Seismic tomography and inverse problems Flashcards
seismic interferometry
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
in cross-correlation: Green’s function, and time-reversed Green’s function
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
convolution
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
dispersal of seismic waves - reasons and results
dispersion curve comes from outcome of cross-correlation. different frequencies propagate at different speeds. faster velocity for longer period. fundamental mode
signal processing steps (4) to extract signal from ambient noise
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.
what generates ambient seismic noise (3)
signals from plate boundaries
ocean microseisms (7sec or 10-20sec periods)
wind-driven infragravity waves (50-300sec period)
examples of uses of model parameterisations
Goce gravity field
Antarctic basal friction (adaptive mesh)
climate models (parameterise the area of the earth’s surface, or the ocean etc)
consequences of parameterisation
changes computing time, changes accuracy or stability, can introduce bias, may introduce solution non-uniqueness
parameterisation: linear interpolation
interpolate data values between two points, eg mesh values
can be used in both regular and irregular parameterisation
parameterisation: pseudo-linear interpolation
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
parameterisation: splines
splines are piecewise polynomial functions designed to represent continuous curves. controlled by a set of points (nodes)
regular parameterisation: local vs global basis functions, advs and disadvs
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
irregular parameterisation: quadtrees and octrees, uses
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
unstructured parameterisations, two examples
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
static vs dynamic parameterisation
in a dynamic parameterisation, number or characteristics of parameters can vary in response to the model
top-down vs bottom-up parameterisation, irregular or regular, and which is used most?
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.
solving the forward problem: how to reduce computational cost
use high frequency approximation - although this gives less information
Eikonal equation
for wavefronts - a PDE used to describe propagation of waves, accounts for reflection and refraction etc
Runge-Kutta solver
a solver for ray tracing - numerical solver for calculating the propagation of seismic waves through earth
formulating the inverse problem: linearity, iterations
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
inverse problem: why can G not be inverted, what to do instead
G is not square, cannot be inverted
find equation for delta (m), includes the generalised inverse - this is square and invertible
inverse problem: how to solve solution non-uniqueness
regularisation solves solution non-uniqueness
damping - reduces large perturbations (?)
smoothing - favours smoother solutions
name 2 methods for solving large systems of linear equations
Jacobi’s method, Gauss-Seidel method
Jacobi’s method for solving large systems of simultaneous equations
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
Gauss Seidel method for solving large systems of linear equations
how it compares to Jacobi’s method
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
solving large systems of linear equations: relaxation methods
relaxation methods offer the possibility of faster convergence
depend on parameter omega (equations in notes)
