> gravity ); axis equal, axis(ax1); 5

6 %fontsize(1), orient tall, wysiwyg; % % Problem 3 % latitudes of two observation points latpas = /6; latchv = /6; % evenly spaced latitude points latsmooth = linspace(,9,1); iwgs = ; if iwgs==1 % World Geodetic System, 1984 (WGS84) % Reference: Blakely, Potential Theory in Gravity and Magnetic Applications (1995), p.136 glatsmooth = * ( *sin(latsmooth/deg).^2)./ sqrt( else glatsmooth = * ( *sin(latsmooth/deg).^ *sin(latsmoot end % interpolate to determine the reference gravities (latitude correction) gpas = interp1(latsmooth,glatsmooth,latpas); gchv = interp1(latsmooth,glatsmooth,latchv); figure; hold on; grid on; plot(latsmooth, glatsmooth); plot(latpas,gpas, ro ); plot(latchv,gchv, ro ); text(latpas,gpas-.1, PAS ); text(latchv,gchv-.1, CHV ); xlabel( Latitude ); ylabel( Reference gravity (m s^{-2}) ); title( Reference gravity for WGS84 (see Blakely, 1995, p. 136) ); % incorporate the elevations (free-air anomaly) and compute the gravities hpas = 25; hchv = 16; %g1pas = gpas - 2*hPAS*gPAS/a; %g1chv = gchv - 2*hCHV*gCHV/a; g1pas = gpas + fa_known*hpas*1e-5; g1chv = gchv + fa_known*hchv*1e-5; % incorporate the effects of the infinite mass sheet (Bouguer) % Assume that the slab bottom is at the height of Pasadena, % and the slab top is at sea level %rho = ; % density of slab kg/m^3 rho = 267; % density of slab kg/m^3 hslab = ; % bottom of slab g2pas = g1pas + 2*pi*G*rho*(hPAS-hslab); g2chv = g1chv + 2*pi*G*rho*(hCHV-hslab); 6

7 disp( Reference gravities, considering latitudes: ); disp([ PAS : num2str(sprintf( %.6f, gpas)) m s^-2 ]); disp([ CHV : num2str(sprintf( %.6f, gchv)) m s^-2 ]); disp([ PAS - CHV : num2str(sprintf( %.3e, gpas-gchv)) m s^-2 =... num2str(sprintf( %.1f, (gpas-gchv)*1e5)) mgal ]); disp( Reference gravities, considering latitudes and elevations: ); disp([ PAS : num2str(sprintf( %.6f, g1pas)) m s^-2 ]); disp([ CHV : num2str(sprintf( %.6f, g1chv)) m s^-2 ]); disp([ PAS - CHV : num2str(sprintf( %.3e, g1pas-g1chv)) m s^-2 =... num2str(sprintf( %.1f, (g1pas-g1chv)*1e5)) mgal ]); disp( --> So the gravity at PAS should be higher, ); disp( since the elevation effect outweighs the latitude effect. ); disp([ Reference gravities, considering latitudes, elevations, and Bouguer (rho =... num2str(sprintf( %.1f, rho)) kg m^-3): ]); disp([ PAS : num2str(sprintf( %.6f, g2pas)) m s^-2 ]); disp([ CHV : num2str(sprintf( %.6f, g2chv)) m s^-2 ]); disp([ PAS - CHV : num2str(sprintf( %.3e, g2pas-g2chv)) m s^-2 =... num2str(sprintf( %.1f, (g2pas-g2chv)*1e5)) mgal ]); disp( --> So gravity at either place might be higher, depending on rho. ); 7

8 Relative gravity at base station (mgal) Correction for drift at base station, 11 Apr 26 1:2: 1: 1:1 1:2 1:3 Time Relative gravity (mgal) Gravity data for S. Mudd : 1:1 1:2 1:3 Time Gravity relative to base station (mgal) Gravity data for S. Mudd : 1:1 1:2 1:3 Time Height relative to base station (m) : 1:1 1:2 1:3 Time Measured height (m) Gravity > heights 5 1 Predicted height (m) Measured free air gravity (mgal) Heights > gravity Predicted free air gravity (mgal) Figure 1: Various plots for Problems 1 and 2. 8

9 9.84 Gravity for reference earth ellipsoid 9.83 Reference gravity (m s 2 ) CHV PAS Latitude Figure 2: Reference gravity as a function of latitude, based on the WGS84 reference system. The curve is a plot of Equation (2). Gravity is weaker at the Equator, where the surface is farther from the earth s center of mass, due to the bulge of the earth. 9

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