Forward modelling • The key to waveform tomography is the calculation of Green’s functions (the point source responses) • Wide range of modelling methods available • Very fast methods are limited (e.g., 1D, or no multiple scattering, or no turning waves, etc) • Very complete methods are prohibitively expensive (e.g., full 3D methods, with anisotropy, attenuation etc)
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Forward modelling The key to waveform tomography is the calculation of Green’s functions (the point source responses) Wide range of modelling methods available.
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Forward modelling• The key to waveform tomography is the calculation of Green’s functions (the point source responses)
• Wide range of modelling methods available
• Very fast methods are limited (e.g., 1D, or no multiple scattering, or no turning waves, etc)
• Very complete methods are prohibitively expensive (e.g., full 3D methods, with anisotropy, attenuation etc)
Our choice is 2D, isotropic, acoustic, two-way wave equation by frequency domain finite differences
Frequency domain finite differences
• You don’t always need all the frequencies for the inverse problem
• There are easy savings for multiple source problems
• You don’t always need a long time window
• In-elastic attenuation is easy to model
• Any dispersion law for attenuation / velocity is possible
Frequency domain finite differences
Return to the frequency domain acoustic wave equation, including an arbitrary source term:
Velocity is complex, attenuating, and dispersive:
Frequency domain finite differences
Reducing (for now) to one-dimension:
(imagine waves propagating on a string …)
On a 1D grid, the particle displacements are stored as a list of numbers, or vector:
Frequency domain finite differences
On a 1D grid, the particle displacements are stored as a list of numbers, or vector. The first space derivative is approximated by
Frequency domain finite differences
An alternative way of representing the differencing is operator is a differencing stencil
This generates the derivative operation at each point as we slide it over the grid, and multiply and sum the corresponding displacement values
Frequency domain finite differences
The second derivative differencing stencil looks like:
Frequency domain finite differences
The finite difference problem in the frequency domain:
• must satisfy the wave equation simultaneously at all grid points