geology 5670/6670 inverse theory 26 jan 2015 © a.r. lowry 2015 read for wed 28 jan: menke ch 4...
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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( )
€
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
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