Basic Problems - 1

Forward Modeling & Backward Modeling

Forward Modeling： Source, Model parameters; mathematical model, observed data

Backford Modeling：Observed data, mathematical model, source, model parameters

inversion is based on forward modeling. is it a simplest way to implement forward modeling???

Green Function

\[Lu=0/-f(x,t)\]

L is 2-order diffferential oprator.

In gravity/magnetic method: $L=\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$

Electromagnetic: $L=\Delta+k^2$, $k=\omega^2\mu\epsilon/c^2+i4\pi\omega\mu r/c^2$

Seismic: $L=\Delta-\frac{1}{v^2}\frac{\partial^2}{\partial t^2}$

The physical meaning of Green function

source function & basic solution
if BC & Initial Condition exists, Green funtion is related to those conditions.
use Green function to solve any field simulated by a point source.

\[u(x)=\iiint_\Omega G(x,x_0)f(x_0)d\Omega+\iint_{\partial \Omega}G(x,x_0)\frac{\partial u(x_0)}{\partial n}ds-\\\iint_{\partial\Omega}u(x_0)\frac{\partial G(x,x_0)}{\partial n}ds\]

How to use Green function to solve a specific problem?

(1) find a proper integration(a specific form) to represent as the solution of the problem.

(2) get the solution of G and $\frac{\partial G}{\partial n}$

(3) put G and $\frac{\partial G}{\partial n}$ into the integration. Then we’ll get the solution.

Numerical Differentiation

Taylor expansion

it’s simple:

\[f(x+h)=f(x)+f'(x)h+\frac{f''(x)}{2}h^2+...\]

Differential

Forward Differential: $f'(x)\approx\frac{f(x+h)-f(x)}{h}+o(h)$

Backward Differential: $f'(x)\approx\frac{f(x)-f(x-h)}{h}+o(h)$

Central Differential: $f'(x)\approx\frac{f(x+h)-f(x-h)}{2h}+o(h^2)$

Explicit: $u_{j+1}=u_j-f(u_j,t_j)\Delta t$

Implicit & Crank-Nocolson form

Numerical Integration

\[I=\int_a^bf(x)dx\\ I\approx\sum_{i=0}^n\Delta xf(x_i)\]

Midpoint formula:

\[I\approx\int_{i=0}^{n-1}\Delta xf(\frac{x_i+x_{x+1}}2)\]

Simpson Formula

\[I\approx \sum_{i=0}^{n-1}\frac{\Delta x}{6}[f(x_i)+4f(\frac{x_i+x_{x+1}}2)+f(x_{i+1})]\]

Inversion of Matrix && Singular value

small sigular value will cause problems, we need to handle it !

\[1/\lambda\to\frac{\lambda}{\lambda^2+\alpha}\]

Norm

1-Norm

2-norm(it’s important)

文档信息

本文作者：Shenyao Jin
本文链接：https://shenyaojin.github.io/2022/06/08/Basic-Problems/
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Basic Problem in computational geophysics