G(m)=d mathematical model d data m model G operator d=G(m true )+  = d true +  Forward problem: find d given m Inverse problem (discrete parameter estimation): find m given d Discrete linear inverse problem: Gm=d

Continuous inverse problem:  g(s,x)m(x)dx=d(s) g is the kernel Convolution equation:  g(s-x)m(x)dx=d(s) baba baba

Example: linear regression for ballistic trajectory y(t)=m 1 +m 2 t-0.5m 3 t 2 m 1 initial altitude, m 2 initial vertical velocity, m 3 effective gravitational acceleration Y(t) t o o o o  1 t t 1 2 m 1 y 1  1 t t 2 2 m 2 y 2  1 t t 3 2 m 3 = y 3  …...  1 t m -0.5t m 2 y m

m=[x  ] G(m)=t t i =||S., i -x|| 2 /c+  (arrival time of wave at station i) Nonlinear problem! Earthquake location  

T = ∫ 1/v(s)ds = ∫u(s)ds T j = ∑ G ij u i i=1 Traveltime tomography j j-th ray

 (x) Gravity h d(s) d(s) =G  h m(x) dx / [(x-s) 2 +h 2 ] 3/2 =  g(x-s) m(x) dx ∞ -∞ ∞ -∞ h(x) d(s) d(s) = G  m(x)  dx / [(x-s) 2 +m(x) 2 ] 3/2 nonlinear in m(x) ∞ -∞ 

Existence: maybe no model that fits data (bad model, noisy data) Uniqueness: maybe several (infinite?) number of models that fit data Instability: small change in data leading to large change in estimate Analysis: What 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?