1. The Basics of Inversion
Short Course
The Basics of Inversion
Bill Menke
Columbia University

2. The power of simple linear models
Lectures
The power of simple linear models
Probability and what it has to do with data analysis
3. Inferences using Least-Squares
4. Examples

3. MatLab Scripts for all calculations in these lectures

4. The power of simple, linear models
Lecture 1
The power of simple, linear models

5. data, d. … what you measure model parameter, m
data, d … what you measure model parameter, m … what you want to know conceptual model … links the two

6. At the start of a project, spend a few moments identifying. data
At the start of a project, spend a few moments identifying data model parameters conceptual model think about the strengths and weaknesses of each

7. Example – seismic tomography of the earth’s mantle
Example – seismic tomography of the earth’s mantle data: traveltimes of shear waves but the seismometer measured wiggles … model parameter: shear velocity in mantle but I really wanted to know temperature shear velocity is just a proxy for temperature conceptual model ray theory, an approximation to how vibrations travel through the earth

8. example: river flow and rain

9. watershed with rain and transport
data: discharge on a succession of days, d1, d2, d3, … model parameters: rain on a succession of days m1, m2, m3, …
conceptual model:
watershed with rain and transport
di = fcn( mi, mi-1, mi-2, …)
Causal: discharge depends only on present and past rain

10. example: CO2 and combustion of fossil fuels

11. data: CO2 on a succession of days,
data: CO2 on a succession of days, d1, d2, d3, … model parameters: combustion rate on a succession of days m1, m2, m3, …
conceptual model: global carbon cycle (transport, storage in biosphere and oceans, etc)
di = fcn( mi, mi-1, mi-2, …)

12. example: gravity anomaly and the earth’s density

13. data: strength of gravity on a 2D grid of points on the earth’s surface d1, d2, d3, … model parameters: density of the earth on a 3D grid of points in the earth’s interior m1, m2, m3, …
conceptual model: Newton’s third law: mass causes gravitational attraction
di = fcn(mi+1, mi, mi-1, …)
not causal

14. example: straight line relationshipd x

15. data: d1, d2, d3, … model parameters: the slope and intercept of the line m1, m2
conceptual model: data are on a line:
di = m1 + m2 xi
or if you prefer
di = a + b xi

16. In all these examples the data are linearly related to the model parameters (at least to first approximation)

17. Data: d= model parameters: m= Linear model: d = Gm d1 d2 … dN m1 m2 …
“data kernel” matrix

18. Straight line relationship
d = G m 1 x1 x2 … xN d1 d2 … dN m1 m2 =
Note that the data kernel matrix embodies the “geometry” of the measurements, that is, the x’s at which the d’s were measured

19. d: gravity anomaly in vertical direction
Gravity anomalies
d: gravity anomaly in vertical direction
m: density anomaly of a small cube of volume Dv di xi mi
qij yi
gravitational constant
Newton’s Inverse-square Law:
di = Scubes g Dv mj cosqij/ |xi - yj|2

20. Gravity anomalies d = G m Newton’s law: zij = g Dv cosqij / |xi - yj|2…
Z1M
Z21
Z22
Z2M
ZN1
ZN2
ZNM m1 m2 … mM d1 d2 … dN =
vertical component of gravity anomaly measured at position xi
density anomaly
of small cube of volume Dv located at position yi
once again, G embodies “geometry”
Newton’s law: zij = g Dv cosqij / |xi - yj|2

21. Thinking About Error error = observed data – predicted data e = dobs – dpre = dobs – Gmest always plot your data and look at the error!

22. Guess values for a, b ypre = aguess + bguessx
Prediction error =
observed minus predicted
e = dobs - dpre
Total error: sum of squared predictions errors
E = Σ ei2
= eT e
aguess=2.0
bguess=2.4

23. Systematically examine combinations of (a, b) on a 101101 grid
Minimum total
error E is here
Note E is not zero
apre
Error Surface
bpre

24. Error Surface Note Emin is not zero Here are best-fitting a, b
best-fitting line
Error Surface

25. Note some range of values where the error is about the same as the minimun value, Emin
Emin is here
Error pretty close to Emin everywhere in here
All a’s in this range and b’s in this range have pretty much the same error
Error Surface

26. conclusion the shape of the error surface controls the accuracy by which (a,b) can be estimated

27. What controls the shape of the error surface
What controls the shape of the error surface? Let’s examine effect of increasing the error in the data

28. The minimum error increases, but the shape of the error surface is pretty much the same
Error in data = 0.5
Emin = 0.20 -5 5
Error in data = 5.0
Emin = 23.5 -5 5

29. What controls the shape of the error surface
What controls the shape of the error surface? Let’s examine effect of shifting the x-position of the data

30. Big change by simply shifting x-values of the data
Region of low error is now tilted
(High b, low a) has low error
(Low b, high a) has low error
But (high b, high a) and (low a, low b) have high error 10 5

31. Meaning of tilted region of low error error in (apre, bpre) are correlated

32. Uncorrelated estimates of intercept and slope
Best-fit line
Best-fit line
Best fit intercept
erroneous intercept
When the data straddle the origin, if you tweak the intercept up, you can’t compensate by changing the slope

33. Negatively correlation of intercept and slope
Same slope s Best-fit line
Best-fit line
Low slope line
erroneous intercept
Best fit intercept
When the data are all to the right of the origin, if you tweak the intercept up, you must lower the slope to compensate

34. Positive correlation of intercept and slope
erroneous intercept
Best fit intercept
Best fit intercept
Same slope as best-fit line
Best-fit line
When the data are all to the left of the origin, if you tweak the intercept up, you must raise the slope to compensate

35. data near origin possibly good control on intercept but lousy control on slope
small
big

36. data far from origin lousy control on intercept but possibly good control on slope
big
small

37. Set up for standard Least Squares
yi = a + b xi
y x a
y2 = x b
… … …
yN xN
d = G m

38. The formula for the least-squares solution for the general linear problem is known:
mest = [GTG]-1 GT d
derived by a standard minimization procedure using calculus
Find the m that minimizes E(m) with E=eTe and e=d-Gm

39. Why least-squares. E=Si ei2 Why not least-absolute length
Why least-squares? E=Si ei2 Why not least-absolute length? E=Si |ei| Or something else?

40. Least-Squares Least Absolute Value