How to find 2d Spatial Spectrum from cross correlation?

enter image description here

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)
I want to generate the figure as Fig. 01 below Fig.01 However, by using improved code I get the figure with higher resolution as Fig. 02 which is different from Fig. 01. Fig.02

I've added another two figures to compare with the Fig. 01. When consider the range [-k, k], the plot looks like Fig. 03 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. Fig. 04

I've put the update Figure as follows: Fig. 05 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.


Solution 1:

It looks to me like you're zooming in on the central lobe. This would also explain why the scale doesn't go from 0 to 1.

If I change these likes:

kx = np.linspace(-20*k,20*k,nx)  # space vector
ky=  np.linspace(-20*k,20*k,ny)   # space vector

then I get

My version of the picture

which looks closer to what you're looking for.

To improve the resolution, I've done a bit of rewriting to get this new picture. See updated code below.

NOTE: I am still not certain this is doing the right thing.

Higher resolution version of image


Code I used

# Code from https://stackoverflow.com/questions/70768384/right-method-for-finding-2-d-spatial-spectrum-from-cross-spectral-densities

import numpy as np
from scipy import signal
import matplotlib.pyplot as plt
import cmath

# Set up data
# 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]

if (len(x) != len(y)):
    raise Exception('X and Y lengthd differ')

n = len(x)
dx = np.array(x);  M = len(dx)
dy = np.array(y) ; N = len(dy)

np.random.seed(12345)
raw_data=np.reshape(np.random.rand(10000*n),(10000,n))

f = 10  # frequency in Hz
c = 50  # wave speed 50, 80, 100, 200  m/s
k = 2.0*np.pi*f/c  # wavenumber
kx = np.linspace(-20*k,20*k,n*10)  # space vector
ky=  np.linspace(-20*k,20*k,n*10)   # space vector


# 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)
            #real_csd = np.real(Pxy)
            total_csd.append(Pxy)                     # output as list
            
    return np.array(total_csd)

## Spatial Spectra:- DFT of the csd along two dimension

def DFT2D(data):
    #data = np.asarray(data)
    dft2d = np.zeros((len(kx),len(ky)), 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


# 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, n))

dft = DFT2D(fcsd)  # Data or cross-correlation matrix
spec = np.abs(dft) #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")