Seismic tomography and inverse problems

Question 1

seismic interferometry

Answer 1

generating virtual sources by cross-correlating observations from pairs of receivers

also called ‘Green’s function retrieval’

provides a virtual signal and produces a virtual impulse. if a signal arrives at A then B, cross correlating provides the signal as if it was emitted from A and arrived at B

Question 2

in cross-correlation: Green’s function, and time-reversed Green’s function

Answer 2

Green’s function: impulse response, function of earth properties etc

cross correlation of two Green’s functions gives the Green’s function of the virtual signal

time-reversed: two simultaneous sources at two possible virtual source positions

Question 3

convolution

Answer 3

convolution of two time series is combining two signals to produce a third signal, by time-reversing one and them and shifting it across the other

Question 4

dispersal of seismic waves - reasons and results

Answer 4

dispersion curve comes from outcome of cross-correlation. different frequencies propagate at different speeds. faster velocity for longer period. fundamental mode

Question 5

signal processing steps (4) to extract signal from ambient noise

Answer 5

1) cross-correlation, to get similarity as a function of time difference

2) stacking, to reduce noise and amplify signal

3) Fourier transform, to get from time domain to frequency domain, use FFT since discrete FT has high computing cost

4) filtering, can be done in frequency domain. multiply by bandpass filter, then transform back to time domain

also tapering to reduce edge effects.

Question 6

what generates ambient seismic noise (3)

Answer 6

signals from plate boundaries

ocean microseisms (7sec or 10-20sec periods)

wind-driven infragravity waves (50-300sec period)

Question 7

examples of uses of model parameterisations

Answer 7

Goce gravity field

Antarctic basal friction (adaptive mesh)

climate models (parameterise the area of the earth’s surface, or the ocean etc)

Question 8

consequences of parameterisation

Answer 8

changes computing time, changes accuracy or stability, can introduce bias, may introduce solution non-uniqueness

Question 9

parameterisation: linear interpolation

Answer 9

interpolate data values between two points, eg mesh values

can be used in both regular and irregular parameterisation

Question 10

parameterisation: pseudo-linear interpolation

Answer 10

not just a straight line drawn between points, but instead takes into account other information

may be a polynomial shape instead of straight line, may weight certain data points more, etc

Question 11

parameterisation: splines

Answer 11

splines are piecewise polynomial functions designed to represent continuous curves. controlled by a set of points (nodes)

Question 12

regular parameterisation: local vs global basis functions, advs and disadvs

Answer 12

model is defined by linear sum of basis functions, each with a weighting coefficient

local basis function: change in function (phi) affects small region of the model

global basis function: change in phi has an affect throughout the whole model, eg spherical harmonic - eg GOCE gravity field. naturally smooth, but can introduce unreal structure in regions of no data constraint

Question 13

irregular parameterisation: quadtrees and octrees, uses

Answer 13

quadtrees: each cell split into 4 (2D) or 8 (3D). used for modelling mantle convection processes

hierarchies of resolution depending on model features and detail.

danger of edge-effects

Question 14

unstructured parameterisations, two examples

Answer 14

Eg triangular mesh. Can use data distribution to construct a mesh

Delauney triangulation: avoid small-angle triangles

Voronoi cells: regions have a max distance to a particular node, eg grains in synthetic rock

Question 15

static vs dynamic parameterisation

Answer 15

in a dynamic parameterisation, number or characteristics of parameters can vary in response to the model

Question 16

top-down vs bottom-up parameterisation, irregular or regular, and which is used most?

Answer 16

irregular

top-down: larger cells divided up as needed, according to data density. most normally used

bottom-up: smaller cells are joined together, mostly to avoid over-parameterisation.

Question 17

solving the forward problem: how to reduce computational cost

Answer 17

use high frequency approximation - although this gives less information

Question 18

Eikonal equation

Answer 18

for wavefronts - a PDE used to describe propagation of waves, accounts for reflection and refraction etc

Question 19

Runge-Kutta solver

Answer 19

a solver for ray tracing - numerical solver for calculating the propagation of seismic waves through earth

Question 20

formulating the inverse problem: linearity, iterations

Answer 20

non-linear, which is not ideal for large datasets

assume weakly non-linear (locally linear)

then do multiple iterations (Newton-Raphson analogy) to minimise misfit - a linear regression approach

Question 21

inverse problem: why can G not be inverted, what to do instead

Answer 21

G is not square, cannot be inverted

find equation for delta (m), includes the generalised inverse - this is square and invertible

Question 22

inverse problem: how to solve solution non-uniqueness

Answer 22

regularisation solves solution non-uniqueness

damping - reduces large perturbations (?)

smoothing - favours smoother solutions

Question 23

name 2 methods for solving large systems of linear equations

Answer 23

Jacobi’s method, Gauss-Seidel method

Question 24

Jacobi’s method for solving large systems of simultaneous equations

Answer 24

Ax = b

assume matrix A has no zeros on its diagonal, turn it into a diagonal matrix + remainder (A must be diagonally dominant)

A = D + R

equation for iterative solution, guaranteed to converge if A is diagonally dominant

Question 25

Gauss Seidel method for solving large systems of linear equations

how it compares to Jacobi’s method

Answer 25

Ax = b

split matrix A into upper and lower triangles, get an iterative process
A = L + U

different to Jacobi’s method in that new values for x are used as soon as they become available. a better iterative method, requires fewer iterations, saves on memory

Question 26

solving large systems of linear equations: relaxation methods

Answer 26

relaxation methods offer the possibility of faster convergence

depend on parameter omega (equations in notes)