Part 5: The Frequency Domain

In this exercise, you will implement solutions to the same problem, except in the frequency domain. Many of your existing functions will be very useful.

In the frequency domain, the (time-shifted) Ricker wavelet is defined as

\[\hat w(\nu) = \frac{2}{\sqrt{\pi}} \frac{\nu^2}{\nu_0^3}\exp\{-\frac{\nu^2}{\nu_0^2}\}\exp\{-i2\pi\nu t_0\}\]

where \(\nu\) is the frequency in Hz.

Problem 5.1

Implement a frequency version of your ricker_wavelet function. Plot the absolute value of \(\hat w(\nu)\) for a reasonable range of \(\nu\) and verify that the peak occurs at \(\pm \nu_0\).

Problem 5.2

Derive the Helmholtz (frequency domain) differential equation equivalent to the time domain wave equation from the first problem set. How do the mass, attenuation, and stiffness matrices relate to this equation?


Be careful of the Fourier transform convention. The Ricker wavelet above uses \(\hat f(\nu) = \int_0^T f(t)\exp\{-i2\pi\nu t\}\textrm{d}t\) as the convention. Then, \(f(t) = \int_{-\infty}^{\infty} \hat f(\nu)\exp\{i2\pi\nu t\}\textrm{d}\nu\) is the inverse transform.

Problem 5.3

Using your point_source and construct_matrices functions, write a function helmholtz_solver(C, source, nu, config) which solves the Helmholtz equation, that you developed in Problem 5.2, at a single frequency nu. Write a function forward_operator_frequency which behaves like the time-domain forward_operator function, except that it implements

\[\hat{\mathcal{F}[m]} = \hat u,\]


\[S\hat{\mathcal{F}[m]} = \hat d.\]

Your record_data function should also be useful.

Problem 5.4

Derive the equivalent adjoint state method for the frequency domain. What is the adjoint equation? What are the adjoint sources? Write a function adjoint_operator_frequency which implements the adjoint state method, using your helmholtz_solver,

\[\hat{F^*}\hat r_\text{ext} = \delta m.\]


Be very careful with your conjugates.

Problem 5.5

Derive the equivalent linear forward model, and using your helmholtz_solver function, implement a linear_forward_operator_frequency function for applying

\[\hat F\delta m = \hat u_1.\]

Problem 5.6

Verify that the adjoint relationship is satisfied for these operators. Again, be careful with your conjugates.

Problem 5.7

Implement gradient descent for the frequency domain. Why don’t you get useful answers using a single frequency (this is exceptional for 1D)?

What happens if you formulate the optimization such that you are using multiple frequencies at once? How do your results compare to those from the time domain?

Bonus Problems

Bonus Problem 5.8: Derive the frequecy domain version of the Ricker wavelet.

Bonus Problem 5.9: Derive how you can implement these same frequency domain operations using only time-domain solvers. Can you implement this? What extra ‘tweaks’ do you need to do to ensure that your time-based frequency operators pass the adjoint test?

Bonus Problem 5.10: Numerically verify that your time- and frequency-domain solvers satisfy the discrete Plancharel’s theorem.