Geology 5670/6670Inverse Theory

26 Jan 2015

© A.R. Lowry 2015Read for Wed 28 Jan: Menke Ch 4 (69-88)

Last time: Ordinary Least Squares ( Statistics)

• The expected value of is:

• Based on this, if (N – M) is “large” we can estimate unknown data variance as:

• If data variance is known a priori, we can calculate the chi-squared parameter of fit as:

(so called because a sum of squared r.v.’s follows a

distribution). This can be used to evaluate fit and adjust parameterization…

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eT

e = Emin

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Emin = N − M( )σ2

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˜ σ 2 =Emin

N − M

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2 =

ei2

i=1

N

∑σ 2 N − M( )

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min2 =1

The 2 parameter is commonly used to evaluate data fit & optimize the choice of number of parameters: 1) If , can safely add more model parameters 2) If , too many parameters (model is fitting noise).

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min2 >1

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min2 <1

Solution appraisal: What is the range of solutions?

Assume: zero-mean, Gaussian, uncorrelated errors

Estimate: Confidence intervals expressed as %: 100(1–)% (i.e., is probability the true value falls in the conf interval).

Case 1: Data error variance is known (= 2)

-z +z1- /2/2€

mi = ˜ m i ± zσ mi

Desired confidence interval is ±z of the normal (z) distribution function

Can get this from standard statistical tables or codes

Suppose we want the 95% confidence interval:Typically we use the F-distribution for F = 1 – /2

95% conf 1 – = 0.95

1 – /2 = 0.975

Looking up on a standard table, find F(z) = 0.975 when z = 1.96 (i.e. not quite 2).

Case 2: Use estimated error variance from

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˜ σ 2 =Emin

N − M

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mi = ˜ m i ± t1−α

2

N−M σ mi

(Look up the t-distribution as you would z-distribution in math probability tables, or use corresponding routines in Matlab or other stat codes).

For a multi-parameter linear model, in reality we have confidence regions: hyperellipsoids in a multidimensional model parameter space

m1

m2

Emin

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˜ m 1€

˜ m 2 QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

E.g. Mars…

Another example:Gravity modelingfrom the Lowry &Pérez-Gussinyéapproach to jointinversion of crustalthickness & VP/VS.

Here, confidenceintervals were estimated byvarying H & K atone seismic site(a nonlinearproblem, so notperfectly elliptical).

Note that some nonlinearproblems can have rather pathological misfit error functions…Especially if sampling issub-optimal.

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m − ˜ m ( )T

GT

G m − ˜ m ( ) ≤ Mσ 2F1- α-1 M ,∞( )

Confidence on Linear sol’ns…To estimate confidence regions from contours of (Gm)TGm:

Example 1: Given known 2, the confidence region is defined by

where is the inverse F-distribution with M, DOF Example: to get 95% confidence for M = 10,

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F1- -1 M ,∞( )

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F 0.95-1 10,∞( ) =1.83⇒ m − ˜ m ( )

TGTG m − ˜ m ( ) ≤18.3σ 2