Geosciences 567: CHAPTER (RR/GZ) CHAPTER : INTRODUCTION Inverse Theory: What It Is and What It Does Inverse theory, at least as I choose to deine it, is the ine art o estimating model parameters rom data It requires a knowledge o the orward model capable o predicting data i the model parameters were, in act, already known Anyone who attempts to solve a problem in the sciences is probably using inverse theory, whether or not he or she is aware o it Inverse theory, however, is capable (at least when properly applied) o doing much more than just estimating model parameters It can be used to estimate the quality o the predicted model parameters It can be used to determine which model parameters, or which combinations o model parameters, are best determined It can be used to determine which data are most important in constraining the estimated model parameters It can determine the eects o noisy data on the stability o the solution Furthermore, it can help in experimental design by determining where, what kind, and how precise data must be to determine model parameters Inverse theory is, however, inherently mathematical and as such does have its limitations It is best suited to estimating the numerical values o, and perhaps some statistics about, model parameters or some known or assumed mathematical model It is less well suited to provide the undamental mathematics or physics o the model itsel I like the example Albert Tarantola gives in the introduction o his classic book on inverse theory He says, you can always measure the captain s age (or instance by picking his passport), but there are ew chances or this measurement to carry much inormation on the number o masts o the boat You must have a good idea o the applicable orward model in order to take advantage o inverse theory Sooner or later, however, most practitioners become rather anatical about the beneits o a particular approach to inverse theory Consider the ollowing as an example o how, or how not, to apply inverse theory The existence or nonexistence o a God is an interesting question Inverse theory, however, is poorly suited to address this question However, i one assumes that there is a God and that She makes angels o a certain sie, then inverse theory might well be appropriate to determine the number o angels that could it on the head o a pin Now, who said practitioners o inverse theory tend toward the anatical? In the rest o this chapter, I will give some useul deinitions o terms that will come up time and again in inverse theory, and give some examples, mostly rom enke s book, o how to set up orward problems in an attempt to clearly identiy model parameters rom data Inverse Problem Theory, by Albert Tarantola, Elsevier Scientiic Publishing Company, 987
Geosciences 567: CHAPTER (RR/GZ) Useul Deinitions Let us begin with some deinitions o things like orward and inverse theory, models and model parameters, data, etc Forward Theory: The (mathematical) process o predicting data based on some physical or mathematical model with a given set o model parameters (and perhaps some other appropriate inormation, such as geometry, etc) Schematically, one might represent this as ollows: model parameters model predicted data As an example, consider the two-way vertical travel time t o a seismic wave through layers o thickness h i and velocity v i Then t is given by hi t = () v i= The orward problem consists o predicting data (travel time) based on a (mathematical) model o how seismic waves travel Suppose that or some reason thickness was known or each layer (perhaps rom drilling) Then only the velocities would be considered model parameters One would obtain a particular travel time t or each set o model parameters one chooses i Inverse Theory: The (mathematical) process o predicting (or estimating) the numerical values (and associated statistics) o a set o model parameters o an assumed model based on a set o data or observations Schematically, one might represent this as ollows: data model predicted (or estimated) model parameters As an example, one might invert the travel time t above to determine the layer velocities Note that one needs to know the (mathematical) model relating travel time to layer thickness and velocity inormation Inverse theory should not be expected to provide the model itsel odel: The model is the (mathematical) relationship between model parameters (and other auxiliary inormation, such as the layer thickness inormation in the previous example) and the data It may be linear or nonlinear, etc
Geosciences 567: CHAPTER (RR/GZ) odel Parameters: The model parameters are the numerical quantities, or unknowns, that one is attempting to estimate The choice o model parameters is usually problem dependent, and quite oten arbitrary For example, in the case o travel times cited earlier, layer thickness is not considered a model parameter, while layer velocity is There is nothing sacred about these choices As a urther example, one might choose to cast the previous example in terms o slowness s i, where: s i = / v i () Travel time t is a nonlinear unction o layer velocities but a linear unction o layer slowness As you might expect, it is much easier to solve linear than nonlinear inverse problems A more serious problem, however, is that linear and nonlinear ormulations may result in dierent estimates o velocity i the data contain any noise The point I am trying to impress on you now is that there is quite a bit o reedom in the way model parameters are chosen, and it can aect the answers you get! Data: Data are simply the observations or measurements one makes in an attempt to constrain the solution o some problem o interest Travel time in the example above is an example o data There are, o course, many other examples o data Some examples o inverse problems (mostly rom enke) ollow: edical tomography Earthquake location Earthquake moment tensor inversion Earth structure rom surace or body wave inversion Plate velocities (kinematics) Image enhancement Curve itting Satellite navigation Factor analysis 3 Possible Goals o an Inverse Analysis Now let us turn our attention to some o the possible goals o an inverse analysis These might include: Estimates o a set o model parameters (obvious) Bounds on the range o acceptable model parameters 3 Estimates o the ormal uncertainties in the model parameters 4 How sensitive is the solution to noise (or small changes) in the data? 5 Where, and what kind, o data are best suited to determine a set o model parameters? 6 Is the it between predicted and observed data adequate? 3
Geosciences 567: CHAPTER (RR/GZ) 7 Is a more complicated (ie, more model parameters) model signiicantly better than a more simple model? Not all o these are completely independent goals It is important to realie, as early as possible, that there is much more to inverse theory than simply a set o estimated model parameters Also, it is important to realie that there is very oten not a single correct answer Unlike a mathematical inverse, which either exists or does not exist, there are many possible approximate inverses These may give dierent answers Part o the goal o an inverse analysis is to determine i the answer you have obtained is reasonable, valid, acceptable, etc This takes experience, o course, but you have begun the process Beore going on with how to ormulate the mathematical methods o inverse theory, I should mention that there are two basic branches o inverse theory In the irst, the model parameters and data are discrete quantities In the second, they are continuous unctions An example o the irst might occur with the model parameters we seek being given by the moments o inertia o the planets: model parameters = I, I, I 3,, I (3) and the data being given by the perturbations in the orbital periods o satellites: data = T, T, T 3,, T N (4) An example o a continuous unction type o problem might be given by velocity as a unction o depth: and the data given by a seismogram o ground motion model parameters = v() (5) data = d(t) (6) Separate strategies have been developed or discrete and continuous inverse theory There is, o course, a air bit o overlap between the two In addition, it is oten possible to approximate continuous unctions with a discrete set o values There are potential problems (aliasing, or example) with this approach, but it oten makes otherwise intractable problems tractable enke s book deals exclusively with the discrete case This course will certainly emphasie discrete inverse theory, but I will also give you a little o the continuous inverse theory at the end o the semester 4 Nomenclature Now let us introduce some nomenclature In these notes, vectors will be denoted by boldace lowercase letters, and matrices will be denoted by boldace uppercase letters 4
Geosciences 567: CHAPTER (RR/GZ) Suppose one makes N measurements in a particular experiment We are trying to determine the values o model parameters Our nomenclature or data and model parameters will be data: d = [d, d, d 3,, d N ] T (7) model parameters: m = [m, m, m 3,, m ] T (8) where d and m are N and dimensional column vectors, respectively, and T denotes transpose The model, or relationship between d and m, can have many orms These can generally be classiied as either explicit or implicit, and either linear or nonlinear Explicit means that the data and model parameters can be separated onto dierent sides o the equal sign For example, d = m + 4m (9) and d = m + 4m m () are two explicit equations Implicit means that the data cannot be separated on one side o an equal sign with model parameters on the other side For example, and d (m + m ) = () d (m + m m ) = ( ) are two implicit equations In each example above, the irst represents a linear relationship between the data and model parameters, and the second represents a nonlinear relationship In this course we will deal exclusively with explicit type equations, and predominantly with linear relationships Then, the explicit linear case takes the orm d = Gm (3) where d is an N-dimensional data vector, m is an -dimensional model parameter vector, and G is an N matrix containing only constant coeicients The matrix G is sometimes called the kernel or data kernel or even the Green s unction because o the analogy with the continuous unction case: 5
Geosciences 567: CHAPTER (RR/GZ) d(x) = G(x, t)m(t) dt ( 4) Consider the ollowing discrete case example with two observations (N = ) and three model parameters ( = 3): d = m + m 4m 3 d = m + m + 3m 3 (5) which may be written as d d = 4 3 m m m 3 (6) or simply d = Gm (3) where d = [d, d ] T m = [m, m, m 3 ] T and G = 4 3 (7) Then d and m are and 3 column vectors, respectively, and G is a 3 matrix with constant coeicients On the ollowing pages I will give some examples o how orward problems are set up using matrix notation See pages 6 o enke or these and other examples 6
Geosciences 567: CHAPTER (RR/GZ) 5 Examples o Forward Problems 5 Example : Fitting a Straight Line (See Page o enke) T (temperature) slope = b a (depth) Suppose that N temperature measurements T i are made at depths i in the earth The data are then a vector d o N measurements o temperature, where d = [T, T, T 3,, T N ] T The depths i are not data Instead, they provide some auxiliary inormation that describes the geometry o the experiment This distinction will be urther clariied below Suppose that we assume a model in which temperature is a linear unction o depth: T = a + b The intercept a and slope b then orm the two model parameters o the problem, m = [a, b] T According to the model, each temperature observation must satisy T = a + b: T = a + b T = a + b T N = a + b N These equations can be arranged as the matrix equation Gm = d: T T = TN a b N 7
Geosciences 567: CHAPTER (RR/GZ) 5 Example : Fitting a Parabola (See Page o enke) T (temperature) (depth) I the model in example is changed to assume a quadratic variation o temperature with depth o the orm T = a + b + c, then a new model parameter is added to the problem, m = [a, b, c] T The number o model parameters is now = 3 The data are supposed to satisy T = a + b + c T = a + b + c T N = a + b N + c N These equations can be arranged into the matrix equation T T = TN N a b c N This matrix equation has the explicit linear orm Gm = d Note that, although the equation is linear in the data and model parameters, it is not linear in the auxiliary variable The equation has a very similar orm to the equation o the previous example, which brings out one o the underlying reasons or employing matrix notation: it can oten emphasie similarities between supericially dierent problems 8
Geosciences 567: CHAPTER (RR/GZ) 53 Example 3: Acoustic Tomography (See Pages 3 o enke) Suppose that a wall is assembled rom a rectangular array o bricks (Figure rom enke, below) and that each brick is composed o a dierent type o clay I the acoustic velocities o the dierent clays dier, one might attempt to distinguish the dierent kinds o bricks by measuring the travel time o sound across the various rows and columns o bricks, in the wall The data in this problem are N = 8 measurements o travel times, d = [T, T, T 3,, T 8 ] T The model assumes that each brick is composed o a uniorm material and that the travel time o sound across each brick is proportional to the width and height o the brick The proportionality actor is the brick s slowness s i, thus giving = 6 model parameters, m = [s, s, s 3,, s 6 ] T, where the ordering is according to the numbering scheme o the igure as The travel time o acoustic waves (dashed lines) through the rows and columns o a square array o bricks is measured with the acoustic source S and receiver R placed on the edges o the square The inverse problem is to iner the acoustic properties o the bricks (which are assumed to be homogeneous) and the matrix equation is row : T = hs + hs + hs 3 + hs 4 row : T = hs 5 + hs 6 + hs 7 + hs 8 column 4: T 8 = hs 4 + hs 8 + hs + hs 6 T T = h T8 s s s6 Here the bricks are assumed to be o width and height h 9
Geosciences 567: CHAPTER (RR/GZ) 54 Example 4: Seismic Tomography An example o the impact o inverse methods in the geosciences: Northern Caliornia A large amount o data is available, much o it redundant Patterns in the data can be interpreted qualitatively Inversion results quantiy the patterns Perhaps, more importantly, inverse methods provide quantitative inormation on the resolution, standard error, and "goodness o it" We cannot overemphasie the "impact" o colorul graphics, or both good and bad Inverse theory is not a magic bullet Bad data will still give bad results, and, interpretation o even good results requires breadth o understanding in the ield Inverse theory does provide quantitative inormation on how well the model is "determined," importance o data, and model errors Another example: improvements in "imaging" subduction ones 55 Example 5: Convolution Convolution is widely signiicant as a physical concept and oers an advantageous starting point or many theoretical developments One way to think about convolution is that it describes the action o an observing instrument when it takes a weighted mean o some physical quantity over a narrow range o some variable All physical observations are limited in this way, and or this reason alone convolution is ubiquitous (paraphrased rom Bracewell, The Fourier Transorm and Its Applications, 964) It is widely used in time series analysis as well to represent physical processes The convolution o two unctions (x) and g(x) represented as (x)*g(x) is (u) g(x u) du (8) For discrete inite unctions with common sampling intervals, the convolution is h k = m i= i g k i < k < m + n ( 9) A FORTRAN computer program or convolution would look something like: L=+N DO I=,L H(I)= DO I=, DO J=,N H(I+J )=H(I+J )+G(I)*F(J)
Geosciences 567: CHAPTER (RR/GZ) Convolution may also be written using matrix notation as = + m n m n n n h h h g g g ( ) In the matrix orm, we recognie our amiliar equation Gm = d (ignoring the conusing notation dierences between ields, when, or example, g above would be m ), and we can deine deconvolution as the inverse problem o inding m = G d Alternatively, we can also reormulate the problem as G T Gm = G T d and ind the solution as m = [G T G] [G T d] 6 Final Comments The purpose o the previous examples has been to help you ormulate orward problems in matrix notation It helps you to clearly dierentiate model parameters rom other inormation needed to calculate predicted data It also helps you separate data rom everything else Getting the orward problem set up in matrix notation is essential beore you can invert the system The logical next step is to take the orward problem given by d = Gm (3) and invert it or an estimate o the model parameters m est as m est = G inverse d () We will spend a lot o eort determining just what G inverse means when the inverse does not exist in the mathematical sense o GG inverse = G inverse G = I () where I is the identity matrix The next order o business, however, is to shit our attention to a review o the basics o matrices and linear algebra as well as probability and statistics in order to take ull advantage o the power o inverse theory