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adminSticky Note- the process of inversion is shown in .docx
- 1. admin Sticky Note - the process of inversion is shown in the figure on next page 1-A and in Fig. 2.1 (GeomathBook.pdf, p. 22/153) - ie., it is an artform with subjective and quantitative objective components admin Sticky Note In math, science, and engineering, the 'inverse problem' of relating data observations to unknown elements or coefficients of a model is a 'process' commonly called 'inversion'. However, other names may be used depending on the application - eg., it may be called => blackbox analysis in physics => factor analysis in social sciences => forward and inverse modeling in geophysics => or simply modeling in any dicipline => trial-and-error analysis => deconvolution in signal processing and engineering => ????? in geodesy? => etc. admin Sticky Note AX is the linear or non-linear forward model,
- 2. so that AX = B + 'errors', where B is the observed data set, admin Sticky Note A is the system input or design matrix, admin Sticky Note and X is the impluse response or set of unknown blackbox coefficients that the inversion seeks to determine admin Sticky Note The essence of any inverse problem is the forward model = AX, which in practice is always processed by linear digital operations. admin Sticky Note SENSITIVITY ANALYSIS - pretty much where most modern developments in inversion analysis are concentrated. admin Sticky Note the most meaningful set of solution coefficients admin Sticky Note translating (ie., rationalizing) the best set of solution coefficients into 'new' information on the conceptual model - usually augmented with digital graphics.
- 3. admin Sticky Note Inversion examples of conceptual and mathematical models are shown on page 2-A below and in Fig. 2.2 (GeomathBook.pdf, p. 23/153) admin Sticky Note Conceptual models are imagined simplifications of reality admin Sticky Note Mathematical models are the further simplified expressions of conceptual models admin Sticky Note joint inversions would obtain density and velocity solutions that satisfy both gravity and seismic observations admin Sticky Note always admin Sticky Note always admin Sticky Note
- 4. always admin Sticky Note => eg., the data set is improperly 'scaled' relative to computer's working precision => for example, the inverse problem is improperly posed to find angstrom scale variations in observations that are known only to nearest kilometer, etc. admin Sticky Note Figure on page 8-A below shows examples of model 'appropriateness' relative to inversion 'objectives' admin Sticky Note Inversion (ie., modeling) objectives dictate the mathematical model that should be invoked - or model 'appropriateness' In other words, the forward model should - not be overly complex numerically so as to waste
- 5. computational resources, - or too simple numerically so as not to achieve the inversion objective. - Rather, it should be just right or 'optimal' so that the inversion objective is achieved with 'minimum' resource expenditure. admin Sticky Note <= for example, the gravity profile on the left may be modeled with equivalent accuracy by the 3 mathematical models generalized in the bottom 3 panels with varying numbers of unknowns, m. admin Sticky Note if one wanted to simply interpolate the gravity signal at unmapped x-coodinates, for example, then the horizontal cyclinder model would be most appropriate, because only m = 1 unknown (eg., the density contrast Δρ- value) needs to be determined from the n-data points admin Sticky Note However, if the objective is to drill the source, which is a relatively expensive operation, then a more complex half-space solution might be warranted involving m = 5+ line segment deviations from the horizontal to better resolve subsurface details of the source's geometry The cost of the half-space solutions would be at least 5+ times
- 6. the cost of the horizontal cylinder solution, but probably worth it given the great costs of the drilling application. admin Sticky Note On the other hand, if the objective is to mine or excavate the source, which is the most expensive way to exploit the subsurface, then a 2-D checkerboard model of the subsurface might be warranted involving the determination of at least m = 5 x 10 = 50 prism densities. The cost of the checkerboard solutions would be at least 50 times the cost of the horizontal cylinder solution, but probably worth it given the extreme costs of subsurface mining and excavation. admin Sticky Note In general, the 'underdetermined' inversion where m > n is frankly the 'norm' in every mapping application . admin Sticky Note Thus, inversion involves=> - conceptualizing or fantasizing an appropriate forward model, - quantitatively determining the solution coefficients, - critiquing the solution for the most meaningful coefficients, and - converting or rationalizing the solution coefficients in terms of the conceptual model
- 7. People who are adept at the art form of conceptualizing models to relate to data, and transforming the numerical results of inversion into meaningful rationalizations or stories are highly sought after and rewarded. The objective, quantitative components of inversion are relatively trivial and well established since Gauss' time (~ 1800). admin Sticky Note The relative advantages and limitation of trial-and-error inversion are summarized on the next page. admin Sticky Note Trial-and-error inversion is perhaps the most successful and widely used method in human history - it also seems to find wide application throughout nature by virtue of Darwin's Law of Natural Selection whereby organisms adapt to their environment. admin Sticky Note i is the row subscript, and
- 8. j the column subscript admin Sticky Note don't really need to take a course in linear algebra anymore to become facile with matrix operations - just become familiar with a linear equation solver like matlab, mathematica, etc. admin Sticky Note ie. the matrix expression of unity admin Sticky Note the determinant is the value or number associated with a square matrix that is used to define the characteristic polynomial of the matrix. admin Sticky Note the number of terms in the determinant of an n-order matrix is n-factorial or n! - where for example, 10! = 3,628,800 is the number of terms needing evaluation for a tenth-order matrix. admin Sticky Note
- 9. Sarrus' rule=> the determinant for n>2 is the sum of he products along the right-pointing diagonals minus the sum of the products along the left-pointing diagonals admin Sticky Note X and B are also sometimes called n-tuples admin Sticky Note Cramer's Rule from ca. 1798 admin Sticky Note this example is also considered in Fig. 4.2 (GeomathBook.pdf, p. 32/153) admin Sticky Note we get=> admin Sticky Note where=> admin Sticky Note a buried 2-D horizontal cylinder extending infinitely into and out of the page
- 10. admin Sticky Note note that the elements aij and bi are known, whereas the element xj is unknown admin Sticky Note ie, the design matrix elements aij are evaluated with the unknown variables x1 and x2 set to unity - or (x1 = Δρ) = 1 = (x2 = C) admin Sticky Note |A'|/|A| = admin Sticky Note error = [(0.500 - 0.444)/(0.5)]100 = 11.2% admin Sticky Note we do not 'invert' anything here, but rather 'repeat the inversion' admin Sticky Note
- 11. ~ 7% error admin Sticky Note ~ 11% error admin Sticky Note inversion of admin Sticky Note compute A @ d1 = 0 km and d2 = 20 km for the system=> admin Sticky Note coefficients of the A-matrix admin Sticky Note coefficients of he Identity matrix admin Sticky Note we 'triangulate' the system using elementary row operations admin Sticky Note ie., the a12-element was zeroed out by multiplying the top row by -(4.440/75.474) and subtracting the result from the bottom row, etc.
- 12. admin Sticky Note why can you not use elementary column operations to triangulate the system? admin Sticky Note for the n-order system admin Sticky Note ie., the system of equations is diagonalized admin Sticky Note in the days before computerized equation solvers were available like Matlab, Mathematica, etc., the 'rank' concept had considerable practical significance for minimizing manual processing of singular systems admin Sticky Note real-world problems are always underdetermined!!!
- 13. admin Sticky Note more generally, in modern least squares applications, we have A => AtA and B => AtB (the weighted observation vector) admin Sticky Note Gaussian elimination algorithm admin Sticky Note Matrix determinant algorithm admin Sticky Note B is the column matrix containing values of the 'dependent' variable, and A is the column matrix (missing the transpose symbol) containing values of the 'independent' variable admin Sticky Note the least squares solution will be developed using the principle of 'mathematical induction' - ie., if 1) when a statement is true for for a natural number n = k, then it will also be true for its successor n = k+1;
- 14. 2) and the statement is true for n = 1; then the statement will be true for every natural number n. To prove a statement by induction, parts 1)- and 2)-above must be proved. admin Sticky Note in linear regression, the slope x1 is the only unknown admin Sticky Note typo=> x2-above should be x1 admin Sticky Note ie., as long as AtA ≠ 0 admin Sticky Note typo=> delete the + symbol in the above term - ie., it should be the product=> x1ai and not the sum=> x1+ai admin Sticky Note typo=> the above unknowns x2 & x3 should be=> x1 & x2
- 15. admin Sticky Note ie., j-indices refer to transpose elements, and k-indices refer to normal vector elements admin Sticky Note δij = 0 for all i ≠ j and δij = 1 for all i = j admin Sticky Note Note that for any number of unknowns m and number of observations n=> A(n×m)X(m×1) = B(n×1) versus AtA(m×m)X(m×1) = AtB(m×1)
- 16. admin Sticky Note but a faster approach that avoids finding (AtA)-1 directly is to use the Cholesky factorization of (AtA) admin Sticky Note see Lawson & Hansen (1976) admin Sticky Note this is a telescoping series or equation where the x3-solution is needed to solve for the x2-solution, etc. admin Sticky Note in evaluating a system of m = 100 unknowns, the use of the #1-option below requires m3 =106 operations; #2-option below reduces the operations by a third to 2m3/3 = 666,667; and #3-option below reduces the operations by two-thirds to 2m3/6 = 333,333 admin
- 17. Sticky Note or 5,050 elements for m =100 unknowns admin Sticky Note in the 1980s, OSU's IBM 3081 mainframe computer could evaluate roughly 1,000 unknowns in-core, which only held a maximum (AtA)-packed array of about 500,500 elements admin Sticky Note however, at the expense of CPU-time, the number of unknowns could be expanded as necessary by updating the packed (AtA)- array on an external (out-of-core) storage device as unknowns are added to the system. admin Sticky Note Assignment #4=> complete exercise 2.1=> EARTHSC_5642_Ex2-1_08Feb15 admin Sticky Note ie., rank (AtA) is < m admin Sticky Note ie., rank (AtA) is approximately < m admin
- 18. Sticky Note Assignment #5=> complete exercise 2.2=> EARTHSC_5642_Ex2-2_08Feb15 admin Sticky Note and overdetermined so that n > m admin Sticky Note note that var(X) = σX2 and var(B) = σB2 admin Sticky Note r2 is sometimes also called 'coherency' or the 'coherency coefficient' admin Sticky Note Assignment #6=> What is the matrix expression for the correlation coefficient r in terms of the matrices A, X, and B that also accounts for the sign of r? admin Sticky Note - under the square root symbol is a number that is just the i-th diagonal element of the n x n matrix A(AtA)-1At - this element is the weight for the confidence interval (CI) or error bar on the prediction bi^hat
- 19. - see example in Fig. VI.5 of the handout admin Sticky Note or Σ(bi^hat - bi^bar)2 ---> 0 admin Sticky Note Assignment #7=> complete exercise 2.3=> EARTHSC_5642_Ex2-3_09Feb15 admin Sticky Note - even though the synthetic data (ie., the predictions from a solution) may fit the observed data, the error bars on the solution may be unacceptably large - ie., sufficiently large that the solution is useless for anything else than matching the observations! - and the solution is said to be unstable or poorly conditioned or near-singular admin Sticky Note - the coefficients of the A-matrix take up most of the resources in solving the inverse problem - thus, rather than re-parameterizing the problem for a new set of A-coefficients, the pressure is on to manipulate or manage the original set of coefficients for a better performing solution
- 20. - classically, this has been done by considering the eigenvalue/eigenvector decomposition of AtA admin Sticky Note in matlab, the condition number is=> COND = λmax/λmin and the reciprocal condition is given by=> RCOND = λmin/λmax admin Sticky Note of course, for least squares problems, this system really is=> AtAX = AtB admin Sticky Note two ways to check your code for finding the eigenvalues of AtA admin Sticky Note so what are the COND or RCOND values here?
- 21. admin Sticky Note x1 x2 = A = 4 8 x3 x4 8 4 admin Sticky Note - arrow intersects the ellipse halfway between (4, 8) and (8, 4) with semi-major axis length=> λmax = 12 and semi-minor axis length=> λmin = -4 - where the directions of the two axes are given by the two corresponding eigenvectors - ie., for λmax = 12 the eigenvector is=> Xmax = (x1 x2)t = (1 1)t - so that the slope of the semi-major axis is=> tan-1(1/1) = 45o - and for λmin = -4 the eigenvector is=> Xmin = (-1 1)t - so that the slope of the semi-minor axis is=> tan-1(-1/1) = 135o = 45o + 90o admin Sticky Note for A1 = 6 8 8 6 - the rows are becoming more similar and the matrix increasingly near-singular - as reflected by the eigenvalues=>
- 22. λmax = 14, λmin = -2, and related COND admin Sticky Note for A2 = 4 8 4 8 - the matrix is singular with eigenvalues=> λmax = 12, λmin = 0, and COND---> ∞ - ie., the second row adds no new info to the system admin Sticky Note for A3 = -4 8 8 4 - the matrix is orthogonal with eigenvalues=> λmax = 8.95 = -λmin, and COND = RCOND - so that the solution using A-1 is stable in the sense that small changes in the A-coefficients do not result in large erratic variations in the predictions admin Sticky Note
- 23. correction=> At = VΛtUt where U = AVΛ-1 since Λ-1 = (Λt)-1 so that to evaluate U we don't need to evaluate the (n x n) system AAt admin Sticky Note of admin Sticky Note correction=> K = - log(RCOND) admin Sticky Note - modeling a (10 x 10)-array of satellite-observed total field magnetic anomaly values in nT at 125 km altitude with a 1o- station spacing - by a (10 x 10)-array of point dipoles at the earth's surface with a 1o-station spacing - in the 1975 International Geomagnetic Reference Field (IGRF- 75)
- 24. -- the inversion obtained point dipole magnetic susceptibilities so that the field of the dipoles fit the observed magnetic anomalies of insert A with least squares accuracy admin Sticky Note - in the remaining 4 inserts, the system's conditioning was investigated using=> K = -log(RCOND) as a function of various inversion parameters - for example, insert B shows that increasing the altitude of the observations degrades the solution increasingly because the system is becoming more near-singular and unstable admin Sticky Note - in insert B, the solutions were obtained for an applied differentially-reduced-to-pole (DRTP) field with intensity Fe = 60,000 nT - this simplification minimized considering the inversion effects from varying IGRF intensities, declinations, and inclinations admin Sticky Note - for insert C, the (10o x 10o)-arrays are moved along a constant longitude - with the observed anomaly values at 125 km altitude, and - a DTRP field of Fe = 60,000 nT - the system becomes increasingly ill condition as it approaches the earth's poles
- 25. admin Sticky Note - insert D shows the conditioning effects of moving the problem along the swath between the geographic equator and 10oN in the IGRF-75 (curve A), - along the swath between the geographic equator and 10oS in the IGRF-75 (curve B), and - along the swath between the geographic equator and 10oS in the DRTP (curve C) - the observed magnetic anomalies were all taken at 125 km altitude admin Sticky Note - in insert E, curve A shows the conditioning effects at 400 km altitude in the DRTP of varying the observation grid spacing for a constant 1o source grid spacing - ie., this system is best conditioned for observation grid spacings equal to or greater than roughly 2.5 times the source spacing admin Sticky Note - curve B of insert E, on the other hand, shows the conditioning effects at 400 km altitude in the DRTP of varying the source grid spacing for a constant 1o observation grid spacing - ie., this system is best conditioned where the source grid spacing is equal to roughly 1.5 to 2 times the observation spacing
- 26. admin Sticky Note ie. the=> GLI admin Sticky Note H is the GLI that is sometimes also referred to as the 'pseudo' or 'Penrose' inverse admin Sticky Note ie., the design matrix A is multiplied from the right by the GLI to see how close the product is to the n-th order identity matrix In admin Sticky Note ie., the design matrix A is multiplied from the left by the GLI to see how close the product is to the m-th order identity matrix Im admin Sticky Note correction=> minimum admin
- 27. Sticky Note ie., where m < n admin Sticky Note where the GLI=> H = (AtA)-1At admin Sticky Note correction=> quantitatively admin Sticky Note ie., the info density matrix S provides insight on how the observations are related to the inferred forward model of the system admin Sticky Note ie., the info density matrix S provides insight on the observations that may be culled from or added to the system for a more effective solution admin Sticky Note ie., the info density matrix S provides insight on designing appropriate surveys, experiments, field work, etc. admin Sticky Note
- 28. - note that the info density matrix S is a function only of the coefficients of the design matrix A, and not the magnitudes of the coefficients of the observation vector B - ie., solving the system where all the B-coefficients have been set to zero yields the exact same COND as does the solution for the original set of B-coefficients admin Sticky Note where the GLI=> H = (AtA)-1At admin Sticky Note ie., where m > n admin Sticky Note ie., where (m = n) or (m > n) admin Sticky Note ie., the corresponding k-th variable admin Sticky Note ie., by deleting either xi or xj, etc. admin
- 29. Sticky Note Inversion of free oscillation earthquake data for a core-to- crustal surface profile of earth densities, where - the densities ρ(ro) are estimated at target depths ro that range from - ro = (0.10)re near the earth's center in the top-left insert to - ro = (0.99)re near the earth's surface in the bottom-right insert, - and re is the earth's mean radius of about 6,372 km admin Sticky Note for each insert, the horizontal R-axis extends from the earth's center at R = 0re to its mean surface at R = 1re admin Sticky Note ρ(ro = 1re) ≈ 3 g/cm3 admin Sticky Note ρ(ro = 0re) ≈ 13 g/cm3 admin Sticky Note m(dots) = 35 admin
- 30. Sticky Note where the co-variance is=> cvar(xk^hat) = σ2(VpΛp-2 Vpt ) admin Sticky Note or 'standardizing' admin Sticky Note The resolution limits of inversion depend on=> admin Sticky Note on page 83/100 below admin Sticky Note on page 83/100 below admin Sticky Note for underdetermined systems admin Sticky Note - the eigenvalues are arranged in decreasing magnitude order
- 31. - in general, these spectra tend to be insightful where the number of unknowns is relatively small like in these examples - for large numbers of unknowns, the choice of a cut-off values can be hard to make - for gridded solutions in particular, the eigenvalue spectrum tends to decay smoothly without a prominent break in slope admin Sticky Note OPTIONAL Assignment #8=> complete exercise 2.4=> EARTHSC_5642_Ex2-4_15Feb15 admin Sticky Note in the computationally long-winded approach of Fig. 2.6=> admin Sticky Note on page 84/100 admin Sticky Note on page 85/100 admin Sticky Note correction=> CONSTRAINED LINEAR INVERSION
- 32. admin Sticky Note ie., the IDEAL WORLD of mostly textbooks admin Sticky Note ie., the REAL WORLD that includes improperly scaled inversions admin Sticky Note where X = A-1B admin Sticky Note where X = (AtA)-1AtB admin Sticky Note where (m - n) of the unknowns must be expressed in terms of the remaining unknowns or reassigned admin Sticky Note Note that the GLI always exists for obtaining solutions from REAL WORLD systems admin Sticky Note Note that the conditioning of the AX-system has nothing to do with how the magnitudes of the observed B-coefficients vary
- 33. admin Sticky Note where the GLI=> H = [AtA)-1At] admin Sticky Note Thus, the Information Density matrix S = AH = A[AtA)-1At] ≈ UUt = In may be useful in cases II, IV, and V, for isolating optimal observation B-coefficients to relate to a chosen forward model AX admin Sticky Note and the IResolution matrix R = HA = [AtA)-1At]A ≈ VVt = Im may be useful in cases III, IV, and V, for isolating optimal X- coefficients to relate to a chosen forward model AX admin Sticky Note - in general, cases IV and V are the REAL WORLD cases most commonly encountered by researchers, - and in need of sensitivity analysis to establish optimally performing solutions admin Sticky Note
- 34. in the Soc. for Industrial and Applied Mathematics (SIAM) journal=> Technometrics admin Sticky Note that is drawn from the Gaussian distribution=> N(μ=0, σ) with σ2 = EV - using EV to stablize a system is equivalent to the classical approach of computing each A-coefficient with random noise added from the N(0, σ)-distribution admin Sticky Note or error of fit (EOF) admin Sticky Note with λj‘ = (λj + EV) admin Sticky Note ie., X’ , can be computed in about a tenth of the effort it took to compute the A-coefficients admin Sticky Note on pages 90/100 and 91/100, respectively admin Sticky Note
- 35. Figures II.1 thru II.6 have been deleted admin Sticky Note The rest of the notes to the end of this 5642Lectures_2_4.pdf are provided for completeness only... - the utility of these notes is becoming limited as modern data processing increasingly relies on linear systems and spectral analysis - thus, we will move on to consider the most computationally efficient and elegant linear system of all=> the spectral model admin Sticky Note Insert A shows a (16 x 16)-array of satellite magnetic observations with 2o-station spacing over India at 400 km asl that we want to relate by least squares inversion to the magnetic susceptibities of a (16 x 16)-array of point dipoles with 2o- station spacing at 100 km bsl assuming the IGRF-75 updated to 1980 admin Sticky Note Insert B shows that the solution provides an exact match within working precision to observed magnetic anomalies in A - however, the condition number as inferred by K = -log(RCOND) ≈ 14 suggests that the solution contains almost no the significant figures
- 36. admin Sticky Note Insert C is a map of the DRTP anomalies produced using the very poorly conditioned solution of insert B - ie., changing the A-coefficients to process the solution for anything (eg., interpolations, continuations, etc.) beyond estimating the original observations yields essentially useless results ie., garbage - even though the predictions are nearly perfect in matching the observed anomalies admin Sticky Note Insert I is the map of the DRTP anomalies produced with a better conditioned solution based on using the optimal EV = 10- 7 that minimizes - the misfit in the observed data according to curve C in Fig. 3- below, and - the map-to-map differences in the predictions with changing EV according to curve B in Fig. 3-below admin Sticky Note Insert H provides an effective error map on the predictions in insert I that was obtained by subtracting the predictions at EV = 10-6 from those at EV = 10-8 admin
- 37. Sticky Note Curve B = sum of squared residuals (SSR) = Σi[(Pred. Map @ EVi) – (Pred. Map @ EV=1)]2 for the map of predictions (Pred. Map) at various EV-values admin Sticky Note Curve C = Σi[(Pred. Obs. @ EVi) – Obs.)]2 admin Sticky Note Curve D = K = -log(RCOND) admin Sticky Note Curve E = variance of solutions @ EVi admin Sticky Note put cursor on top-left corner to see range of 'optimal' EV=> |---------------| 10-6 admin Sticky Note Curves B and C provide an effective trade-off diagram for estimating an 'optimal' EV, whereas Curves D and E are only marginally effective
- 38. admin Sticky Note Principles of Linear Programming <===> Game Theory admin Sticky Note convex solution space that satisfies the a-thru-e inequalities of eq. 4.1