Convolution

This example shows how to use the pylops.signalprocessing.Convolve1D and pylops.signalprocessing.Convolve2D operators to perform convolution between two signals.

Such operators can be used in the forward model of several common application in signal processing that require filtering of an input signal for the instrument response. Similarly, removing the effect of the instrument response from signal is equivalent to solving linear system of equations based on Convolve1D or Convolve2D operators. This problem is generally referred to as Deconvolution.

A very practical example of deconvolution can be found in the geophysical processing of seismic data where the effect of the source response (i.e., airgun or vibroseis) should be removed from the recorded signal to be able to better interpret the response of the subsurface.

import numpy as np
import matplotlib.pyplot as plt
import pylops

from pylops.utils.wavelets import ricker

plt.close('all')

We will start by creating a zero signal of lenght \(nt\) and we will place a unitary spike at its center. We also create our filter to be applied by means of pylops.signalprocessing.Convolve1D operator. Following the seismic example mentioned above, the filter is a Ricker wavelet with dominant frequency \(f_0 = 30 Hz\).

nt = 1001
dt = 0.004
t = np.arange(nt)*dt
x = np.zeros(nt)
x[int(nt/2)] = 1
h, th, hcenter = ricker(t[:101], f0=30)

Cop = pylops.signalprocessing.Convolve1D(nt, h=h, offset=hcenter, dtype='float32')
y = Cop*x

xinv = Cop / y

fig, ax = plt.subplots(1, 1, figsize=(10, 3))
ax.plot(t, x, 'k', lw=2, label=r'$x$')
ax.plot(t, y, 'r', lw=2, label=r'$y=Ax$')
ax.plot(t, xinv, '--g', lw=2, label=r'$x_{ext}$')
ax.set_title('Convolve in 1st direction', fontsize=14, fontweight='bold')
ax.legend()
ax.set_xlim(1.9, 2.1)

We show now that also a filter with mixed phase (i.e., not centered around zero) can be applied and inverted for using the pylops.signalprocessing.Convolve1D operator.

Cop = pylops.signalprocessing.Convolve1D(nt, h=h, offset=hcenter - 3, dtype='float32')
y = Cop*x
y1 = Cop.H*x
xinv = Cop / y

fig, ax = plt.subplots(1, 1, figsize=(10, 3))
ax.plot(t, x, 'k', lw=2, label=r'$x$')
ax.plot(t, y, 'r', lw=2, label=r'$y=Ax$')
ax.plot(t, y1, 'b', lw=2, label=r'$y=A^Hx$')
ax.plot(t, xinv, '--g', lw=2, label=r'$x_{ext}$')
ax.set_title('Convolve in 1st direction', fontsize=14, fontweight='bold')
ax.set_xlim(1.9, 2.1)
ax.legend()

Finally we repeat a similar exercise but using two dimensional signals and filters taking advantage of the pylops.signalprocessing.Convolve2D operator.

nt = 51
nx = 81
dt = 0.004
t = np.arange(nt)*dt
x = np.zeros((nt, nx))
x[int(nt/2), int(nx/2)] = 1

nh = [11, 5]
h = np.ones((nh[0], nh[1]))

Cop = pylops.signalprocessing.Convolve2D(nt * nx, h=h, offset=(int(nh[0]) / 2, int(nh[1]) / 2),
                                         dims=(nt, nx), dtype='float32')
y = Cop*x.flatten()
xinv = Cop / y

y = y.reshape(nt, nx)
xinv = xinv.reshape(nt, nx)

fig, axs = plt.subplots(1, 3, figsize=(10, 3))
fig.suptitle('Convolve in 1st direction of 2d data', fontsize=14,
             fontweight='bold', y=0.95)
axs[0].imshow(x, cmap='gray', vmin=-1, vmax=1)
axs[1].imshow(y, cmap='gray', vmin=-1, vmax=1)
axs[2].imshow(xinv, cmap='gray', vmin=-1, vmax=1)
axs[0].set_title('x')
axs[0].axis('tight')
axs[1].set_title('y')
axs[1].axis('tight')
axs[2].set_title('xlsqr')
axs[2].axis('tight')
plt.tight_layout()
plt.subplots_adjust(top=0.8)

fig, ax = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle('Convolve in 1st direction of 2d data', fontsize=14,
             fontweight='bold', y=0.95)
ax[0].plot(x[int(nt/2), :], 'k', lw=2, label=r'$x$')
ax[0].plot(y[int(nt/2), :], 'r', lw=2, label=r'$y=Ax$')
ax[0].plot(xinv[int(nt/2), :], '--g', lw=2, label=r'$x_{ext}$')
ax[1].plot(x[:, int(nx/2)], 'k', lw=2, label=r'$x$')
ax[1].plot(y[:, int(nx/2)], 'r', lw=2, label=r'$y=Ax$')
ax[1].plot(xinv[:, int(nx/2)], '--g', lw=2, label=r'$x_{ext}$')
ax[0].legend()
ax[0].set_xlim(30, 50)
ax[1].legend()
ax[1].set_xlim(10, 40)
plt.tight_layout()
plt.subplots_adjust(top=0.8)

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