SOEE3250/5675/5115 Inverse Theory Lecture 2; notes by G. Houseman

Transcription

1 SOEE3250/5675/5115 Inverse Theory Lecture 2; notes by G. Houseman Topics covered in this lecture: Matrix diagonalisation Quadratic forms Examples from gravity and seismology Model parameterisation Model parameters Forward modelling Model Testing The misfit vector

2 Matrix diagonalisation Eigenvectors define a natural way to describe the space on which a matrix operates. Define the matrix V to have columns that are the eigenvectors of A. Then V T V = VV T = 1 (because any two eigenvectors are orthogonal). Thus V T = V -1 If is the diagonal matrix composed of the eigenvalues: AV = V Thus V T AV = V T V = or A = V V T If y = Ax, then y = V V T x or (V T y) = (V T x), Action of the matrix A on x is equivalent to that of diagonal matrix on the rotated coordinate system x' = V T x.

3 Quadratic forms The function x T Ax is a scalar quantity that has a quadratic dependence on the components of x, and is therefore referred to as a quadratic form. Since A = V V T x T Ax = x T V V T x = x T x In the rotated coordinate system x =V T x the quadratic form thus reduces to: x T x = i x i 2 Consider now the equation x T Ax = 1, or i x i 2 = 1 The solutions to this equation, considered as a set of points in N- dimensional space have a simple geometrical interpretation: If all of the i are positive, the resulting surface is an N- dimensional ellipsoid. The semi-axis in each component x i ' is of length i -1/2.

4 Examples from Gravity A geophysical model can be used to make predictions if it can be specified by a number of known parameters and a set of mathematical rules (the laws of physics) that depend only on the parameters and other externally known quantities. E.g. a gravity anomaly is caused by some hypothetical density anomaly. A simple model is hypothesised. Key parameters are, r, z. The gravity anomaly can be predicted from this model. gravity anomaly z r density z = 0 surface of Earth Exercise: what is the equation for g(, r, z)?

5 Gravity vs distance along profile N S Thousands x 0

6 Gravity profile across a fault g( x)=2g ρ t [ π 2 +tan 1 ( x x 0 h )] amplitude of gravity anomaly is: g g =2πGρt max slope of gravity anomaly is: 2 G ρt h the ratio of amplitude to maximum slope is = h

7 Raypaths are perturbed by the velocity anomalies. The relative position of fast and slow anomalies is offset for rays coming from different directions. Raypaths which traverse the slow (fast) anomaly arrive late (early). Seismic Tomography

8 Model Parameterization There are two aspects to defining a geophysical model: The model parameterization is the term used to describe the kind of model that we construct. e.g., a model of subsurface density, might be described using 2D polygonal structures of variable shape and size, each of which has a constant density. The model parameters then are the spatial coordinates of the polygonal bodies, and their densities. Alternatively, we might just divide the subsurface into regular rectangular blocks of fixed size and shape, each with its own constant density. The model parameters are then just the set of model densities. Alternatively, we might define a set of subsurface nodes at fixed separation, and represent the density as a smoothly varying polynomial surface that is constrained by a set of nodal density values (i.e. the parameters). A model parameterisation is unsatisfactory if there is no set of parameters that allows the data constraints to be predicted.

9 Model Parameters The second aspect of defining a geophysical model is simpler and more objective: Given a model parameterisation, provide a set of numerical values for the model parameters. Sometimes the set of model parameters m = {m i }, i = 1,N is referred to as the model. This is a convenient shorthand when most of the effort is spent in trying to determine what should be the correct values in m. Comparing different models in this context refers simply to the comparison of two different sets of {m i }. The best m is referred to as the solution model. If we accept a certain kind of model parameterisation then we can be very quantitative and objective about getting the best m. But if the best m is not good enough, then the model parameterisation must be redesigned. Some parameterisations are more robust than others; they are capable of adequately representing a greater range of conceivable realities, but all models are limited one way or another.

10 Forward Modelling In common use before Inverse Theory was developed and widely applied, forward modelling remains a useful preliminary to many inversion exercises. The principle is simple: Use a priori data, or your favourite geological preconceptions, to construct an initial model Compare the model predictions with the data tweak the model parameters, based on geophysical understanding, trial and error, or even a random number generator, and observe how the model changes. select the best model that you have managed to find. Forward modelling is often surprisingly successful in producing a model that satisfies the data, but its main disadvantage is lack of objectivity. There is no assurance that you have found the optimum model, and no clear indication of the range of acceptable models.

11 Model Testing If we have reduced the problem to one of determining the best set of model parameters ("the best model"), what is the criterion for a good model? or better, or best model? Is the model consistent with known data? What data are available? How reliable are the data? Are all the data equally reliable? What range of models is acceptable (within accuracy of the data)? In order to be useful a model must be testable somehow against independent data. Our criterion for goodness of the model is: how well do the model predictions agree with the data? Inverse Theory is all about defining a precise quantifiable method of selecting the best model, and (equally importantly) determining the range of acceptable models.

12 Is the Model Right? At one level, the answer to this is always: NO! The model is always an approximation to reality. The actual cause of a gravity anomaly will never be a perfect sphere. The question should be: Is the model good enough? The topic of inverse theory is all about how we decide the answer to this question. And decide it in an objective quantitative way. A model is like a theory in the sense that we can't ever prove that it is right. We can only disprove it. Science is based on the idea that we systematically and objectively test our scientific theories in an attempt to disprove them. Those that we cannot disprove, we accept (for the time being) as theories. Inverse theory starts from the same premise, that we test the model using objective measurements (data) in order to determine whether it is good enough. If the model doesn't explain the data adequately, then the model must be revised.

13 Misfit vector Suppose we have a set of observations (the data): d and we use our model to produce a set of matching predictions: p If the problem is linear we generally can write: p = Gm where m is the set of model parameters, and G is the gradient matrix. More generally, p = f(m) How well does m match to d? The simplest definition of how well a model satisfies the data is provided in terms of the misfit vector: (d - p) d p=[ d 1 d 2 d N ] G[ m1 m 2 m M ]