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Discussion Question #1: Find an article about a current event that discusses a change in the supply or demand of a product. For example, has there been a weather event that has affected certain food crops or the availablility of energy? Has a new consumer product been introduced? Or has a government regulation affected the production of a product? Provide a brief description of the article’s content and explain how economics can be used to analyze the situation and predict changes in equilibrium prices and quantities. Will the change likely persist over time, or is it temporary?  admin Sticky Note ie., the SLFIi > 0 identify positively or directly correlative (X-peak)-to-(Y-peak) associations admin Sticky Note ie., the SLFIi < 0 identify positively or directly correlative (X-valley)-to-(Y-valley) associations admin Sticky Note to fruther simplify the positively correlated feature associations in the X- and Y-datasets, replot the SLFIi as=> - map-A for all SLFIi > 0, and - map-B for all SLFIi < 0, where the admin Sticky Note to fruther simplify the negaitively correlated feature associations in the X- and Y-datasets, replot the DLFIi as=> - map-C for all DLFIi > 0, and - map-D for all DLFIi < 0, where the admin Sticky Note ie., the DLFIi > 0 identify negatively or inversely correlative (X-peak)-to-(Y-valley) associations admin Sticky Note ie., the DLFIi < 0 identify negatively or inversely correlative (X-valley)-to-(Y-peak) associations admin Sticky Note To illustrate WCA, consider the signals T1 and T2 plotted in Fig. 4-below with their respective amplitude means (AM) removed. - the signals' other statistical attributes also are highly disparate, including the - amplitude standard deviations (ASD) and - amplitude ranges of (min, max)-values (AR) - the signals also have a negligible correlation coefficient (CC) that seems to be supported graphically by an apparent lack of feature associations admin Sticky Note - the 128-point signal T1 is the superposition of the regional background signal (B1) and the higher frequency residual signal (R1), and - the 128-point signal T2 is the superposition of the regional background signal (B2) and the higher frequency residual signal (R2), where - CC(B1, B2) = 0, and - the first half of R1 is positively correlated with the first half of R2, and - the other two halves are negatively correlated so that - CC(R1, R2) = 0. admin Sticky Note - this example is from von Frese et al. (1997) and - also is summarized in Chap. 7.3 [p. (135-138)/153] of the GeomathBook.pdf admin Sticky Note - to facilitate graphical correlation analysis of apples (eg., T1) to oranges (eg., T2), the signals were normalized as described on the next page-below=> admin Sticky Note Accordingly, in Fig. 5, the normalized T1 (ie., NT1) and T2 (ie., NT2) signals have dimensionless - ASD(NT1) = ASD(NT2) = 2.0, and - normalization factors (NF) so that T1 = [NF(T1) x NT1] and T2 = [NF(T2) x NT2] a.

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Discussion Question #1: Find an article about a current event that discusses a change in the supply or demand of a product. For example, has there been a weather event that has affected certain food crops or the availablility of energy? Has a new consumer product been introduced? Or has a government regulation affected the production of a product? Provide a brief description of the article’s content and explain how economics can be used to analyze the situation and predict changes in equilibrium prices and quantities. Will the change likely persist over time, or is it temporary? admin Sticky Note ie., the SLFIi > 0 identify positively or directly correlative (X- peak)-to-(Y-peak) associations admin Sticky Note ie., the SLFIi < 0 identify positively or directly correlative (X- valley)-to-(Y-valley) associations admin Sticky Note to fruther simplify the positively correlated feature associations in the X- and Y-datasets, replot the SLFIi as=> - map-A for all SLFIi > 0, and - map-B for all SLFIi < 0, where the
admin Sticky Note to fruther simplify the negaitively correlated feature associations in the X- and Y-datasets, replot the DLFIi as=> - map-C for all DLFIi > 0, and - map-D for all DLFIi < 0, where the admin Sticky Note ie., the DLFIi > 0 identify negatively or inversely correlative (X-peak)-to-(Y-valley) associations admin Sticky Note ie., the DLFIi < 0 identify negatively or inversely correlative (X-valley)-to-(Y-peak) associations admin Sticky Note To illustrate WCA, consider the signals T1 and T2 plotted in Fig. 4-below with their respective amplitude means (AM) removed. - the signals' other statistical attributes also are highly disparate, including the - amplitude standard deviations (ASD) and
- amplitude ranges of (min, max)-values (AR) - the signals also have a negligible correlation coefficient (CC) that seems to be supported graphically by an apparent lack of feature associations admin Sticky Note - the 128-point signal T1 is the superposition of the regional background signal (B1) and the higher frequency residual signal (R1), and - the 128-point signal T2 is the superposition of the regional background signal (B2) and the higher frequency residual signal (R2), where - CC(B1, B2) = 0, and - the first half of R1 is positively correlated with the first half of R2, and - the other two halves are negatively correlated so that - CC(R1, R2) = 0. admin Sticky Note - this example is from von Frese et al. (1997) and - also is summarized in Chap. 7.3 [p. (135-138)/153] of the GeomathBook.pdf admin Sticky Note - to facilitate graphical correlation analysis of apples (eg., T1) to oranges (eg., T2), the signals were normalized as described
on the next page-below=> admin Sticky Note Accordingly, in Fig. 5, the normalized T1 (ie., NT1) and T2 (ie., NT2) signals have dimensionless - ASD(NT1) = ASD(NT2) = 2.0, and - normalization factors (NF) so that T1 = [NF(T1) x NT1] and T2 = [NF(T2) x NT2] admin Sticky Note - Note that normalization does not affect CC(T1, T2). - However, this plot of the normalized signals suggests a possible regional phase shift for WCA investigation - Thus. wavenumber correlation filters (WCF) were developed and applied to NT1 & NT2 to extract the minimally correlated wavenumber components output respectively in Fig.s 6- & 7- below admin Sticky Note - or set σz = σx and μz = μx to normalize Y to X, so that zi(Y) = NY can be plotted on the same axes or contour intervals, etc. as those of X - or normalize both X and Y to common σz and μz so that zi(X) and zi(Y) may be plotted with common graphical parameters, etc.
admin Sticky Note RMSE is the root-mean-squared error admin Sticky Note - an even better estimate of B1 (ie., EB1) is possible with (-0.1 < CC(k) < 0.1) admin Sticky Note - subtracting EB1 from T1, and EB2 from T2 yields pretty good estimated residuals ER1 and ER2, respectively - however=> CC(ER1, ER2) = 0 admin Sticky Note ER1 is signal A(x) in figure on next page admin Sticky Note ER2 is signal B(x) in figure on next page admin Sticky Note => ER1 admin
Sticky Note => ER2 admin Sticky Note (ER1-valley)-to-(ER2-valley) associations admin Sticky Note (ER1-peak)-to-(ER2-peak) associations admin Sticky Note (ER1-peak)-to-(ER2-valley) associations admin Sticky Note (ER1-valley)-to-(ER2-peak) associations admin Sticky Note As another example of WCF from von Frese et al. (1997), consider the satellite magnetic observations in nT from the two Magsat mission orbital tracks across the arctic to northern Finnland in Fig. 14.A-below admin Sticky Note - these orbital track segments of lithospheric magnetic anomaly data at 400 km altitude are within about 10 km or so of each other - thus the lithospheric anomaly estimates should be highly correlated,
- but the CC = 0.705, which suggests an initial noise contribution of roughly=> no = √[(1/CC2) – 1] ≈ 0.6468 or about 65% admin Sticky Note - to estimate the more correlative wavenumber components in the data tracks, a WCF for CC(k) > 0.6 was applied with output shown in Fig. 14.B-below admin Sticky Note - these positively correlated signals reflect ocean-continent boundaries and other lithospheric features - also, the improved CC suggests a noise contribution of only=> n1 ≈ 0.0707 or about 7.1%, or - a noise reduction of [(no - n1)/no]100 ≈ 89% admin Sticky Note - however, to avoid throwing out the baby with the bath water, it is always prudent to check out the rejected signal components as shown in Fig. 14.C-below - the positive residual correlation was checked by WCF for CC(k) > 0.3, - which output the regionally correlated components in Fig. 14.D-below
admin Sticky Note - these regionally correlated signals seem to ignore continent- ocean boundaries and other lithospheric features, - thus, they were attributed mostly to external geomagnetic field effects and admin Sticky Note - the residual signals of Fig. 14.C shown here also lack prominent lithospheric affinities, and - their amplitudes are marginal relative to the amplitudes of the presumed lithospheric magnetic anomalies in Fig. 14.B-above admin Sticky Note - thus, pair-averages of the strong, positively correlated anomalies in Fig. 14.B-above were taken as least squares estimates of the lithospheric anomalies that the two Magsat tracks observed - here, the pair-averaged magnetic anomalies of the lithosphere are mapped by the solid-line profile, whereas - the dotted-line profile gives the pair-wise RMS-differences as error estimates on the corresponding lithospheric magnetic anomaly estimates admin Sticky Note A further WCF example from von Frese et al. (1997) considers
five track-pairs of geoidal undulations from ascending orbits of the US Navy's Geosat-GM altimetry mission over a region east of the Gunerus Ridge which extends offshore of East Antarctica's admin Sticky Note - the marine altimetry estimates geoidal undulations in cm- amplitude units (AU) - vertical differentiation of the geoidal undulations can yield gravity anomalies for constraints on modeling the subjacent lithosphere admin Sticky Note - the tracks in these 5 pairs are separated by mean distances of about 2 to 5 km over a mean sea floor depth of about 4 km - so that geological features at these scales and larger should yield strong, positively correlated features between the altimetry tracks - but the mean CC between the track-pairs is CCM = 0.832, which implies a noise level of about=> no = √[(1/CC2) – 1] ≈ 44.93% admin Sticky Note - to reduce the noise level, the track-pairs were WCF for CC(k) > 0.80 for the output in Fig. 16-below admin
Sticky Note - the WCF-output suggests an reduced noise level of about => n1 ≈ 29.1%, which is in a noise reduction of roughly => [(0.4493 - 0.2909)/0.4493]100 ≈ 35% admin Sticky Note - here, the rejected, broadband non-correlative features presumably include=> - crustal signals that are smaller than the track spacing, and - dynamic signals from temporal and spatial variations of the - orbit errors, - ocean currents, - waves and ice, - measurement and data reduction errors, and - other non-lithospheric effects admin Sticky Note Basement - not Bedrock...which refers to the overlying Phanerozoic sedimentary rocks
admin Sticky Note WCA of the gravity and magnetic anomaly maps of Ohio - adapted from von Frese et al., 1997, Spectral correlation of magnetic and gravity anomalies of Ohio, Geophysics, v. 62, 365-380. admin Sticky Note compositional constraints on the basement rocks from 149 basement-penetrating boreholes admin Sticky Note interpretation of the anomaly maps is based on modeling of statewide profiles A-thru-E admin Sticky Note interpretation examples for profiles A and B are shown below admin Sticky Note A-profile admin Sticky Note Correlative features between the two datasets=> - reduce interpretational ambiguity
- and improve interpretational reliability admin Sticky Note CC < 0 admin Sticky Note CC < 0 admin Sticky Note CC > 0 admin Sticky Note CC < 0 admin Sticky Note B-profile admin Sticky Note CC > 0 admin Sticky Note CC < 0 admin Sticky Note CC > 0
admin Sticky Note CC > 0 admin Sticky Note CC > 0 admin Sticky Note Basement geology inferred from the magnetic and gravity anomaly data admin Sticky Note physical basis for correlating magnetic and gravity anomalies admin Sticky Note where there is - a correlation spectrum CC(k), there also is - an intercept spectrum A(k), and - slope spectrum S(k) involving the ratio Δj/Δσ of physical property contrasts admin Sticky Note
A(k) admin Sticky Note S(k) admin Sticky Note pseudo-anomalies admin Sticky Note raw (ie., unnormalized) pseudo-anomalies derived from the Bouguer gravity and total magnetic anomalies admin Sticky Note admin Sticky Note admin Sticky Note here the normalization factor NF = SF admin Sticky Note admin Sticky Note normalized pseudo anomalies for enhance visual-spatial correlation analysis
admin Sticky Note normalized FVD gravity and RTP magnetic anomalies WCF for (+1 ≤ k ≤ 0) to emphasize positive anomaly correlations admin Sticky Note admin Sticky Note admin Sticky Note admin Sticky Note admin Sticky Note admin Sticky Note SLFI > 0 emphasizes peak-to-peak gravity and magnetic features
admin Sticky Note admin Sticky Note SLFI > 0 emphasizes peak-to-peak gravity and magnetic feature associations admin Sticky Note admin Sticky Note SLFI < 0 emphasizes valley-to-valley gravity and magnetic feature associations admin Sticky Note admin Sticky Note SLFI < 0 emphasizes valley-to-valley gravity and magnetic feature associations admin Sticky Note SLFI > 1 standard deviation (SD) emphasizes the strongest peak-to-peak gravity and magnetic feature associations
admin Sticky Note SLFI < -1 standard deviation (SD) emphasizes the strongest valley-to-valley gravity and magnetic feature associations admin Sticky Note normalized FVD gravity and RTP magnetic anomalies WCF for (-1 ≤ k ≤ 0) to emphasize negative anomaly correlations admin Sticky Note comparison of WCF normalized FVD gravity anomalies with DLFI based on the LFI difference=> LFI(FVDG) - LFI(RTP) to hi-lite inverse feature associations in the FVD gravity anomalies admin Sticky Note comparison of WCF normalized RTP magnetic anomalies with DLFI based on the LFI difference=> LFI(FVDG) - LFI(RTP) to hi-lite inverse feature associations in the RTP magnetic anomalies
admin Sticky Note comparison of normalized FVD gravity anomalies with DLFI > 0 to hi-lite=> FVDG-peak to RTPM-valley associations admin Sticky Note comparison of normalized RTP magnetic anomalies with DLFI > 0 to hi-lite=> FVDG-peak to RTPM-valley associations admin Sticky Note comparison of normalized FVD gravity anomalies with DLFI < 0 to hi-lite=> FVDG-valley to RTPM-peak associations admin Sticky Note comparison of normalized RTP magnetic anomalies with DLFI < 0 to hi-lite=>
FVDG-valley to RTPM-peak associations admin Sticky Note DLFI > 1 standard deviation (SD) emphasizes the strongest FVDG-peak to RTPM-valley anomaly associations admin Sticky Note DLFI < -1 standard deviation (SD) emphasizes the strongest FVDG-valley to RTPM-peak anomaly associations admin Sticky Note normalized FVD gravity and RTP magnetic anomalies WCF for (-0.3 ≤ k ≤ +0.3) to emphasize the null-correlated anomaly features admin Sticky Note #4 admin Sticky Note #4
admin Sticky Note #4 admin Sticky Note #4 admin Sticky Note #4 admin Sticky Note #4 admin Sticky Note - regions of the basement rocks where the WCA emphasizes possible occurrences of 4 physical property combinations - the edges of these regions were mapped from the zero contours of the SVD(g) and SVD(RTPm) anomalies admin Sticky Note negative=> [Δm(-), Δσ(+)] admin Sticky Note negative=> [Δm(+), Δσ(-)] admin Sticky Note
positive=> [Δm(+), Δσ(+)] admin Sticky Note positive=> [Δm(-), Δσ(-)] admin Sticky Note null admin Sticky Note null admin Sticky Note - differentiate wrt m=> admin Sticky Note = jωl admin Sticky Note = (jωl)p admin Sticky Note = (jωk)p - thus, the total horizontal p-order derivative can be evaluated
from=> = √[(jωl)p + (jωk)p] - in particular, for p = 2 the horizontal curvature may be obtained with zero contours that estimate feature edges, and thus provide edge detection admin Sticky Note - this is an elementary linear differential equation of order p with constant coefficients admin Sticky Note - typically found in Chap. 2 of common textbooks on ordinary differential equations admin Sticky Note => iff A = 0 admin Sticky Note ideal SVD-response admin Sticky Note ideal VD-response for p = 4 admin Sticky Note
lousy (ie., crude) SVD-response - perhaps derived from the data domain convolution of a simplified first derivative operator=> (-1, 0, 1), - where the SVD(1D)=> (-1, 0, 1(۞)-1, 0, 1) and ۞ denotes convolution admin Sticky Note - much improved response of a SVD-operator such as developed in Fig. 1-below on next page admin Sticky Note - derivative filters generally behave like high-pass/low-cut filters admin Sticky Note Elkins 7-by-7 point SVD-grid or -mask operator admin Sticky Note zero row admin Sticky Note zero column admin Sticky Note - the weights at radii ri=> r1 = Δx,
r2 = Δx√2, r3 = Δx2, etc. admin Sticky Note real admin Sticky Note imaginary with=> i = j = √(-1) admin Sticky Note imaginary with=> i = j = √(-1) admin Sticky Note - note that the order 'p' previously used is now 'n' - ie., p-order => n-order helpful for assignment 5.3 admin
Sticky Note regional gravity effect admin Sticky Note total gravity effect of 5-cylinders + regional admin Sticky Note individual gravity effects of the 5 cylinders admin Sticky Note - or estimate the 5 cylinder densities and regional slope and intercept values from the total signal by inversion - and modify the A-coefficients for coefficients that obain the desired integral/derivative properties - by forward modeling of the modified system assuming the other cylinder parameters are known admin Sticky Note - or simply take the FFT of the total signal, - filter it using the appropriate integral/derivative transfer function coefficients - and IFFT the modified spectrum to obtain the data domain estimates of the desired integral/derivative properties admin Sticky Note The FFT is the superior approach in terms of numerical efficiency, accuracy, and the minimum assumptions needed
admin Sticky Note To evaluate the integral/derivative properties of the total signal, we could=> - perform graphical operations with significant computational labor - or equivalent convolutions with significant computational effort and error Ralph Sticky Note cylinder #1 Ralph Sticky Note cylinder #2 Ralph Sticky Note cylinder #3 Ralph Sticky Note cylinder #4 Ralph Sticky Note cylinder #5 admin Sticky Note
plus regional admin Sticky Note mgal admin Sticky Note rind applied at both ends to minimze Gibbs' error Ralph Sticky Note - if the investigator correctly assumes that these anomaly peaks are all due to the gravity effects of horizontal cylinder mass variations, - then at the anomaly peaks or maximum amplitudes (MA), the depths to the central axes (zc) of the sources may be estimated from the ratios of the anomalies to their respective FVD anomalies - ie., zc = [g#1(MA)] / [∂g#1(MA)/∂z] admin Sticky Note mgal/km admin Sticky Note where the analytic FVD for the i-th cylinder is=>
∂[gz(x)]i/∂z = K(Δρi)Ri2(x2 – zi2)/(x2 + zi2)2 so that for all i = 1-thru-5 cylinders and the regional, the total analytic FHD (= ftn) is=> ∂[gz(x)]/∂z = 1∑5 ∂[gz(x)]i/∂z + [∂gz(regional)/∂z = ???] admin Sticky Note first vertical derivative (FVD) of the gravity effects of the 5- cylinders plus regional Ralph Sticky Note - the peak FVD gravity anomaly [∂g#1(MA)/∂z] so that zc = [g#1(MA)] / [∂g#1(MA)/∂z] = -2 km admin Sticky Note where the Analytic Ftn=> SVD = ∂{1∑5 ∂[gz(x)]i/∂z}/∂z + [∂{∂gz(regional)/∂z}/∂z = ???] admin Sticky Note mgal/km2 admin Sticky Note second vertical derivative (SVD) of the gravity effects of the 5-
cylinders plus regional admin Sticky Note in addition, the maximum amplitudes (MA) - locate the x-coordinates of the central axes of the cylinders, - of p-order may be compared against their (p±1)-order components for the depths zi of the central axes, and - the zero crossings of the SVD about the MAs estimate the diameters (= 2Ri) of the related cylinders Ralph Sticky Note - thus from the FFT of the gravity anomalies, we may determine for each cylinder the=> - x-coordinate of its central axis, - z-coordinate or depth of its central axis, and - diameter of the source (eg., D#1 = 4 km) - the above results effectively constrain the A-coefficients so that least-squares estimates of the density contrasts (∆ρ) may be obtained admin Sticky Note - integration of the FFT-SVD completely recovers the total
gravity signal on page 47/59 - compared to differentiation, integration is a very stable process admin Sticky Note analytic ftn=> ∫∫∂/∂z[(∂gz/∂z) ∂z] ∂z = gz admin Sticky Note what would you get if you integrated gz one more time? admin Sticky Note Assignment #11=> complete exercise 5.3=> EARTHSC_5642_Ex5-3_17Mar15 John Sticky Note note that the teacher integrated twice John Sticky Note to produce geoid anylation or the sense of the gravity potencial.
admin Sticky Note -3 km admin Sticky Note +3 km admin Sticky Note - 'downward' continuation amplifies the high-frequency components - thus, it essentially behaves like a high-pass/low-cut filter admin Sticky Note - 'upward' continuation attenuates the high-frequency components - thus, it behaves essentially like a high-cut/low-pass filter, whereas admin Sticky Note the Nyquist limit of coefficient values admin Sticky Note - nowadays, however, multiple altitude grids of potential field data are becoming increasingly available from surface, airborne, and satellite surveys, - for which 'interval' continuation operators can be developed to evaluate data values at the intermediate altitudes between the grids
- that also honor the boundary grid values - the 'interval' grid continuation operator is described on pages (117-119)/153 of the GeomathBook.pdf admin Sticky Note - the inverse operations of 'upward' and 'downward' continuation are most reliable only over elevations within a few grid intervals of the observations - due to measurement errors and the non-uniquess of continuation - thus, to be certain of data behavior at more distant elevations, there is no recourse but to survey admin Sticky Note - array size=> 480 x 220 = 115,200 anomaly values - station interval = 2 km - from NRC (Keller et al., 1980) admin Sticky Note - note prominent edge effects - better window carpentry can reduce these edge effects
admin Sticky Note - upward continuation substantially reduces and smooths the anomaly gradients with increasing altitude Exercise 5.3 Fast Fourier Transform (FFT) A) Show that the gravity effect of the horizontal cylinder=> gz = [41.93 Δρ(R2/z)]/[(d2/z2) + 1] satisfies Laplace’s equation=> ∂2gz/∂d2 + ∂2gz/∂z2 = 0. For the gravity effects of the 5 buried horizontal cylinders B) Compute, list, and plot the related amplitude and phase spectra. C) Inverse transform the Fourier coefficients and compare the synthesized signal with the original. What are the sources of the mismatches? D) Compute, list and plot the first horizontal derivative ∂gz/∂d from the FFT of (gz). E) How do the results in D-above compare with the analytical horizontal first derivative gravity effects of the buried
horizontal cylinders? F) Compute, list, and plot the second horizontal derivative ∂2gz/∂d2 from the FFT of (gz). G) How do the results in F-above compare with the analytical horizontal second derivative gravity effects of the buried horizontal cylinders? H) Compute, list and plot the first vertical derivative ∂gz/∂z from the FFT of (gz). I) How do the results in H-above compare with the analytical vertical first derivative gravity effects of the buried horizontal cylinders? J) Compute, list, and plot the second vertical derivative ∂2gz/∂z2 from the FFT of (gz). K) How do the results in J-above compare with the analytical vertical second derivative gravity effects of the buried horizontal cylinders? L) Compute, list, and plot (gz) from the FFT of the analytical (∂2gz/∂z2) in K-above. M) How do the FFT estimates from L-above compare with the gravity effects of the 5 buried horizontal cylinders obtained in Exercise 2.1? Assignment TIP to consider: Just take an fft of the data and use the MATLAB "angle" command on the output. This will calculate the phase spectra. The "abs" command calculates the magnitude.
Look page 51/59 in pdf note %% Ex2-1 A) % Compute the 5 gravity profiles d=[-34 -20 0 20 35]; % in km z=[2 3 5 10 10]; % in km R=[2 1 3 4 3.5]; % in km dro=[3.0 5.0 1.0 2.0 1.5]; % in gm/cm^3 gz = (41.93*dro.*R.^2./z)./(d.^2./z.^2+1); d_itv=-100:1:100; gz1=(41.93*dro(1)*R(1)^2/z(1))./(d_itv.^2./z(1)^2+1); gz2=(41.93*dro(2)*R(2)^2/z(2))./(d_itv.^2./z(2)^2+1); gz3=(41.93*dro(3)*R(3)^2/z(3))./(d_itv.^2./z(3)^2+1); gz4=(41.93*dro(4)*R(4)^2/z(4))./(d_itv.^2./z(4)^2+1); gz5=(41.93*dro(5)*R(5)^2/z(5))./(d_itv.^2./z(5)^2+1); d1=d_itv+d(1); d2=d_itv+d(2); d3=d_itv+d(3); d4=d_itv+d(4); d5=d_itv+d(5); index1=find(abs(d1)<=64); index2=find(abs(d2)<=64); index3=find(abs(d3)<=64); index4=find(abs(d4)<=64); index5=find(abs(d5)<=64); % Plot the 5 gravity superimposed figure(1) plot(d1(index1), gz1(index1),'green'); hold on; plot(d2(index2), gz2(index2),'red'); hold on; plot(d3(index3), gz3(index3),'blue'); hold on; plot(d4(index4), gz4(index4),'yellow'); hold on; plot(d5(index5), gz5(index5),'black');
hold off title('Gravity Profiles', 'FontSize', 20); xlabel('Distance (km)', 'FontSize', 15); ylabel('Gravity (mgal)', 'FontSize', 15); legend('Caylinder 1','Cylinder 2','Cylinder 3','Cylinder 4','Cylinder 5'); %% Ex2-1 B) % Compute the total gravity effect of the 5 cylinders by summing their % effects at each observation point on the profile g=gz1(index1)+gz2(index2)+gz3(index3)+gz4(index4)+gz5(inde x5); % mean value average = sum(g)/length(g); % standard deviation std=sqrt(sum((g-average).^2)/length(g)); % Plot the total gravity effect figure(2); plot(d1(index1),g); hold on; plot(d1(index1),repmat(average,length(index1),1)); title('Total Gravity From Five Cylinders', 'FontSize', 20); xlabel('Distance (km)', 'FontSize', 15); ylabel('Gravity (mgal)', 'FontSize', 15); %% Ex2-1 C) A1=((41.93*R(1)^2/z(1))./(d_itv.^2./z(1)^2+1))' B1=gz1' dro1=(inv(A1'*A1))*A1'*B1; A2=((41.93*R(2)^2/z(2))./(d_itv.^2./z(2)^2+1))'; B2=gz2'; dro2=(inv(A2'*A2))*A2'*B2;
A3=((41.93*R(3)^2/z(3))./(d_itv.^2./z(3)^2+1))'; B3=gz3'; dro3=(inv(A3'*A3))*A3'*B3; A4=((41.93*R(4)^2/z(4))./(d_itv.^2./z(4)^2+1))'; B4=gz4'; dro4=(inv(A4'*A4))*A4'*B4; A5=((41.93*R(5)^2/z(5))./(d_itv.^2./z(5)^2+1))'; B5=gz5'; dro5=(inv(A5'*A5))*A5'*B5; %% Ex2-1 D) % lower lower triangular matrix L L1 = chol(A1'*A1,'lower'); L2 = chol(A2'*A2,'lower'); L3 = chol(A3'*A3,'lower'); L4 = chol(A4'*A4,'lower'); L5 = chol(A5'*A5,'lower'); % coefficients of P P1=(inv(L1))*A1'*B1; P2=(inv(L2))*A2'*B2; P3=(inv(L3))*A3'*B3; P4=(inv(L4))*A4'*B4; P5=(inv(L5))*A5'*B5; % least square estimates of dro dro1_CF=(inv(L1*L1'))*L1*P1; dro2_CF=(inv(L2*L2'))*L2*P2; dro3_CF=(inv(L3*L3'))*L3*P3; dro4_CF=(inv(L4*L4'))*L4*P4; dro5_CF=(inv(L5*L5'))*L5*P5; %%Update 2.1%computation of the ATA matrix for inverse solution from gravity signal A1=((41.93*R(1).^2/z(1))./(((D+34).^2./z(1).^2)+1)); A2=((41.93*R(2).^2/z(2))./(((D+20).^2./z(2).^2)+1));
A3=((41.93*R(3).^2/z(3))./(((D).^2./z(3).^2)+1)); A4=((41.93*R(4).^2/z(4))./(((D-20).^2./z(4).^2)+1)); A5=((41.93*R(5).^2/z(5))./(((D-35).^2./z(5).^2)+1)); A= [A1' A2' A3' A4' A5']; B=gzsum'; x=inv(A'*A)*A'*B; %%%%% Cholesky factors ATA=A'*A; ATB=A'*B; R1=chol(ATA); Z1=(inv(R1))'*ATB; X1=(inv(R1))*Z1;

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Discussion Question #1Find an article about a current event tha.docx

  • 1. Discussion Question #1: Find an article about a current event that discusses a change in the supply or demand of a product. For example, has there been a weather event that has affected certain food crops or the availablility of energy? Has a new consumer product been introduced? Or has a government regulation affected the production of a product? Provide a brief description of the article’s content and explain how economics can be used to analyze the situation and predict changes in equilibrium prices and quantities. Will the change likely persist over time, or is it temporary? admin Sticky Note ie., the SLFIi > 0 identify positively or directly correlative (X- peak)-to-(Y-peak) associations admin Sticky Note ie., the SLFIi < 0 identify positively or directly correlative (X- valley)-to-(Y-valley) associations admin Sticky Note to fruther simplify the positively correlated feature associations in the X- and Y-datasets, replot the SLFIi as=> - map-A for all SLFIi > 0, and - map-B for all SLFIi < 0, where the
  • 2. admin Sticky Note to fruther simplify the negaitively correlated feature associations in the X- and Y-datasets, replot the DLFIi as=> - map-C for all DLFIi > 0, and - map-D for all DLFIi < 0, where the admin Sticky Note ie., the DLFIi > 0 identify negatively or inversely correlative (X-peak)-to-(Y-valley) associations admin Sticky Note ie., the DLFIi < 0 identify negatively or inversely correlative (X-valley)-to-(Y-peak) associations admin Sticky Note To illustrate WCA, consider the signals T1 and T2 plotted in Fig. 4-below with their respective amplitude means (AM) removed. - the signals' other statistical attributes also are highly disparate, including the - amplitude standard deviations (ASD) and
  • 3. - amplitude ranges of (min, max)-values (AR) - the signals also have a negligible correlation coefficient (CC) that seems to be supported graphically by an apparent lack of feature associations admin Sticky Note - the 128-point signal T1 is the superposition of the regional background signal (B1) and the higher frequency residual signal (R1), and - the 128-point signal T2 is the superposition of the regional background signal (B2) and the higher frequency residual signal (R2), where - CC(B1, B2) = 0, and - the first half of R1 is positively correlated with the first half of R2, and - the other two halves are negatively correlated so that - CC(R1, R2) = 0. admin Sticky Note - this example is from von Frese et al. (1997) and - also is summarized in Chap. 7.3 [p. (135-138)/153] of the GeomathBook.pdf admin Sticky Note - to facilitate graphical correlation analysis of apples (eg., T1) to oranges (eg., T2), the signals were normalized as described
  • 4. on the next page-below=> admin Sticky Note Accordingly, in Fig. 5, the normalized T1 (ie., NT1) and T2 (ie., NT2) signals have dimensionless - ASD(NT1) = ASD(NT2) = 2.0, and - normalization factors (NF) so that T1 = [NF(T1) x NT1] and T2 = [NF(T2) x NT2] admin Sticky Note - Note that normalization does not affect CC(T1, T2). - However, this plot of the normalized signals suggests a possible regional phase shift for WCA investigation - Thus. wavenumber correlation filters (WCF) were developed and applied to NT1 & NT2 to extract the minimally correlated wavenumber components output respectively in Fig.s 6- & 7- below admin Sticky Note - or set σz = σx and μz = μx to normalize Y to X, so that zi(Y) = NY can be plotted on the same axes or contour intervals, etc. as those of X - or normalize both X and Y to common σz and μz so that zi(X) and zi(Y) may be plotted with common graphical parameters, etc.
  • 5. admin Sticky Note RMSE is the root-mean-squared error admin Sticky Note - an even better estimate of B1 (ie., EB1) is possible with (-0.1 < CC(k) < 0.1) admin Sticky Note - subtracting EB1 from T1, and EB2 from T2 yields pretty good estimated residuals ER1 and ER2, respectively - however=> CC(ER1, ER2) = 0 admin Sticky Note ER1 is signal A(x) in figure on next page admin Sticky Note ER2 is signal B(x) in figure on next page admin Sticky Note => ER1 admin
  • 6. Sticky Note => ER2 admin Sticky Note (ER1-valley)-to-(ER2-valley) associations admin Sticky Note (ER1-peak)-to-(ER2-peak) associations admin Sticky Note (ER1-peak)-to-(ER2-valley) associations admin Sticky Note (ER1-valley)-to-(ER2-peak) associations admin Sticky Note As another example of WCF from von Frese et al. (1997), consider the satellite magnetic observations in nT from the two Magsat mission orbital tracks across the arctic to northern Finnland in Fig. 14.A-below admin Sticky Note - these orbital track segments of lithospheric magnetic anomaly data at 400 km altitude are within about 10 km or so of each other - thus the lithospheric anomaly estimates should be highly correlated,
  • 7. - but the CC = 0.705, which suggests an initial noise contribution of roughly=> no = √[(1/CC2) – 1] ≈ 0.6468 or about 65% admin Sticky Note - to estimate the more correlative wavenumber components in the data tracks, a WCF for CC(k) > 0.6 was applied with output shown in Fig. 14.B-below admin Sticky Note - these positively correlated signals reflect ocean-continent boundaries and other lithospheric features - also, the improved CC suggests a noise contribution of only=> n1 ≈ 0.0707 or about 7.1%, or - a noise reduction of [(no - n1)/no]100 ≈ 89% admin Sticky Note - however, to avoid throwing out the baby with the bath water, it is always prudent to check out the rejected signal components as shown in Fig. 14.C-below - the positive residual correlation was checked by WCF for CC(k) > 0.3, - which output the regionally correlated components in Fig. 14.D-below
  • 8. admin Sticky Note - these regionally correlated signals seem to ignore continent- ocean boundaries and other lithospheric features, - thus, they were attributed mostly to external geomagnetic field effects and admin Sticky Note - the residual signals of Fig. 14.C shown here also lack prominent lithospheric affinities, and - their amplitudes are marginal relative to the amplitudes of the presumed lithospheric magnetic anomalies in Fig. 14.B-above admin Sticky Note - thus, pair-averages of the strong, positively correlated anomalies in Fig. 14.B-above were taken as least squares estimates of the lithospheric anomalies that the two Magsat tracks observed - here, the pair-averaged magnetic anomalies of the lithosphere are mapped by the solid-line profile, whereas - the dotted-line profile gives the pair-wise RMS-differences as error estimates on the corresponding lithospheric magnetic anomaly estimates admin Sticky Note A further WCF example from von Frese et al. (1997) considers
  • 9. five track-pairs of geoidal undulations from ascending orbits of the US Navy's Geosat-GM altimetry mission over a region east of the Gunerus Ridge which extends offshore of East Antarctica's admin Sticky Note - the marine altimetry estimates geoidal undulations in cm- amplitude units (AU) - vertical differentiation of the geoidal undulations can yield gravity anomalies for constraints on modeling the subjacent lithosphere admin Sticky Note - the tracks in these 5 pairs are separated by mean distances of about 2 to 5 km over a mean sea floor depth of about 4 km - so that geological features at these scales and larger should yield strong, positively correlated features between the altimetry tracks - but the mean CC between the track-pairs is CCM = 0.832, which implies a noise level of about=> no = √[(1/CC2) – 1] ≈ 44.93% admin Sticky Note - to reduce the noise level, the track-pairs were WCF for CC(k) > 0.80 for the output in Fig. 16-below admin
  • 10. Sticky Note - the WCF-output suggests an reduced noise level of about => n1 ≈ 29.1%, which is in a noise reduction of roughly => [(0.4493 - 0.2909)/0.4493]100 ≈ 35% admin Sticky Note - here, the rejected, broadband non-correlative features presumably include=> - crustal signals that are smaller than the track spacing, and - dynamic signals from temporal and spatial variations of the - orbit errors, - ocean currents, - waves and ice, - measurement and data reduction errors, and - other non-lithospheric effects admin Sticky Note Basement - not Bedrock...which refers to the overlying Phanerozoic sedimentary rocks
  • 11. admin Sticky Note WCA of the gravity and magnetic anomaly maps of Ohio - adapted from von Frese et al., 1997, Spectral correlation of magnetic and gravity anomalies of Ohio, Geophysics, v. 62, 365-380. admin Sticky Note compositional constraints on the basement rocks from 149 basement-penetrating boreholes admin Sticky Note interpretation of the anomaly maps is based on modeling of statewide profiles A-thru-E admin Sticky Note interpretation examples for profiles A and B are shown below admin Sticky Note A-profile admin Sticky Note Correlative features between the two datasets=> - reduce interpretational ambiguity
  • 12. - and improve interpretational reliability admin Sticky Note CC < 0 admin Sticky Note CC < 0 admin Sticky Note CC > 0 admin Sticky Note CC < 0 admin Sticky Note B-profile admin Sticky Note CC > 0 admin Sticky Note CC < 0 admin Sticky Note CC > 0
  • 13. admin Sticky Note CC > 0 admin Sticky Note CC > 0 admin Sticky Note Basement geology inferred from the magnetic and gravity anomaly data admin Sticky Note physical basis for correlating magnetic and gravity anomalies admin Sticky Note where there is - a correlation spectrum CC(k), there also is - an intercept spectrum A(k), and - slope spectrum S(k) involving the ratio Δj/Δσ of physical property contrasts admin Sticky Note
  • 14. A(k) admin Sticky Note S(k) admin Sticky Note pseudo-anomalies admin Sticky Note raw (ie., unnormalized) pseudo-anomalies derived from the Bouguer gravity and total magnetic anomalies admin Sticky Note admin Sticky Note admin Sticky Note here the normalization factor NF = SF admin Sticky Note admin Sticky Note normalized pseudo anomalies for enhance visual-spatial correlation analysis
  • 15. admin Sticky Note normalized FVD gravity and RTP magnetic anomalies WCF for (+1 ≤ k ≤ 0) to emphasize positive anomaly correlations admin Sticky Note admin Sticky Note admin Sticky Note admin Sticky Note admin Sticky Note admin Sticky Note SLFI > 0 emphasizes peak-to-peak gravity and magnetic features
  • 16. admin Sticky Note admin Sticky Note SLFI > 0 emphasizes peak-to-peak gravity and magnetic feature associations admin Sticky Note admin Sticky Note SLFI < 0 emphasizes valley-to-valley gravity and magnetic feature associations admin Sticky Note admin Sticky Note SLFI < 0 emphasizes valley-to-valley gravity and magnetic feature associations admin Sticky Note SLFI > 1 standard deviation (SD) emphasizes the strongest peak-to-peak gravity and magnetic feature associations
  • 17. admin Sticky Note SLFI < -1 standard deviation (SD) emphasizes the strongest valley-to-valley gravity and magnetic feature associations admin Sticky Note normalized FVD gravity and RTP magnetic anomalies WCF for (-1 ≤ k ≤ 0) to emphasize negative anomaly correlations admin Sticky Note comparison of WCF normalized FVD gravity anomalies with DLFI based on the LFI difference=> LFI(FVDG) - LFI(RTP) to hi-lite inverse feature associations in the FVD gravity anomalies admin Sticky Note comparison of WCF normalized RTP magnetic anomalies with DLFI based on the LFI difference=> LFI(FVDG) - LFI(RTP) to hi-lite inverse feature associations in the RTP magnetic anomalies
  • 18. admin Sticky Note comparison of normalized FVD gravity anomalies with DLFI > 0 to hi-lite=> FVDG-peak to RTPM-valley associations admin Sticky Note comparison of normalized RTP magnetic anomalies with DLFI > 0 to hi-lite=> FVDG-peak to RTPM-valley associations admin Sticky Note comparison of normalized FVD gravity anomalies with DLFI < 0 to hi-lite=> FVDG-valley to RTPM-peak associations admin Sticky Note comparison of normalized RTP magnetic anomalies with DLFI < 0 to hi-lite=>
  • 19. FVDG-valley to RTPM-peak associations admin Sticky Note DLFI > 1 standard deviation (SD) emphasizes the strongest FVDG-peak to RTPM-valley anomaly associations admin Sticky Note DLFI < -1 standard deviation (SD) emphasizes the strongest FVDG-valley to RTPM-peak anomaly associations admin Sticky Note normalized FVD gravity and RTP magnetic anomalies WCF for (-0.3 ≤ k ≤ +0.3) to emphasize the null-correlated anomaly features admin Sticky Note #4 admin Sticky Note #4
  • 20. admin Sticky Note #4 admin Sticky Note #4 admin Sticky Note #4 admin Sticky Note #4 admin Sticky Note - regions of the basement rocks where the WCA emphasizes possible occurrences of 4 physical property combinations - the edges of these regions were mapped from the zero contours of the SVD(g) and SVD(RTPm) anomalies admin Sticky Note negative=> [Δm(-), Δσ(+)] admin Sticky Note negative=> [Δm(+), Δσ(-)] admin Sticky Note
  • 21. positive=> [Δm(+), Δσ(+)] admin Sticky Note positive=> [Δm(-), Δσ(-)] admin Sticky Note null admin Sticky Note null admin Sticky Note - differentiate wrt m=> admin Sticky Note = jωl admin Sticky Note = (jωl)p admin Sticky Note = (jωk)p - thus, the total horizontal p-order derivative can be evaluated
  • 22. from=> = √[(jωl)p + (jωk)p] - in particular, for p = 2 the horizontal curvature may be obtained with zero contours that estimate feature edges, and thus provide edge detection admin Sticky Note - this is an elementary linear differential equation of order p with constant coefficients admin Sticky Note - typically found in Chap. 2 of common textbooks on ordinary differential equations admin Sticky Note => iff A = 0 admin Sticky Note ideal SVD-response admin Sticky Note ideal VD-response for p = 4 admin Sticky Note
  • 23. lousy (ie., crude) SVD-response - perhaps derived from the data domain convolution of a simplified first derivative operator=> (-1, 0, 1), - where the SVD(1D)=> (-1, 0, 1(۞)-1, 0, 1) and ۞ denotes convolution admin Sticky Note - much improved response of a SVD-operator such as developed in Fig. 1-below on next page admin Sticky Note - derivative filters generally behave like high-pass/low-cut filters admin Sticky Note Elkins 7-by-7 point SVD-grid or -mask operator admin Sticky Note zero row admin Sticky Note zero column admin Sticky Note - the weights at radii ri=> r1 = Δx,
  • 24. r2 = Δx√2, r3 = Δx2, etc. admin Sticky Note real admin Sticky Note imaginary with=> i = j = √(-1) admin Sticky Note imaginary with=> i = j = √(-1) admin Sticky Note - note that the order 'p' previously used is now 'n' - ie., p-order => n-order helpful for assignment 5.3 admin
  • 25. Sticky Note regional gravity effect admin Sticky Note total gravity effect of 5-cylinders + regional admin Sticky Note individual gravity effects of the 5 cylinders admin Sticky Note - or estimate the 5 cylinder densities and regional slope and intercept values from the total signal by inversion - and modify the A-coefficients for coefficients that obain the desired integral/derivative properties - by forward modeling of the modified system assuming the other cylinder parameters are known admin Sticky Note - or simply take the FFT of the total signal, - filter it using the appropriate integral/derivative transfer function coefficients - and IFFT the modified spectrum to obtain the data domain estimates of the desired integral/derivative properties admin Sticky Note The FFT is the superior approach in terms of numerical efficiency, accuracy, and the minimum assumptions needed
  • 26. admin Sticky Note To evaluate the integral/derivative properties of the total signal, we could=> - perform graphical operations with significant computational labor - or equivalent convolutions with significant computational effort and error Ralph Sticky Note cylinder #1 Ralph Sticky Note cylinder #2 Ralph Sticky Note cylinder #3 Ralph Sticky Note cylinder #4 Ralph Sticky Note cylinder #5 admin Sticky Note
  • 27. plus regional admin Sticky Note mgal admin Sticky Note rind applied at both ends to minimze Gibbs' error Ralph Sticky Note - if the investigator correctly assumes that these anomaly peaks are all due to the gravity effects of horizontal cylinder mass variations, - then at the anomaly peaks or maximum amplitudes (MA), the depths to the central axes (zc) of the sources may be estimated from the ratios of the anomalies to their respective FVD anomalies - ie., zc = [g#1(MA)] / [∂g#1(MA)/∂z] admin Sticky Note mgal/km admin Sticky Note where the analytic FVD for the i-th cylinder is=>
  • 28. ∂[gz(x)]i/∂z = K(Δρi)Ri2(x2 – zi2)/(x2 + zi2)2 so that for all i = 1-thru-5 cylinders and the regional, the total analytic FHD (= ftn) is=> ∂[gz(x)]/∂z = 1∑5 ∂[gz(x)]i/∂z + [∂gz(regional)/∂z = ???] admin Sticky Note first vertical derivative (FVD) of the gravity effects of the 5- cylinders plus regional Ralph Sticky Note - the peak FVD gravity anomaly [∂g#1(MA)/∂z] so that zc = [g#1(MA)] / [∂g#1(MA)/∂z] = -2 km admin Sticky Note where the Analytic Ftn=> SVD = ∂{1∑5 ∂[gz(x)]i/∂z}/∂z + [∂{∂gz(regional)/∂z}/∂z = ???] admin Sticky Note mgal/km2 admin Sticky Note second vertical derivative (SVD) of the gravity effects of the 5-
  • 29. cylinders plus regional admin Sticky Note in addition, the maximum amplitudes (MA) - locate the x-coordinates of the central axes of the cylinders, - of p-order may be compared against their (p±1)-order components for the depths zi of the central axes, and - the zero crossings of the SVD about the MAs estimate the diameters (= 2Ri) of the related cylinders Ralph Sticky Note - thus from the FFT of the gravity anomalies, we may determine for each cylinder the=> - x-coordinate of its central axis, - z-coordinate or depth of its central axis, and - diameter of the source (eg., D#1 = 4 km) - the above results effectively constrain the A-coefficients so that least-squares estimates of the density contrasts (∆ρ) may be obtained admin Sticky Note - integration of the FFT-SVD completely recovers the total
  • 30. gravity signal on page 47/59 - compared to differentiation, integration is a very stable process admin Sticky Note analytic ftn=> ∫∫∂/∂z[(∂gz/∂z) ∂z] ∂z = gz admin Sticky Note what would you get if you integrated gz one more time? admin Sticky Note Assignment #11=> complete exercise 5.3=> EARTHSC_5642_Ex5-3_17Mar15 John Sticky Note note that the teacher integrated twice John Sticky Note to produce geoid anylation or the sense of the gravity potencial.
  • 31. admin Sticky Note -3 km admin Sticky Note +3 km admin Sticky Note - 'downward' continuation amplifies the high-frequency components - thus, it essentially behaves like a high-pass/low-cut filter admin Sticky Note - 'upward' continuation attenuates the high-frequency components - thus, it behaves essentially like a high-cut/low-pass filter, whereas admin Sticky Note the Nyquist limit of coefficient values admin Sticky Note - nowadays, however, multiple altitude grids of potential field data are becoming increasingly available from surface, airborne, and satellite surveys, - for which 'interval' continuation operators can be developed to evaluate data values at the intermediate altitudes between the grids
  • 32. - that also honor the boundary grid values - the 'interval' grid continuation operator is described on pages (117-119)/153 of the GeomathBook.pdf admin Sticky Note - the inverse operations of 'upward' and 'downward' continuation are most reliable only over elevations within a few grid intervals of the observations - due to measurement errors and the non-uniquess of continuation - thus, to be certain of data behavior at more distant elevations, there is no recourse but to survey admin Sticky Note - array size=> 480 x 220 = 115,200 anomaly values - station interval = 2 km - from NRC (Keller et al., 1980) admin Sticky Note - note prominent edge effects - better window carpentry can reduce these edge effects
  • 33. admin Sticky Note - upward continuation substantially reduces and smooths the anomaly gradients with increasing altitude Exercise 5.3 Fast Fourier Transform (FFT) A) Show that the gravity effect of the horizontal cylinder=> gz = [41.93 Δρ(R2/z)]/[(d2/z2) + 1] satisfies Laplace’s equation=> ∂2gz/∂d2 + ∂2gz/∂z2 = 0. For the gravity effects of the 5 buried horizontal cylinders B) Compute, list, and plot the related amplitude and phase spectra. C) Inverse transform the Fourier coefficients and compare the synthesized signal with the original. What are the sources of the mismatches? D) Compute, list and plot the first horizontal derivative ∂gz/∂d from the FFT of (gz). E) How do the results in D-above compare with the analytical horizontal first derivative gravity effects of the buried
  • 34. horizontal cylinders? F) Compute, list, and plot the second horizontal derivative ∂2gz/∂d2 from the FFT of (gz). G) How do the results in F-above compare with the analytical horizontal second derivative gravity effects of the buried horizontal cylinders? H) Compute, list and plot the first vertical derivative ∂gz/∂z from the FFT of (gz). I) How do the results in H-above compare with the analytical vertical first derivative gravity effects of the buried horizontal cylinders? J) Compute, list, and plot the second vertical derivative ∂2gz/∂z2 from the FFT of (gz). K) How do the results in J-above compare with the analytical vertical second derivative gravity effects of the buried horizontal cylinders? L) Compute, list, and plot (gz) from the FFT of the analytical (∂2gz/∂z2) in K-above. M) How do the FFT estimates from L-above compare with the gravity effects of the 5 buried horizontal cylinders obtained in Exercise 2.1? Assignment TIP to consider: Just take an fft of the data and use the MATLAB "angle" command on the output. This will calculate the phase spectra. The "abs" command calculates the magnitude.
  • 35. Look page 51/59 in pdf note %% Ex2-1 A) % Compute the 5 gravity profiles d=[-34 -20 0 20 35]; % in km z=[2 3 5 10 10]; % in km R=[2 1 3 4 3.5]; % in km dro=[3.0 5.0 1.0 2.0 1.5]; % in gm/cm^3 gz = (41.93*dro.*R.^2./z)./(d.^2./z.^2+1); d_itv=-100:1:100; gz1=(41.93*dro(1)*R(1)^2/z(1))./(d_itv.^2./z(1)^2+1); gz2=(41.93*dro(2)*R(2)^2/z(2))./(d_itv.^2./z(2)^2+1); gz3=(41.93*dro(3)*R(3)^2/z(3))./(d_itv.^2./z(3)^2+1); gz4=(41.93*dro(4)*R(4)^2/z(4))./(d_itv.^2./z(4)^2+1); gz5=(41.93*dro(5)*R(5)^2/z(5))./(d_itv.^2./z(5)^2+1); d1=d_itv+d(1); d2=d_itv+d(2); d3=d_itv+d(3); d4=d_itv+d(4); d5=d_itv+d(5); index1=find(abs(d1)<=64); index2=find(abs(d2)<=64); index3=find(abs(d3)<=64); index4=find(abs(d4)<=64); index5=find(abs(d5)<=64); % Plot the 5 gravity superimposed figure(1) plot(d1(index1), gz1(index1),'green'); hold on; plot(d2(index2), gz2(index2),'red'); hold on; plot(d3(index3), gz3(index3),'blue'); hold on; plot(d4(index4), gz4(index4),'yellow'); hold on; plot(d5(index5), gz5(index5),'black');
  • 36. hold off title('Gravity Profiles', 'FontSize', 20); xlabel('Distance (km)', 'FontSize', 15); ylabel('Gravity (mgal)', 'FontSize', 15); legend('Caylinder 1','Cylinder 2','Cylinder 3','Cylinder 4','Cylinder 5'); %% Ex2-1 B) % Compute the total gravity effect of the 5 cylinders by summing their % effects at each observation point on the profile g=gz1(index1)+gz2(index2)+gz3(index3)+gz4(index4)+gz5(inde x5); % mean value average = sum(g)/length(g); % standard deviation std=sqrt(sum((g-average).^2)/length(g)); % Plot the total gravity effect figure(2); plot(d1(index1),g); hold on; plot(d1(index1),repmat(average,length(index1),1)); title('Total Gravity From Five Cylinders', 'FontSize', 20); xlabel('Distance (km)', 'FontSize', 15); ylabel('Gravity (mgal)', 'FontSize', 15); %% Ex2-1 C) A1=((41.93*R(1)^2/z(1))./(d_itv.^2./z(1)^2+1))' B1=gz1' dro1=(inv(A1'*A1))*A1'*B1; A2=((41.93*R(2)^2/z(2))./(d_itv.^2./z(2)^2+1))'; B2=gz2'; dro2=(inv(A2'*A2))*A2'*B2;
  • 37. A3=((41.93*R(3)^2/z(3))./(d_itv.^2./z(3)^2+1))'; B3=gz3'; dro3=(inv(A3'*A3))*A3'*B3; A4=((41.93*R(4)^2/z(4))./(d_itv.^2./z(4)^2+1))'; B4=gz4'; dro4=(inv(A4'*A4))*A4'*B4; A5=((41.93*R(5)^2/z(5))./(d_itv.^2./z(5)^2+1))'; B5=gz5'; dro5=(inv(A5'*A5))*A5'*B5; %% Ex2-1 D) % lower lower triangular matrix L L1 = chol(A1'*A1,'lower'); L2 = chol(A2'*A2,'lower'); L3 = chol(A3'*A3,'lower'); L4 = chol(A4'*A4,'lower'); L5 = chol(A5'*A5,'lower'); % coefficients of P P1=(inv(L1))*A1'*B1; P2=(inv(L2))*A2'*B2; P3=(inv(L3))*A3'*B3; P4=(inv(L4))*A4'*B4; P5=(inv(L5))*A5'*B5; % least square estimates of dro dro1_CF=(inv(L1*L1'))*L1*P1; dro2_CF=(inv(L2*L2'))*L2*P2; dro3_CF=(inv(L3*L3'))*L3*P3; dro4_CF=(inv(L4*L4'))*L4*P4; dro5_CF=(inv(L5*L5'))*L5*P5; %%Update 2.1%computation of the ATA matrix for inverse solution from gravity signal A1=((41.93*R(1).^2/z(1))./(((D+34).^2./z(1).^2)+1)); A2=((41.93*R(2).^2/z(2))./(((D+20).^2./z(2).^2)+1));
  • 38. A3=((41.93*R(3).^2/z(3))./(((D).^2./z(3).^2)+1)); A4=((41.93*R(4).^2/z(4))./(((D-20).^2./z(4).^2)+1)); A5=((41.93*R(5).^2/z(5))./(((D-35).^2./z(5).^2)+1)); A= [A1' A2' A3' A4' A5']; B=gzsum'; x=inv(A'*A)*A'*B; %%%%% Cholesky factors ATA=A'*A; ATB=A'*B; R1=chol(ATA); Z1=(inv(R1))'*ATB; X1=(inv(R1))*Z1;