# 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

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?

Note

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

and

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`

,

Hint

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

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.