I try to implement the spatial spectrum from the above equation (attached)
Where kX, kY are the grid points in k space, C(w,r) - cross spectral densities between the i'th and j'th sensor(here it is a matrix of size ns * ns >no. of sensors). x, y are distances between the sensors. (nk - grid density for kx, ky)
I look for suitable python implementation of the above equation. I've 34 sensors which generates data of size [row*column]=[n*34]. At first, I've found the cross spectral densities (CSD) of among the data of each sensor. Then 2D DFT is performed of the CSD values to get the spatial spectrum.
*) I'm not sure whether the procedure is correct or not. **) Does the python implementation procedure correct? ***) Also, if someone provides some relevant tutorial/link, it will also be helpful for me.
import numpy as np
from scipy import signal
import matplotlib.pyplot as plt
import cmath
# Finding cross spectral density (CSD)
fs=500
def csdMat(data):
rows, cols = data.shape
total_csd = []
for i in range(cols):
for j in range(cols):
f, Pxy = signal.csd(data[:,i], data[:,j], fs, nperseg=512)
abs_csd = np.abs(Pxy)
total_csd.append(abs_csd) # output as list
csd_mat = np.array(total_csd)
return csd_mat
## Spatial Spectra:- DFT of the csd along two dimension
def DFT2D(data):
#data = np.asarray(data)
dft2d = np.zeros((M,N), dtype=complex)
for k in range(len(kx)):
for l in range(len(ky)):
sum_matrix = 0.0
for m in range(M):
for n in range(N):
e = cmath.exp(- 1j * ((kx[k] * dx[m]) / len(dx) + (ky[l] * dy[n]) / len(dy)))
sum_matrix += data[m,n] * e
dft2d[k,l] = sum_matrix
return dft2d
raw_data=np.reshape(np.random.rand(10000*34),(10000,34))
# Call the seismic array
#** Open .NPY files as an array
#with open('res_array_1000f_131310.npy', 'rb') as f:
# arr= np.load(f)
#raw_data = arr[0:10000, :]
#CSD of the seismic data
csd = csdMat(raw_data)
print('Shape of CSD data', csd.shape)
# CSD data of a specific frequency
csd_dat=csd[:, 11]
fcsd = np.reshape(csd_dat, (-1, 34))
fcsd.shape
n = 34
f = 10 # frequency in Hz
c = 50 # wave speed 50, 80, 100, 200 m/s
k = 2.0*np.pi*f/c # wavenumber
nx = n # grid density
ny = n
kx = np.linspace(-k,k,nx) # space vector
ky= np.linspace(-k,k,ny) # space vector
# Distance[Meter] between sensors
x = [2.1,2.1,-0.7,-2.1,-2.1,-0.7,-0.7,0.6,-5.7,-8.5,-11.4,-7.7,-6.3,-3.5,-2.1,-3.4,5.4,-5.2,-8.9,-10,-10,5.4,5.4,-0.8,-3.6,-6.2,-6.8,-12.2,-17.1,-19,-18.6,-13.5,14.8,14.8]
y = [6.65,4.15,3.65,5.05,7.25,8.95,11.85,8.95,-2,-0.6,-0.9,1.25,2.9,0.9,-0.1,-1.4,9.2,5.2,4.8,6.1,8.9,13.3,17.1,17.9,13.8,-9.3,-5.2,-3.6,-3.6,-0.9,3.7,3.7,-1.8,5.7]
dx = np.array(x); M = len(dx)
dy = np.array(y) ; N = len(dy)
X,Y = np.meshgrid(kx, ky)
dft = DFT2D(fcsd) # Data or cross-correlation matrix
spec = dft.real # Spectrum or 2D_DFT of data[real part]
spec = spec/spec.max()
plt.figure()
c = plt.imshow(spec, cmap ='seismic', vmin = spec.min(), vmax = spec.max(),
extent =[kx.min(), kx.max(), ky.min(), ky.max()],
interpolation ='nearest', origin ='lower')
plt.colorbar(c)
plt.rcParams.update({'font.size': 18})
plt.xlabel("Wavenumber, $K_x$ [rad/m]", fontsize=18)
plt.ylabel("Wavenumber,$K_y$ [rad/m]", fontsize=18)
plt.title(f'Spatial Spectrum @10Hz', weight="bold")
#c = Wave Speed; 50, 80,100,200
cc = 2*np.pi*f /c *np.cos(np.linspace(0, 2*np.pi, 34))
cs = 2*np.pi*f /c *np.sin(np.linspace(0, 2*np.pi, 34))
plt.plot(cc,cs)
However, by using improved code I get the figure with higher resolution as Fig. 02 which is different from Fig. 01.
I've added another two figures to compare with the Fig. 01. When consider the range [-k, k], the plot looks like Fig. 03
which is analogous [w.r.t. XY-axis] to Fig. 01, I think this figure is OK except some K-space missed. I hope here exist an issue that need to be fixed.
In Fig. 04, we consider the k-space range [-20k, 20k], which looks good but don't have similar axis as of Fig. 01.
I've put the update Figure as follows:
Can anyone help me to generate the figure 01 or similar type? I'm confused about the Figure 02. Can anybody help to make me understand? Thanks in advance.
