Toward SystemLevel Earthquake Science

1
G(m)d mathematical model d data m
model G operator dG(mtrue)?
dtrue ? Forward problem find d given
m Inverse problem (discrete parameter
estimation) find
m given d Discrete linear inverse problem Gmd
2

Continuous inverse problem ?
g(s,x)m(x)dxd(s) g is the
kernel Convolution equation ?
g(s-x)m(x)dxd(s)
b a
b a
3

Example linear regression for ballistic
trajectory y(t)m1m2t-0.5m3t2 m1
initial altitude, m2 initial vertical velocity,
m3 effective gravitational acceleration
o

• 1 t1
  -0.5t12 m1 y1
• 1 t2 -0.5t22
  m2 y2
  
• 1 t3 -0.5t32
  m3 y3
• ..
  .
• 1 tm -0.5tm2
  ym

Y(t)
o
o
o
t
4
Earthquake location
mx ? G(m)t tiS.,i-x2/c? (arrival time
of wave at station i) Nonlinear problem!

????? ?????????????????????????????
5
j
Traveltime tomography
T ? 1/v(s)ds ?u(s)ds Tj ? Gij ui
j-th ray
i1
6
Gravity
d(s)
h
??(x)
8 -8
8 -8
d(s) G ? h m(x) dx / (x-s)2h23/2 ?
g(x-s) m(x) dx
d(s)
h(x)
??
8 -8
d(s) G ? m(x) ?? dx / (x-s)2m(x)23/2
nonlinear in m(x)
7
Existence maybe no model that fits data (bad
model, noisy data)Uniqueness maybe several
(infinite?) number of models that fit
dataInstability small change in data leading
to large change in estimateAnalysisWhat
are the data?Discrete or continuous
data?Sources of noise?What is the mathematical
model?Discrete or continuous model?What
physical laws determine G?Is G linear or
nonlinear?Any issues of existence, uniqueness,
or instability?