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

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

MatLab Scripts for all calculations in these lectures www.ldeo.columbia.edu/users/menkecourses.html

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

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

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

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

example: river flow and rain

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

example: CO2 and combustion of fossil fuels

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, …)

example: gravity anomaly and the earth’s density

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

example: straight line relationship d x

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

data near origin possibly good control on intercept but lousy control on slope small big -5 0 5

data far from origin lousy control on intercept but possibly good control on slope big 0 50 100 small

Set up for standard Least Squares yi = a + b xi y1 1 x1 a y2 = 1 x2 b … … … yN 1 xN d = G m

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

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?

Least-Squares Least Absolute Value