Shift

This example shows how to use the pylops.signalprocessing.Shift operator to apply fractional delay to an input signal. Whilst this operator acts on 1D signals it can also be applied on any multi-dimensional signal on a specific direction of it.

import matplotlib.pyplot as plt
import numpy as np

import pylops

plt.close("all")

Let’s start with a 1D example. Define the input parameters: number of samples of input signal (nt), sampling step (dt) as well as the input signal which will be equal to a ricker wavelet:

nt = 127
dt = 0.004
t = np.arange(nt) * dt
ntwav = 41

wav = pylops.utils.wavelets.ricker(t[:ntwav], f0=20)[0]
wav = np.pad(wav, [0, nt - len(wav)])
WAV = np.fft.rfft(wav, n=nt)

We can shift this wavelet by \(5.5*dt\):

shift = 5.5 * dt
Op = pylops.signalprocessing.Shift(nt, shift, sampling=dt, real=True, dtype=np.float64)
wavshift = Op * wav
wavshiftback = Op.H * wavshift

plt.figure(figsize=(10, 3))
plt.plot(t, wav, "k", lw=2, label="Original")
plt.plot(t, wavshift, "r", lw=2, label="Shifted")
plt.plot(t, wavshiftback, "--b", lw=2, label="Adjoint")
plt.axvline(t[ntwav - 1], color="k")
plt.axvline(t[ntwav - 1] + shift, color="r")
plt.xlim(0, 0.3)
plt.legend()
plt.title("1D Shift")
plt.tight_layout()

We can repeat the same exercise for a 2D signal and perform the shift along the first and second dimensions.

shift = 10.5 * dt

# 1st dir
wav2d = np.outer(wav, np.ones(10))
Op = pylops.signalprocessing.Shift(
    (nt, 10), shift, dir=0, sampling=dt, real=True, dtype=np.float64
)
wav2dshift = (Op * wav2d.ravel()).reshape(nt, 10)
wav2dshiftback = (Op.H * wav2dshift.ravel()).reshape(nt, 10)

fig, axs = plt.subplots(1, 3, figsize=(10, 3))
axs[0].imshow(wav2d, cmap="gray")
axs[0].axis("tight")
axs[0].set_title("Original")
axs[1].imshow(wav2dshift, cmap="gray")
axs[1].set_title("Shifted")
axs[1].axis("tight")
axs[2].imshow(wav2dshiftback, cmap="gray")
axs[2].set_title("Adjoint")
axs[2].axis("tight")
fig.tight_layout()

# 2nd dir
wav2d = np.outer(wav, np.ones(10)).T
Op = pylops.signalprocessing.Shift(
    (10, nt), shift, dir=1, sampling=dt, real=True, dtype=np.float64
)
wav2dshift = (Op * wav2d.ravel()).reshape(10, nt)
wav2dshiftback = (Op.H * wav2dshift.ravel()).reshape(10, nt)

fig, axs = plt.subplots(1, 3, figsize=(10, 3))
axs[0].imshow(wav2d, cmap="gray")
axs[0].axis("tight")
axs[0].set_title("Original")
axs[1].imshow(wav2dshift, cmap="gray")
axs[1].set_title("Shifted")
axs[1].axis("tight")
axs[2].imshow(wav2dshiftback, cmap="gray")
axs[2].set_title("Adjoint")
axs[2].axis("tight")
fig.tight_layout()

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