def correlation_random(rec0=0, rec1=1, verbose=0, plot=0, save=0):
"""
cct,cct_proc,t,ccf,ccf_proc,f = correlation_random(rec0=0,rec1=1,verbose=0,plot=0,save=1)
Compute and plot correlation function based on random source summation.
INPUT:
------
rec0, rec1: indeces of the receivers used in the correlation.
plot: plot when 1.
verbose: give screen output when 1.
save: store individual correlations to OUTPUT/correlations_individual
OUTPUT:
-------
cct, t: Time-domain correlation function and time axis [N^2 s / m^4],[s].
ccf, f: Frequency-domain correlation function and frequency axis [N^2 s^2 / m^4],[1/s].
Last updated: 19 May 2016.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
#- Start time.
t1 = time.time()
#- Input parameters.
p = parameters.Parameters()
#- Spatial grid.
x_line = np.arange(p.xmin, p.xmax, p.dx)
y_line = np.arange(p.ymin, p.ymax, p.dy)
x, y = np.meshgrid(x_line, y_line)
#- Compute number of samples as power of 2.
n = 2.0**np.round(np.log2((p.Twindow) / p.dt))
#- Frequency axis.
df = 1.0 / (n * p.dt)
f = np.arange(0.0, 1.0 / p.dt, df)
omega = 2.0 * np.pi * f
#- Compute time axis
t = np.arange(-0.5 * n * p.dt, 0.5 * n * p.dt, p.dt)
#- Compute instrument response and natural source spectrum.
S, indeces = s.space_distribution()
instrument, natural = s.frequency_distribution(f)
#- Issue some information if wanted.
if verbose == 1:
print 'number of samples: ' + str(n)
print 'maximum time: ' + str(np.max(t)) + ' s'
print 'maximum frequency: ' + str(np.max(f)) + ' Hz'
#- Warnings.
if (p.fmax > 1.0 / p.dt):
print 'WARNING: maximum bandpass frequency cannot be represented with this time step!'
if (p.fmin < 1.0 / (n * p.dt)):
print 'WARNING: minimum bandpass frequency cannot be represented with this window length!'
#==============================================================================
#- March through source locations and compute raw frequency-domain noise traces.
#==============================================================================
#- Set a specific random seed to make simulation repeatable, e.g. for different receiver pair.
np.random.seed(p.seed)
#- Initialise frequency-domain wavefields.
u1 = np.zeros([n, p.Nwindows], dtype=complex)
u2 = np.zeros([n, p.Nwindows], dtype=complex)
G1 = np.zeros([n, p.Nwindows], dtype=complex)
G2 = np.zeros([n, p.Nwindows], dtype=complex)
#- Regularise zero-frequency to avoid singularity in Green function.
omega[0] = 0.01 * 2.0 * np.pi * df
#- March through source indices.
for k in indeces:
#- Green function for a specific source point.
G1[:, 0] = g.green_input(p.x[rec0], p.y[rec0], x[k], y[k], omega, p.dx,
p.dy, p.rho, p.v, p.Q)
G2[:, 0] = g.green_input(p.x[rec1], p.y[rec1], x[k], y[k], omega, p.dx,
p.dy, p.rho, p.v, p.Q)
#- Apply instrument response and source spectrum
G1[:, 0] = G1[:, 0] * instrument * np.sqrt(natural)
G2[:, 0] = G2[:, 0] * instrument * np.sqrt(natural)
#- Copy this Green function to all time intervals.
for i in range(p.Nwindows):
G1[:, i] = G1[:, 0]
G2[:, i] = G2[:, 0]
#- Random phase matrix, frequency steps times time windows.
phi = 2.0 * np.pi * (np.random.rand(n, p.Nwindows) - 0.5)
ff = np.exp(1j * phi)
#- Matrix of random frequency-domain wavefields.
u1 += S[k] * ff * G1
u2 += S[k] * ff * G2
#- March through time windows to add earthquakes.
for win in range(p.Nwindows):
neq = len(p.eq_t[win])
for i in range(neq):
G1 = g.green_input(p.x[rec0], p.y[rec0], p.eq_x[win][i],
p.eq_y[win][i], omega, p.dx, p.dy, p.rho, p.v,
p.Q)
G2 = g.green_input(p.x[rec1], p.y[rec1], p.eq_x[win][i],
p.eq_y[win][i], omega, p.dx, p.dy, p.rho, p.v,
p.Q)
G1 = G1 * instrument * np.sqrt(natural)
G2 = G2 * instrument * np.sqrt(natural)
u1[:, win] += p.eq_m[win][i] * G1 * np.exp(
-1j * omega * p.eq_t[win][i])
u2[:, win] += p.eq_m[win][i] * G2 * np.exp(
-1j * omega * p.eq_t[win][i])
#==============================================================================
#- Processing.
#==============================================================================
#- Apply single-station processing.
u1_proc, u2_proc = proc.processing_single_station(u1, u2, f, verbose)
#- Compute correlation function, raw and processed.
ccf = u1 * np.conj(u2)
ccf_proc = u1_proc * np.conj(u2_proc)
#- Apply correlation processing.
ccf_proc = proc.processing_correlation(ccf_proc, f, verbose)
#==============================================================================
#- Apply the standard bandpass.
#==============================================================================
bandpass = np.zeros(np.shape(f))
Nminmax = np.round((p.bp_fmin) / df)
Nminmin = np.round((p.bp_fmin - p.bp_width) / df)
Nmaxmin = np.round((p.bp_fmax) / df)
Nmaxmax = np.round((p.bp_fmax + p.bp_width) / df)
bandpass[Nminmin:Nminmax] = np.linspace(0.0, 1.0, Nminmax - Nminmin)
bandpass[Nmaxmin:Nmaxmax] = np.linspace(1.0, 0.0, Nmaxmax - Nmaxmin)
bandpass[Nminmax:Nmaxmin] = 1.0
for i in range(p.Nwindows):
ccf[:, i] = bandpass * ccf[:, i]
ccf_proc[:, i] = bandpass * ccf_proc[:, i]
#==============================================================================
#- Time-domain correlation function.
#==============================================================================
#- Some care has to be taken here with the inverse FFT convention of numpy.
cct = np.zeros([n, p.Nwindows], dtype=float)
cct_proc = np.zeros([n, p.Nwindows], dtype=float)
dummy = np.real(np.fft.ifft(ccf, axis=0) / p.dt)
cct[0.5 * n:n, :] = dummy[0:0.5 * n, :]
cct[0:0.5 * n, :] = dummy[0.5 * n:n, :]
dummy = np.real(np.fft.ifft(ccf_proc, axis=0) / p.dt)
cct_proc[0.5 * n:n, :] = dummy[0:0.5 * n, :]
cct_proc[0:0.5 * n, :] = dummy[0.5 * n:n, :]
#==============================================================================
#- Save results if wanted.
#==============================================================================
if save == 1:
#- Store frequency and time axes.
fid = open('OUTPUT/correlations_individual/f', 'w')
np.save(fid, f)
fid.close()
fid = open('OUTPUT/correlations_individual/t', 'w')
np.save(fid, t)
fid.close()
#- Store raw and processed correlations in the frequency domain.
fn = 'OUTPUT/correlations_individual/ccf_' + str(rec0) + '_' + str(
rec1)
fid = open(fn, 'w')
np.save(fid, ccf)
fid.close()
fn = 'OUTPUT/correlations_individual/ccf_proc_' + str(
rec0) + '_' + str(rec1)
fid = open(fn, 'w')
np.save(fid, ccf_proc)
fid.close()
#==============================================================================
#- Plot results if wanted.
#==============================================================================
if plot == 1:
#- Noise traces for first window.
plt.subplot(2, 1, 1)
plt.plot(t, np.real(np.fft.ifft(u1[:, 0])) / p.dt, 'k')
plt.ylabel('u1(t) [N/m^2]')
plt.title('recordings for first time window')
plt.subplot(2, 1, 2)
plt.plot(t, np.real(np.fft.ifft(u2[:, 0])) / p.dt, 'k')
plt.ylabel('u2(t) [N/m^2]')
plt.xlabel('t [s]')
plt.show()
#- Spectrum of the pressure wavefield for first window.
plt.subplot(2, 1, 1)
plt.plot(f, np.abs(np.sqrt(bandpass) * u1[:, 0]), 'k', linewidth=2)
plt.plot(f, np.real(np.sqrt(bandpass) * u1[:, 0]), 'b', linewidth=1)
plt.plot(f, np.imag(np.sqrt(bandpass) * u1[:, 0]), 'r', linewidth=1)
plt.ylabel('u1(f) [Ns/m^2]')
plt.title(
'raw and processed spectra for first window (abs=black, real=blue, imag=red)'
)
plt.subplot(2, 1, 2)
plt.plot(f,
np.abs(np.sqrt(bandpass) * u1_proc[:, 0]),
'k',
linewidth=2)
plt.plot(f,
np.real(np.sqrt(bandpass) * u1_proc[:, 0]),
'b',
linewidth=1)
plt.plot(f,
np.imag(np.sqrt(bandpass) * u1_proc[:, 0]),
'r',
linewidth=1)
plt.ylabel('u1_proc(f) [?]')
plt.xlabel('f [Hz]')
plt.show()
#- Raw time- and frequency-domain correlation for first window.
plt.subplot(2, 1, 1)
plt.semilogy(f, np.abs(ccf[:, 0]), 'k', linewidth=2)
plt.title('raw frequency-domain correlation for first window')
plt.ylabel('correlation [N^2 s^2 / m^4]')
plt.xlabel('f [Hz]')
plt.subplot(2, 1, 2)
plt.plot(t, np.real(cct[:, 0]), 'k')
plt.title('raw time-domain correlation for first window')
plt.ylabel('correlation [N^2 s / m^4]')
plt.xlabel('t [s]')
plt.show()
#- Processed time- and frequency-domain correlation for first window.
plt.subplot(2, 1, 1)
plt.semilogy(f, np.abs(ccf_proc[:, 0]), 'k', linewidth=2)
plt.title('processed frequency-domain correlation for first window')
plt.ylabel('correlation [N^2 s^2 / m^4]*unit(T)')
plt.xlabel('f [Hz]')
plt.subplot(2, 1, 2)
plt.plot(t, np.real(cct_proc[:, 0]))
plt.title('processed time-domain correlation for first window')
plt.ylabel('correlation [N^2 s / m^4]*unit(T)')
plt.xlabel('t [s]')
plt.show()
#- Raw and processed ensemble correlations.
plt.plot(t, np.sum(cct, 1) / np.max(cct), 'k')
plt.plot(t, np.sum(cct_proc, 1) / np.max(cct_proc), 'r')
plt.title(
'ensemble time-domain correlation (black=raw, red=processed)')
plt.ylabel('correlation [N^2 s / m^4]*unit(T)')
plt.xlabel('t [s]')
plt.show()
#- End time.
t2 = time.time()
if verbose == 1:
print 'elapsed time: ' + str(t2 - t1) + ' s'
#==============================================================================
#- Output.
#==============================================================================
return cct, cct_proc, t, ccf, ccf_proc, f
def correlation_function(rec0=0, rec1=1, effective=0, plot=0):
"""
cct, t, ccf, f = correlation_function(rec0=0,rec1=1,effective=0,plot=0)
Compute time- and frequency-domain correlation functions.
INPUT:
------
rec0, rec1: indeces of the receivers used in the correlation.
plot: When plot=1, the source distribution, and the time- and frequency domain correlation functions are plotted.
effective: When effective==1, effective correlations are computed using the propagation correctors stored in OUTPUT/correctors.
The source power-spectral density is then interpreted as the effective one.
OUTPUT:
-------
cct, t: Time-domain correlation function and time axis [N^2 s / m^4],[s].
ccf, f: Frequency-domain correlation function and frequency axis [N^2 s^2 / m^4],[1/s].
Last updated: 27 May 2016.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
p = parameters.Parameters()
x_line = np.arange(p.xmin, p.xmax, p.dx)
y_line = np.arange(p.ymin, p.ymax, p.dy)
x, y = np.meshgrid(x_line, y_line)
f = np.arange(p.fmin - p.fwidth, p.fmax + p.fwidth, p.df)
omega = 2.0 * np.pi * f
t = np.arange(p.tmin, p.tmax, p.dt)
#- Frequency- and space distribution of the source. ---------------------------
S, indices = s.space_distribution(plot)
instrument, natural = s.frequency_distribution(f)
filt = natural * instrument * instrument
#- Read propagation corrector if needed. --------------------------------------
if (effective == 1):
gf = gpc.get_propagation_corrector(rec0, rec1, plot=0)
else:
gf = np.ones(len(f), dtype=complex)
#==============================================================================
#- Compute inter-station correlation function.
#==============================================================================
cct = np.zeros(np.shape(t), dtype=float)
ccf = np.zeros(np.shape(f), dtype=complex)
for idf in range(len(omega)):
P = g.conjG1_times_G2(p.x[rec0], p.y[rec0], p.x[rec1], p.y[rec1], x, y,
omega[idf], p.dx, p.dy, p.rho, p.v, p.Q)
ccf[idf] = gf[idf] * np.conj(np.sum(P * S))
cct = cct + np.real(filt[idf] * ccf[idf] * np.exp(1j * omega[idf] * t))
cct = cct * p.dx * p.dy * p.df
#==============================================================================
#- Plot result.
#==============================================================================
if (plot == 1):
#- Frequency domain.
plt.semilogy(f, np.abs(ccf), 'k')
plt.semilogy(f, np.real(ccf), 'b')
plt.title(
'frequency-domain correlation function (black=abs, blue=real)')
plt.xlabel('frequency [Hz]')
plt.ylabel('correlation [N^2 s^2/m^4]')
plt.show()
#- Time domain.
tt = np.sqrt((p.x[rec0] - p.x[rec1])**2 +
(p.y[rec0] - p.y[rec1])**2) / p.v
cct_max = np.max(np.abs(cct))
plt.plot(t, cct, 'k', linewidth=2.0)
plt.plot([tt, tt], [-1.1 * cct_max, 1.1 * cct_max],
'--',
color=(0.5, 0.5, 0.5),
linewidth=1.5)
plt.plot([-tt, -tt], [-1.1 * cct_max, 1.1 * cct_max],
'--',
color=(0.5, 0.5, 0.5),
linewidth=1.5)
plt.ylim((-1.1 * cct_max, 1.1 * cct_max))
plt.title('correlation function')
plt.xlabel('time [s]')
plt.ylabel('correlation [N^2 s/m^4]')
plt.show()
#==============================================================================
#- Return.
#==============================================================================
return cct, t, ccf, f
def snapshot(rec=0,
t=1.0,
minvalplot=0.0,
maxvalplot=0.0,
plot=False,
save=False,
verbose=False,
dir_precomputed='OUTPUT/'):
"""
snapshot(rec=0,t=1.0, minvalplot=0.0, maxvalplot=0.0, plot=False, save=False, verbose=False, dir_precomputed='OUTPUT/')
Compute and plot correlation wavefield for a specific time t. This requires that the frequency-domain wavefield has been precomputed using the precompute function below.
INPUT:
------
rec: index of receiver.
t: time in s.
minvalplot: minimum of colour scale, ignored when 0.
maxvalplot: maximum of colour scale, ignored when 0.
plot: plot when True.
save: save as pdf when True.
verbose: give screen output when True.
dir_precomputed: directory where precomputed correlation field is located.
OUTPUT:
-------
C: 2D time-domain correlation wavefield [N^2 s / m^4].
x,y: 2D axes [m].
Last updated: 18 July 2019.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
p = parameters.Parameters()
#- Spatial grid.
x_line = np.arange(p.xmin, p.xmax, p.dx)
y_line = np.arange(p.ymin, p.ymax, p.dy)
x, y = np.meshgrid(x_line, y_line)
#- Frequency line.
f = np.arange(p.fmin - p.fwidth, p.fmax + p.fwidth, p.df)
omega = 2.0 * np.pi * f
#- Power-spectral density.
S, indeces = s.space_distribution()
instrument, natural = s.frequency_distribution(f)
filt = natural * instrument * instrument
C = np.zeros(np.shape(x))
#==============================================================================
#- Load forward interferometric wavefields in the frequency domain.
#==============================================================================
fn = dir_precomputed + 'cf_' + str(rec) + '.npy'
Cfull = np.load(fn)
#==============================================================================
#- Compute correlation field for a specific time.
#==============================================================================
for idx in range(len(x_line)):
for idy in range(len(y_line)):
C[idy, idx] = np.real(
np.sum(Cfull[idy, idx, :] * np.exp(1j * omega * t))) * p.df
#==============================================================================
#- Plot.
#==============================================================================
if (plot or save):
plt.rcParams["font.family"] = "serif"
plt.rcParams.update({'font.size': 10})
if (minvalplot == 0.0 and maxvalplot == 0.0):
maxvalplot = 0.8 * np.max(np.abs(C))
minvalplot = -maxvalplot
#- Plot interferometric wavefield. ----------------------------------------
#plt.pcolor(x/1000.0,y/1000.0,C,cmap='RdBu',vmin=minvalplot,vmax=maxvalplot)
plt.pcolor(x / 1000.0,
y / 1000.0,
np.abs(C),
cmap='Greys',
vmin=0.0,
vmax=maxvalplot)
#- Plot receiver positions. -----------------------------------------------
for k in range(p.Nreceivers):
plt.plot(p.x[k] / 1000.0, p.y[k] / 1000.0, 'ko', markersize='5')
plt.text(p.x[k] / 1000.0 + 4.0 * p.dx / 1000.0,
p.y[k] / 1000.0 + 4.0 * p.dx / 1000.0, str(k))
plt.plot(p.x[rec] / 1000.0, p.y[rec] / 1000.0, 'kx')
plt.text(p.x[rec] / 1000.0 + 4.0 * p.dx / 1000.0,
p.y[rec] / 1000.0 + 4.0 * p.dx / 1000.0, str(rec))
#- Embellish the plot. ----------------------------------------------------
plt.colorbar()
plt.axis('image')
plt.title('correlation field, t=' + str(t) + ' s')
plt.xlim((p.xmin / 1000.0, p.xmax / 1000.0))
plt.ylim((p.ymin / 1000.0, p.ymax / 1000.0))
plt.xlabel('x [km]')
plt.ylabel('y [km]')
if plot:
plt.show()
if save:
fn = 'OUTPUT/' + str(t) + '.pdf'
plt.savefig(fn, format='pdf')
plt.clf()
#==============================================================================
#- Return.
#==============================================================================
return C, x, y
def structure_kernel(cct,
t,
rec0=0,
rec1=1,
measurement='cctime',
dir_forward='OUTPUT/',
effective=0,
plot=0):
"""
x,y,K_kappa = structure_kernel(cct, t, rec0=0, rec1=1, measurement='cctime', dir_forward='OUTPUT/', effective=0, plot=0):
Compute structure kernel K_kappa for a frequency-independent source power-spectral density.
INPUT:
------
cct, t: Time-domain correlation function and time axis as obtained from correlation_function.py.
rec0, rec1: Indeces of the receivers used in the correlation.
measurement: Type of measurement used to compute the adjoint source. See adsrc.py for options.
dir_forward: Location of the forward interferometric fields from rec0 and rec1. Must exist.
plot: When plot=1, plot structure kernel.
effective: When effective==1, effective correlations are computed using the propagation correctors stored in OUTPUT/correctors.
The source power-spectral density is then interpreted as the effective one.
OUTPUT:
-------
x,y: Space coordinates.
K: Structure kernel [unit of measurement * 1/N].
Last updated: 11 July 2016.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
p = parameters.Parameters()
x_line = np.arange(p.xmin, p.xmax, p.dx)
y_line = np.arange(p.ymin, p.ymax, p.dy)
x, y = np.meshgrid(x_line, y_line)
f = np.arange(p.fmin - p.fwidth, p.fmax + p.fwidth, p.df)
df = f[1] - f[0]
omega = 2.0 * np.pi * f
K_kappa = np.zeros(np.shape(x))
C1 = np.zeros((len(y_line), len(x_line), len(omega)), dtype=complex)
C2 = np.zeros((len(y_line), len(x_line), len(omega)), dtype=complex)
nx = len(x_line)
ny = len(y_line)
kappa = p.rho * (p.v**2)
#- Frequency- and space distribution of the source. ---------------------------
S, indeces = s.space_distribution(plot=0)
instrument, natural = s.frequency_distribution(f)
filt = natural * instrument * instrument
#- Compute the adjoint source. ------------------------------------------------
a = adsrc.adsrc(cct, t, measurement, plot)
#- Compute number of grid points corresponding to the minimum wavelength. -----
L = int(np.ceil(p.v / (p.fmax * p.dx)))
#- Read propagation corrector if needed. --------------------------------------
if (effective == 1):
gf = gpc.get_propagation_corrector(rec0, rec1, plot=0)
else:
gf = np.ones(len(f), dtype=complex)
#==============================================================================
#- Load forward interferometric wavefields.
#==============================================================================
fn = dir_forward + '/cf_' + str(rec0)
fid = open(fn, 'r')
C1 = np.load(fid)
fid.close()
fn = dir_forward + '/cf_' + str(rec1)
fid = open(fn, 'r')
C2 = np.load(fid)
fid.close()
#==============================================================================
#- Loop over frequencies.
#==============================================================================
for k in range(len(omega)):
w = omega[k]
#- Adjoint fields. --------------------------------------------------------
G1 = -w**2 * g.green_input(x, y, p.x[rec0], p.y[rec0], w, p.dx, p.dy,
p.rho, p.v, p.Q) * gf[k]
G2 = -w**2 * g.green_input(x, y, p.x[rec1], p.y[rec1], w, p.dx, p.dy,
p.rho, p.v, p.Q) * gf[k]
#- Multiplication with adjoint fields. ------------------------------------
K_kappa = K_kappa - 2.0 * np.real(G2 * C1[:, :, k] * np.conj(a[k]) +
G1 * C2[:, :, k] * a[k])
K_kappa = K_kappa / kappa
#==============================================================================
#- Smooth over minimum wavelength.
#==============================================================================
for k in range(L):
K_kappa[1:ny - 2, :] = (K_kappa[1:ny - 2, :] + K_kappa[0:ny - 3, :] +
K_kappa[2:ny - 1, :]) / 3.0
for k in range(L):
K_kappa[:, 1:nx - 2] = (K_kappa[:, 1:nx - 2] + K_kappa[:, 0:nx - 3] +
K_kappa[:, 2:nx - 1]) / 3.0
#==============================================================================
#- Visualise if wanted.
#==============================================================================
if plot == 1:
cmap = plt.get_cmap('RdBu')
plt.pcolormesh(x, y, K_kappa, cmap=cmap, shading='interp')
plt.clim(-np.max(np.abs(K_kappa)) * 0.25, np.max(np.abs(K_kappa)) * 0.25)
plt.axis('image')
plt.colorbar()
plt.title('Structure (kappa) kernel [unit of measurement / m^2]')
plt.xlabel('x [km]')
plt.ylabel('y [km]')
plt.plot(p.x[rec0], p.y[rec0], 'ro')
plt.plot(p.x[rec1], p.y[rec1], 'ro')
plt.show()
return x, y, K_kappa
def snapshot(rec=0,t=1.0, mg_level=5, mg_tol=0.05, minvalplot=0.0, maxvalplot=0.0, plot=0, save=0, verbose=0, precomputed=0, dir_precomputed='OUTPUT/'):
"""
snapshot(rec=0,t=1.0, mg_level=5, mg_tol=0.05, minvalplot=0.0, maxvalplot=0.0, plot=0, save=0, verbose=0, precomputed=0, dir_precomputed='OUTPUT/')
Compute and plot correlation wavefield.
INPUT:
------
rec: index of receiver.
t: time in s.
mg_level: level for multi-grid solver.
mg_tol: tolerance for multi-grid solver.
minvalplot: minimum of colour scale, ignored when 0.
maxvalplot: maximum of colour scale, ignored when 0.
plot: plot when 1.
save: save as png when 1.
verbose: give screen output when 1.
precomputed: set to 1 if precomputed frequency-domain correlation field available (see precompute() below).
dir_precomputed: directory where precomputed correlation field is located.
OUTPUT:
-------
C: 2D time-domain correlation wavefield [N^2 s / m^4].
x,y: 2D axes [m].
Last updated: 27 May 2016.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
p=parameters.Parameters()
#- Spatial grid.
x_line=np.arange(p.xmin,p.xmax,p.dx)
y_line=np.arange(p.ymin,p.ymax,p.dy)
x,y=np.meshgrid(x_line,y_line)
#- Frequency line.
f=np.arange(p.fmin-p.fwidth,p.fmax+p.fwidth,p.df)
omega=2.0*np.pi*f
#- Power-spectral density.
S,indeces=s.space_distribution()
instrument,natural=s.frequency_distribution(f)
filt=natural*instrument*instrument
C=np.zeros(np.shape(x))
#==============================================================================
#- Load forward interferometric wavefields.
#==============================================================================
if precomputed==1:
fn=dir_precomputed+'/cf_'+str(rec)
fid=open(fn,'r')
Cfull=np.load(fid)
fid.close()
for idx in range(len(x_line)):
for idy in range(len(y_line)):
C[idy,idx]=np.real(np.sum(Cfull[idy,idx,:]*np.exp(1j*omega*t)))*p.df
#==============================================================================
#- Compute correlation field for a specific time.
#==============================================================================
else:
#- First multi-grid stage. ------------------------------------------------
if (verbose==1): print 'First multi-grid stage'
#- March through the spatial grid.
for idx in range(0,len(x_line),mg_level):
if (verbose==1): print str(100*float(idx)/float(len(x_line)))+' %'
for idy in range(0,len(y_line),mg_level):
C_proto=np.zeros(len(omega),dtype=complex)
#- March through all sources.
for k in indeces:
C_proto+=g.conjG1_times_G2(x[idy,idx],y[idy,idx],p.x[rec],p.y[rec],x[k],y[k],omega,p.dx,p.dy,p.rho,p.v,p.Q)*S[k]
C_proto=np.conj(C_proto)
#- Transform to time domain.
C[idy,idx]=np.real(np.sum(filt*C_proto*np.exp(1j*omega*t)))
#- Second multi-grid stage. -----------------------------------------------
if (verbose==1): print 'Second multi-grid stage'
c_max=np.max(np.abs(C))
#- March through the spatial grid.
for idx in range(mg_level,len(x_line)-mg_level):
if (verbose==1): print str(100*float(idx)/float(len(x_line)))+' %'
for idy in range(mg_level,len(y_line)-mg_level):
if (np.max(np.abs(C[(idy-mg_level):(idy+mg_level),(idx-mg_level):(idx+mg_level)]))>mg_tol*c_max):
C_proto=np.zeros(len(omega),dtype=complex)
#- March through all sources.
for k in indeces:
C_proto+=g.conjG1_times_G2(x[idy,idx],y[idy,idx],p.x[rec],p.y[rec],x[k],y[k],omega,p.dx,p.dy,p.rho,p.v,p.Q)*S[k]
C_proto=np.conj(C_proto)
#- Transform to time domain.
C[idy,idx]=np.real(np.sum(filt*C_proto*np.exp(1j*omega*t)))
#- Normalisation.
C=C*p.dx*p.dy*p.df
#==============================================================================
#- Plot.
#==============================================================================
if (plot==1 or save==1):
if (minvalplot==0.0 and maxvalplot==0.0):
maxvalplot=0.8*np.max(np.abs(C))
minvalplot=-maxvalplot
font = {'size' : 14,}
#- Plot interferometric wavefield. ----------------------------------------
plt.pcolor(x,y,C,cmap='RdBu',vmin=minvalplot,vmax=maxvalplot)
#- Plot receiver positions. -----------------------------------------------
for k in range(p.Nreceivers):
plt.plot(p.x[k],p.y[k],'kx')
plt.text(p.x[k]+3.0*p.dx,p.y[k]+3.0*p.dx,str(k),fontdict=font)
plt.plot(p.x[rec],p.y[rec],'ro')
plt.text(p.x[rec]+3.0*p.dx,p.y[rec]+3.0*p.dx,str(rec),fontdict=font)
#- Embellish the plot. ----------------------------------------------------
plt.colorbar()
plt.axis('image')
plt.title('correlation field, t='+str(t)+' s')
plt.xlim((p.xmin,p.xmax))
plt.ylim((p.ymin,p.ymax))
plt.xlabel('x [m]')
plt.ylabel('y [m]')
if (plot==1):
plt.show()
if (save==1):
fn='OUTPUT/'+str(t)+'.png'
plt.savefig(fn)
plt.clf()
#==============================================================================
#- Return.
#==============================================================================
return C, x, y
def precompute(rec=0, verbose=False, mode='individual'):
"""
precompute(rec=0,verbose=False,mode='individual')
Compute correlation wavefield in the frequency domain and store for in /OUTPUT for re-use in snapshot kernel computation.
INPUT:
------
rec: index of reference receiver.
verbose: give screen output when True.
mode: 'individual' sums over individual sources. This is very efficient when there are only a few sources. This mode requires that the indeces array returned by source.space_distribution is not empty.
'random' performs a randomised, down-sampled integration over a quasi-continuous distribution of sources. This is more efficient for widely distributed and rather smooth sources.
'combined' is the sum of 'individual' and 'random'. This is efficient when a few point sources are super-imposed on a quasi-continuous distribution.
OUTPUT:
-------
Frequency-domain interferometric wavefield stored in /OUTPUT.
Last updated: 18 July 2019.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
p = parameters.Parameters()
#- Spatial grid.
x_line = np.arange(p.xmin, p.xmax, p.dx)
y_line = np.arange(p.ymin, p.ymax, p.dy)
x, y = np.meshgrid(x_line, y_line)
nx = len(x_line)
ny = len(y_line)
#- Frequency line.
f = np.arange(p.fmin - p.fwidth, p.fmax + p.fwidth, p.df)
omega = 2.0 * np.pi * f
#- Power-spectral density.
S, indeces = s.space_distribution()
instrument, natural = s.frequency_distribution(f)
filt = natural * instrument * instrument
C = np.zeros((len(y_line), len(x_line), len(omega)), dtype=complex)
#==============================================================================
#- Compute correlation field by summing over individual sources.
#==============================================================================
if (mode == 'individual'):
#- March through the spatial grid. ----------------------------------------
for idx in range(nx):
if verbose:
print(str(100 * float(idx) / float(len(x_line))) + ' %')
for idy in range(ny):
#- March through all sources.
for k in indeces:
C[idy, idx, :] += S[k] * filt * g.conjG1_times_G2(
x[idy, idx], y[idy, idx], p.x[rec], p.y[rec], x[k],
y[k], omega, p.dx, p.dy, p.rho, p.v, p.Q)
#- Normalisation.
C = np.conj(C) * p.dx * p.dy
#==============================================================================
#- Compute correlation field by random integration over all sources
#==============================================================================
downsampling_factor = 5.0
n_samples = np.floor(float(nx * ny) / downsampling_factor)
if (mode == 'random'):
#- March through frequencies. ---------------------------------------------
for idf in range(0, len(f), 3):
if verbose: print('f=', f[idf], ' Hz')
if (filt[idf] > 0.05 * np.max(filt)):
#- March through downsampled spatial grid. ------------------------
t0 = time.time()
for idx in range(0, nx, 3):
for idy in range(0, ny, 3):
samples_x = np.random.randint(0, nx, n_samples)
samples_y = np.random.randint(0, ny, n_samples)
G1 = g.green_input(x[samples_y, samples_x],
y[samples_y,
samples_x], x_line[idx],
y_line[idy], omega[idf], p.dx, p.dy,
p.rho, p.v, p.Q)
G2 = g.green_input(x[samples_y,
samples_x], y[samples_y,
samples_x],
p.x[rec], p.y[rec], omega[idf],
p.dx, p.dy, p.rho, p.v, p.Q)
C[idy, idx,
idf] = downsampling_factor * filt[idf] * np.sum(
S[samples_y, samples_x] * G1 * np.conj(G2))
t1 = time.time()
if verbose: print('time per frequency: ', t1 - t0, 's')
#- Normalisation. ---------------------------------------------------------
C = C * p.dx * p.dy
#- Spatial interpolation. -------------------------------------------------
for idx in range(0, nx - 3, 3):
C[:, idx + 1, :] = 0.67 * C[:, idx, :] + 0.33 * C[:, idx + 3, :]
C[:, idx + 2, :] = 0.33 * C[:, idx, :] + 0.67 * C[:, idx + 3, :]
for idy in range(0, ny - 3, 3):
C[idy + 1, :, :] = 0.67 * C[idy, :, :] + 0.33 * C[idy + 3, :, :]
C[idy + 2, :, :] = 0.33 * C[idy, :, :] + 0.67 * C[idy + 3, :, :]
#- Frequency interpolation. -----------------------------------------------
for idf in range(0, len(f) - 3, 3):
C[:, :, idf + 1] = 0.67 * C[:, :, idf] + 0.33 * C[:, :, idf + 3]
C[:, :, idf + 2] = 0.33 * C[:, :, idf] + 0.67 * C[:, :, idf + 3]
#==============================================================================
#- Compute correlation field by random integration over all sources + individual sources
#==============================================================================
downsampling_factor = 5.0
n_samples = np.floor(float(nx * ny) / downsampling_factor)
if (mode == 'combined'):
#--------------------------------------------------------------------------
#- March through frequencies for random sampling. -------------------------
for idf in range(0, len(f), 3):
if verbose: print('f=', f[idf], ' Hz')
if (filt[idf] > 0.05 * np.max(filt)):
#- March through downsampled spatial grid. ------------------------
t0 = time.time()
for idx in range(0, nx, 3):
for idy in range(0, ny, 3):
samples_x = np.random.randint(0, nx, n_samples)
samples_y = np.random.randint(0, ny, n_samples)
G1 = g.green_input(x[samples_y, samples_x],
y[samples_y,
samples_x], x_line[idx],
y_line[idy], omega[idf], p.dx, p.dy,
p.rho, p.v, p.Q)
G2 = g.green_input(x[samples_y,
samples_x], y[samples_y,
samples_x],
p.x[rec], p.y[rec], omega[idf],
p.dx, p.dy, p.rho, p.v, p.Q)
C[idy, idx,
idf] = downsampling_factor * filt[idf] * np.sum(
S[samples_y, samples_x] * G1 * np.conj(G2))
t1 = time.time()
if verbose: print('time per frequency: ', t1 - t0, 's')
#- Spatial interpolation. -------------------------------------------------
for idx in range(0, nx - 3, 3):
C[:, idx + 1, :] = 0.67 * C[:, idx, :] + 0.33 * C[:, idx + 3, :]
C[:, idx + 2, :] = 0.33 * C[:, idx, :] + 0.67 * C[:, idx + 3, :]
for idy in range(0, ny - 3, 3):
C[idy + 1, :, :] = 0.67 * C[idy, :, :] + 0.33 * C[idy + 3, :, :]
C[idy + 2, :, :] = 0.33 * C[idy, :, :] + 0.67 * C[idy + 3, :, :]
#- Frequency interpolation. -----------------------------------------------
for idf in range(0, len(f) - 3, 3):
C[:, :, idf + 1] = 0.67 * C[:, :, idf] + 0.33 * C[:, :, idf + 3]
C[:, :, idf + 2] = 0.33 * C[:, :, idf] + 0.67 * C[:, :, idf + 3]
#--------------------------------------------------------------------------
#- March through the spatial grid for individual sources. -----------------
for idx in range(nx):
if verbose:
print(str(100 * float(idx) / float(len(x_line))) + ' %')
for idy in range(ny):
#- March through all sources.
for k in indeces:
C[idy, idx, :] += S[k] * filt * np.conj(
g.conjG1_times_G2(x[idy, idx], y[idy, idx], p.x[rec],
p.y[rec], x[k], y[k], omega, p.dx,
p.dy, p.rho, p.v, p.Q))
#- Normalisation. ---------------------------------------------------------
C = C * p.dx * p.dy
#==============================================================================
#- Save interferometric wavefield.
#==============================================================================
fn = 'OUTPUT/cf_' + str(rec)
np.save(fn, C)
def movie(time_axis, minvalplot=0.0, maxvalplot=0.0, verbose=0):
"""
movie(time_axis, mg_level=5, mg_tol=0.05, minvalplot=0.0, maxvalplot=0.0, verbose=0)
Compute correlation wavefield and save png figures to /OUTPUT.
INPUT:
------
time_axis: array containing time values in s for which figures will be saved.
minvalplot: minimum of colour scale, ignored when 0.
maxvalplot: maximum of colour scale, ignored when 0.
verbose: give screen output when 1.
OUTPUT:
-------
Snapshots of the interferometric wavefield saved as png files to /OUTPUT
Last updated: 5 May 2016.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
p=parameters.Parameters()
#- Spatial grid.
x_line=np.arange(p.xmin,p.xmax,p.dx)
y_line=np.arange(p.ymin,p.ymax,p.dy)
x,y=np.meshgrid(x_line,y_line)
#- Frequency line.
f=np.arange(p.fmin-p.fwidth,p.fmax+p.fwidth,p.df)
omega=2.0*np.pi*f
#- Power-spectral density.
S,indeces=s.space_distribution()
instrument,natural=s.frequency_distribution(f)
filt=natural*instrument*instrument
#- Check if the indeces are actually available. If not, interrupt.
if len(indeces)==0:
print 'Correlation field cannot be computed because source index array is empty.'
return
#==============================================================================
#- Compute correlation field for specific times and store.
#==============================================================================
C=np.zeros((len(y_line),len(x_line),len(time_axis)))
#- March through the spatial grid. --------------------------------------------
for idx in range(len(x_line)):
if (verbose==1):
print str(100*float(idx)/float(len(x_line)))+' %'
for idy in range(len(y_line)):
C_proto=np.zeros(len(omega),dtype=complex)
#- March through all sources.
for k in indeces:
C_proto+=g.conjG1_times_G2(x[idy,idx],y[idy,idx],p.x[1],p.y[1],x[k],y[k],omega,p.dx,p.dy,p.rho,p.v,p.Q)*S[k]
C_proto=np.conj(C_proto)
#- Transform to time domain.
for t in range(len(time_axis)):
C[idy,idx,t]=np.real(np.sum(filt*C_proto*np.exp(1j*omega*time_axis[t])))
#- Normalisation.
C=C*p.dx*p.dy*p.df
#==============================================================================
#- Save images.
#==============================================================================
for t in range(len(time_axis)):
if (minvalplot==0.0 and maxvalplot==0.0):
maxvalplot=0.8*np.max(np.abs(C[:,:,t]))
minvalplot=-maxvalplot
plt.pcolor(x,y,C[:,:,t],cmap='RdBu',vmin=minvalplot,vmax=maxvalplot)
font = {'family' : 'sansserif', 'color' : 'darkred', 'weight' : 'normal', 'size' : 14,}
plt.plot(p.x[0],p.y[0],'ko')
plt.plot(p.x[1],p.y[1],'ko')
plt.text(0.9*p.x[0],p.y[0],'1',fontdict=font)
plt.text(1.1*p.x[1],p.y[1],'2',fontdict=font)
plt.colorbar()
plt.axis('image')
plt.title('correlation field, t='+str(time_axis[t])+' s')
plt.xlabel('x [m]')
plt.ylabel('y [m]')
fn='OUTPUT/'+str(time_axis[t])+'.png'
plt.savefig(fn)
plt.clf()
def precompute(rec=0,verbose=0,mode='individual'): """ precompute(verbose=0) Compute correlation wavefield in the frequency domain and store for in /OUTPUT for re-use in kernel computation. INPUT: ------ rec: index of reference receiver. verbose: give screen output when 1. mode: 'individual' sums over individual sources. This is very efficient when there are only a few sources. This mode requires that the indeces array returned by source.space_distribution is not empty. 'random' performs a randomised, down-sampled integration over a quasi-continuous distribution of sources. This is more efficient for widely distributed and rather smooth sources. 'combined' is the sum of 'individual' and 'random'. This is efficient when a few point sources are super-imposed on a quasi-continuous distribution. OUTPUT: ------- Frequency-domain interferometric wavefield stored in /OUTPUT. Last updated: 13 July 2016. """ #============================================================================== #- Initialisation. #============================================================================== p=parameters.Parameters() #- Spatial grid. x_line=np.arange(p.xmin,p.xmax,p.dx) y_line=np.arange(p.ymin,p.ymax,p.dy) x,y=np.meshgrid(x_line,y_line) nx=len(x_line) ny=len(y_line) #- Frequency line. f=np.arange(p.fmin-p.fwidth,p.fmax+p.fwidth,p.df) omega=2.0*np.pi*f #- Power-spectral density. S,indeces=s.space_distribution() instrument,natural=s.frequency_distribution(f) filt=natural*instrument*instrument C=np.zeros((len(y_line),len(x_line),len(omega)),dtype=complex) #============================================================================== #- Compute correlation field by summing over individual sources. #============================================================================== if (mode=='individual'): #- March through the spatial grid. ---------------------------------------- for idx in range(nx): if (verbose==1): print str(100*float(idx)/float(len(x_line)))+' %' for idy in range(ny): #- March through all sources. for k in indeces: C[idy,idx,:]+=S[k]*filt*g.conjG1_times_G2(x[idy,idx],y[idy,idx],p.x[rec],p.y[rec],x[k],y[k],omega,p.dx,p.dy,p.rho,p.v,p.Q) #- Normalisation. C=np.conj(C)*p.dx*p.dy #============================================================================== #- Compute correlation field by random integration over all sources #============================================================================== downsampling_factor=5.0 n_samples=np.floor(float(nx*ny)/downsampling_factor) if (mode=='random'): #- March through frequencies. --------------------------------------------- for idf in range(0,len(f),3): if verbose==1: print 'f=', f[idf], ' Hz' if (filt[idf]>0.05*np.max(filt)): #- March through downsampled spatial grid. ------------------------ t0=time.time() for idx in range(0,nx,3): for idy in range(0,ny,3): samples_x=np.random.randint(0,nx,n_samples) samples_y=np.random.randint(0,ny,n_samples) G1=g.green_input(x[samples_y,samples_x],y[samples_y,samples_x],x_line[idx],y_line[idy],omega[idf],p.dx,p.dy,p.rho,p.v,p.Q) G2=g.green_input(x[samples_y,samples_x],y[samples_y,samples_x],p.x[rec], p.y[rec], omega[idf],p.dx,p.dy,p.rho,p.v,p.Q) C[idy,idx,idf]=downsampling_factor*filt[idf]*np.sum(S[samples_y,samples_x]*G1*np.conj(G2)) t1=time.time() if verbose==1: print 'time per frequency: ', t1-t0, 's' #- Normalisation. --------------------------------------------------------- C=C*p.dx*p.dy #- Spatial interpolation. ------------------------------------------------- for idx in range(0,nx-3,3): C[:,idx+1,:]=0.67*C[:,idx,:]+0.33*C[:,idx+3,:] C[:,idx+2,:]=0.33*C[:,idx,:]+0.67*C[:,idx+3,:] for idy in range(0,ny-3,3): C[idy+1,:,:]=0.67*C[idy,:,:]+0.33*C[idy+3,:,:] C[idy+2,:,:]=0.33*C[idy,:,:]+0.67*C[idy+3,:,:] #- Frequency interpolation. ----------------------------------------------- for idf in range(0,len(f)-3,3): C[:,:,idf+1]=0.67*C[:,:,idf]+0.33*C[:,:,idf+3] C[:,:,idf+2]=0.33*C[:,:,idf]+0.67*C[:,:,idf+3] #============================================================================== #- Compute correlation field by random integration over all sources + individual sources #============================================================================== downsampling_factor=5.0 n_samples=np.floor(float(nx*ny)/downsampling_factor) if (mode=='combined'): #-------------------------------------------------------------------------- #- March through frequencies for random sampling. ------------------------- for idf in range(0,len(f),3): if verbose==1: print 'f=', f[idf], ' Hz' if (filt[idf]>0.05*np.max(filt)): #- March through downsampled spatial grid. ------------------------ t0=time.time() for idx in range(0,nx,3): for idy in range(0,ny,3): samples_x=np.random.randint(0,nx,n_samples) samples_y=np.random.randint(0,ny,n_samples) G1=g.green_input(x[samples_y,samples_x],y[samples_y,samples_x],x_line[idx],y_line[idy],omega[idf],p.dx,p.dy,p.rho,p.v,p.Q) G2=g.green_input(x[samples_y,samples_x],y[samples_y,samples_x],p.x[rec], p.y[rec], omega[idf],p.dx,p.dy,p.rho,p.v,p.Q) C[idy,idx,idf]=downsampling_factor*filt[idf]*np.sum(S[samples_y,samples_x]*G1*np.conj(G2)) t1=time.time() if verbose==1: print 'time per frequency: ', t1-t0, 's' #- Spatial interpolation. ------------------------------------------------- for idx in range(0,nx-3,3): C[:,idx+1,:]=0.67*C[:,idx,:]+0.33*C[:,idx+3,:] C[:,idx+2,:]=0.33*C[:,idx,:]+0.67*C[:,idx+3,:] for idy in range(0,ny-3,3): C[idy+1,:,:]=0.67*C[idy,:,:]+0.33*C[idy+3,:,:] C[idy+2,:,:]=0.33*C[idy,:,:]+0.67*C[idy+3,:,:] #- Frequency interpolation. ----------------------------------------------- for idf in range(0,len(f)-3,3): C[:,:,idf+1]=0.67*C[:,:,idf]+0.33*C[:,:,idf+3] C[:,:,idf+2]=0.33*C[:,:,idf]+0.67*C[:,:,idf+3] #-------------------------------------------------------------------------- #- March through the spatial grid for individual sources. ----------------- for idx in range(nx): if (verbose==1): print str(100*float(idx)/float(len(x_line)))+' %' for idy in range(ny): #- March through all sources. for k in indeces: C[idy,idx,:]+=S[k]*filt*np.conj(g.conjG1_times_G2(x[idy,idx],y[idy,idx],p.x[rec],p.y[rec],x[k],y[k],omega,p.dx,p.dy,p.rho,p.v,p.Q)) #- Normalisation. --------------------------------------------------------- C=C*p.dx*p.dy #============================================================================== #- Save interferometric wavefield. #============================================================================== fn='OUTPUT/cf_'+str(rec) fid=open(fn,'w') np.save(fid,C) fid.close()
def correlation_random(rec0=0,rec1=1,verbose=0,plot=0,save=0):
"""
cct,cct_proc,t,ccf,ccf_proc,f = correlation_random(rec0=0,rec1=1,verbose=0,plot=0,save=1)
Compute and plot correlation function based on random source summation.
INPUT:
------
rec0, rec1: indeces of the receivers used in the correlation.
plot: plot when 1.
verbose: give screen output when 1.
save: store individual correlations to OUTPUT/correlations_individual
OUTPUT:
-------
cct, t: Time-domain correlation function and time axis [N^2 s / m^4],[s].
ccf, f: Frequency-domain correlation function and frequency axis [N^2 s^2 / m^4],[1/s].
Last updated: 19 May 2016.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
#- Start time.
t1=time.time()
#- Input parameters.
p=parameters.Parameters()
#- Spatial grid.
x_line=np.arange(p.xmin,p.xmax,p.dx)
y_line=np.arange(p.ymin,p.ymax,p.dy)
x,y=np.meshgrid(x_line,y_line)
#- Compute number of samples as power of 2.
n=2.0**np.round(np.log2((p.Twindow)/p.dt))
#- Frequency axis.
df=1.0/(n*p.dt)
f=np.arange(0.0,1.0/p.dt,df)
omega=2.0*np.pi*f
#- Compute time axis
t=np.arange(-0.5*n*p.dt,0.5*n*p.dt,p.dt)
#- Compute instrument response and natural source spectrum.
S,indeces=s.space_distribution()
instrument,natural=s.frequency_distribution(f)
#- Issue some information if wanted.
if verbose==1:
print 'number of samples: '+str(n)
print 'maximum time: '+str(np.max(t))+' s'
print 'maximum frequency: '+str(np.max(f))+' Hz'
#- Warnings.
if (p.fmax>1.0/p.dt):
print 'WARNING: maximum bandpass frequency cannot be represented with this time step!'
if (p.fmin<1.0/(n*p.dt)):
print 'WARNING: minimum bandpass frequency cannot be represented with this window length!'
#==============================================================================
#- March through source locations and compute raw frequency-domain noise traces.
#==============================================================================
#- Set a specific random seed to make simulation repeatable, e.g. for different receiver pair.
np.random.seed(p.seed)
#- Initialise frequency-domain wavefields.
u1=np.zeros([n,p.Nwindows],dtype=complex)
u2=np.zeros([n,p.Nwindows],dtype=complex)
G1=np.zeros([n,p.Nwindows],dtype=complex)
G2=np.zeros([n,p.Nwindows],dtype=complex)
#- Regularise zero-frequency to avoid singularity in Green function.
omega[0]=0.01*2.0*np.pi*df
#- March through source indices.
for k in indeces:
#- Green function for a specific source point.
G1[:,0]=g.green_input(p.x[rec0],p.y[rec0],x[k],y[k],omega,p.dx,p.dy,p.rho,p.v,p.Q)
G2[:,0]=g.green_input(p.x[rec1],p.y[rec1],x[k],y[k],omega,p.dx,p.dy,p.rho,p.v,p.Q)
#- Apply instrument response and source spectrum
G1[:,0]=G1[:,0]*instrument*np.sqrt(natural)
G2[:,0]=G2[:,0]*instrument*np.sqrt(natural)
#- Copy this Green function to all time intervals.
for i in range(p.Nwindows):
G1[:,i]=G1[:,0]
G2[:,i]=G2[:,0]
#- Random phase matrix, frequency steps times time windows.
phi=2.0*np.pi*(np.random.rand(n,p.Nwindows)-0.5)
ff=np.exp(1j*phi)
#- Matrix of random frequency-domain wavefields.
u1+=S[k]*ff*G1
u2+=S[k]*ff*G2
#- March through time windows to add earthquakes.
for win in range(p.Nwindows):
neq=len(p.eq_t[win])
for i in range(neq):
G1=g.green_input(p.x[rec0],p.y[rec0],p.eq_x[win][i],p.eq_y[win][i],omega,p.dx,p.dy,p.rho,p.v,p.Q)
G2=g.green_input(p.x[rec1],p.y[rec1],p.eq_x[win][i],p.eq_y[win][i],omega,p.dx,p.dy,p.rho,p.v,p.Q)
G1=G1*instrument*np.sqrt(natural)
G2=G2*instrument*np.sqrt(natural)
u1[:,win]+=p.eq_m[win][i]*G1*np.exp(-1j*omega*p.eq_t[win][i])
u2[:,win]+=p.eq_m[win][i]*G2*np.exp(-1j*omega*p.eq_t[win][i])
#==============================================================================
#- Processing.
#==============================================================================
#- Apply single-station processing.
u1_proc,u2_proc=proc.processing_single_station(u1,u2,f,verbose)
#- Compute correlation function, raw and processed.
ccf=u1*np.conj(u2)
ccf_proc=u1_proc*np.conj(u2_proc)
#- Apply correlation processing.
ccf_proc=proc.processing_correlation(ccf_proc,f,verbose)
#==============================================================================
#- Apply the standard bandpass.
#==============================================================================
bandpass=np.zeros(np.shape(f))
Nminmax=np.round((p.bp_fmin)/df)
Nminmin=np.round((p.bp_fmin-p.bp_width)/df)
Nmaxmin=np.round((p.bp_fmax)/df)
Nmaxmax=np.round((p.bp_fmax+p.bp_width)/df)
bandpass[Nminmin:Nminmax]=np.linspace(0.0,1.0,Nminmax-Nminmin)
bandpass[Nmaxmin:Nmaxmax]=np.linspace(1.0,0.0,Nmaxmax-Nmaxmin)
bandpass[Nminmax:Nmaxmin]=1.0
for i in range(p.Nwindows):
ccf[:,i]=bandpass*ccf[:,i]
ccf_proc[:,i]=bandpass*ccf_proc[:,i]
#==============================================================================
#- Time-domain correlation function.
#==============================================================================
#- Some care has to be taken here with the inverse FFT convention of numpy.
cct=np.zeros([n,p.Nwindows],dtype=float)
cct_proc=np.zeros([n,p.Nwindows],dtype=float)
dummy=np.real(np.fft.ifft(ccf,axis=0)/p.dt)
cct[0.5*n:n,:]=dummy[0:0.5*n,:]
cct[0:0.5*n,:]=dummy[0.5*n:n,:]
dummy=np.real(np.fft.ifft(ccf_proc,axis=0)/p.dt)
cct_proc[0.5*n:n,:]=dummy[0:0.5*n,:]
cct_proc[0:0.5*n,:]=dummy[0.5*n:n,:]
#==============================================================================
#- Save results if wanted.
#==============================================================================
if save==1:
#- Store frequency and time axes.
fid=open('OUTPUT/correlations_individual/f','w')
np.save(fid,f)
fid.close()
fid=open('OUTPUT/correlations_individual/t','w')
np.save(fid,t)
fid.close()
#- Store raw and processed correlations in the frequency domain.
fn='OUTPUT/correlations_individual/ccf_'+str(rec0)+'_'+str(rec1)
fid=open(fn,'w')
np.save(fid,ccf)
fid.close()
fn='OUTPUT/correlations_individual/ccf_proc_'+str(rec0)+'_'+str(rec1)
fid=open(fn,'w')
np.save(fid,ccf_proc)
fid.close()
#==============================================================================
#- Plot results if wanted.
#==============================================================================
if plot==1:
#- Noise traces for first window.
plt.subplot(2,1,1)
plt.plot(t,np.real(np.fft.ifft(u1[:,0]))/p.dt,'k')
plt.ylabel('u1(t) [N/m^2]')
plt.title('recordings for first time window')
plt.subplot(2,1,2)
plt.plot(t,np.real(np.fft.ifft(u2[:,0]))/p.dt,'k')
plt.ylabel('u2(t) [N/m^2]')
plt.xlabel('t [s]')
plt.show()
#- Spectrum of the pressure wavefield for first window.
plt.subplot(2,1,1)
plt.plot(f,np.abs(np.sqrt(bandpass)*u1[:,0]),'k',linewidth=2)
plt.plot(f,np.real(np.sqrt(bandpass)*u1[:,0]),'b',linewidth=1)
plt.plot(f,np.imag(np.sqrt(bandpass)*u1[:,0]),'r',linewidth=1)
plt.ylabel('u1(f) [Ns/m^2]')
plt.title('raw and processed spectra for first window (abs=black, real=blue, imag=red)')
plt.subplot(2,1,2)
plt.plot(f,np.abs(np.sqrt(bandpass)*u1_proc[:,0]),'k',linewidth=2)
plt.plot(f,np.real(np.sqrt(bandpass)*u1_proc[:,0]),'b',linewidth=1)
plt.plot(f,np.imag(np.sqrt(bandpass)*u1_proc[:,0]),'r',linewidth=1)
plt.ylabel('u1_proc(f) [?]')
plt.xlabel('f [Hz]')
plt.show()
#- Raw time- and frequency-domain correlation for first window.
plt.subplot(2,1,1)
plt.semilogy(f,np.abs(ccf[:,0]),'k',linewidth=2)
plt.title('raw frequency-domain correlation for first window')
plt.ylabel('correlation [N^2 s^2 / m^4]')
plt.xlabel('f [Hz]')
plt.subplot(2,1,2)
plt.plot(t,np.real(cct[:,0]),'k')
plt.title('raw time-domain correlation for first window')
plt.ylabel('correlation [N^2 s / m^4]')
plt.xlabel('t [s]')
plt.show()
#- Processed time- and frequency-domain correlation for first window.
plt.subplot(2,1,1)
plt.semilogy(f,np.abs(ccf_proc[:,0]),'k',linewidth=2)
plt.title('processed frequency-domain correlation for first window')
plt.ylabel('correlation [N^2 s^2 / m^4]*unit(T)')
plt.xlabel('f [Hz]')
plt.subplot(2,1,2)
plt.plot(t,np.real(cct_proc[:,0]))
plt.title('processed time-domain correlation for first window')
plt.ylabel('correlation [N^2 s / m^4]*unit(T)')
plt.xlabel('t [s]')
plt.show()
#- Raw and processed ensemble correlations.
plt.plot(t,np.sum(cct,1)/np.max(cct),'k')
plt.plot(t,np.sum(cct_proc,1)/np.max(cct_proc),'r')
plt.title('ensemble time-domain correlation (black=raw, red=processed)')
plt.ylabel('correlation [N^2 s / m^4]*unit(T)')
plt.xlabel('t [s]')
plt.show()
#- End time.
t2=time.time()
if verbose==1:
print 'elapsed time: '+str(t2-t1)+' s'
#==============================================================================
#- Output.
#==============================================================================
return cct,cct_proc,t,ccf,ccf_proc,f
def correlation_function(rec0=0,rec1=1,effective=0,plot=0):
"""
cct, t, ccf, f = correlation_function(rec0=0,rec1=1,effective=0,plot=0)
Compute time- and frequency-domain correlation functions.
INPUT:
------
rec0, rec1: indeces of the receivers used in the correlation.
plot: When plot=1, the source distribution, and the time- and frequency domain correlation functions are plotted.
effective: When effective==1, effective correlations are computed using the propagation correctors stored in OUTPUT/correctors.
The source power-spectral density is then interpreted as the effective one.
OUTPUT:
-------
cct, t: Time-domain correlation function and time axis [N^2 s / m^4],[s].
ccf, f: Frequency-domain correlation function and frequency axis [N^2 s^2 / m^4],[1/s].
Last updated: 27 May 2016.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
p=parameters.Parameters()
x_line=np.arange(p.xmin,p.xmax,p.dx)
y_line=np.arange(p.ymin,p.ymax,p.dy)
x,y=np.meshgrid(x_line,y_line)
f=np.arange(p.fmin-p.fwidth,p.fmax+p.fwidth,p.df)
omega=2.0*np.pi*f
t=np.arange(p.tmin,p.tmax,p.dt)
#- Frequency- and space distribution of the source. ---------------------------
S,indices=s.space_distribution(plot)
instrument,natural=s.frequency_distribution(f)
filt=natural*instrument*instrument
#- Read propagation corrector if needed. --------------------------------------
if (effective==1):
gf=gpc.get_propagation_corrector(rec0,rec1,plot=0)
else:
gf=np.ones(len(f),dtype=complex)
#==============================================================================
#- Compute inter-station correlation function.
#==============================================================================
cct=np.zeros(np.shape(t),dtype=float)
ccf=np.zeros(np.shape(f),dtype=complex)
for idf in range(len(omega)):
P=g.conjG1_times_G2(p.x[rec0],p.y[rec0],p.x[rec1],p.y[rec1],x,y,omega[idf],p.dx,p.dy,p.rho,p.v,p.Q)
ccf[idf]=gf[idf]*np.conj(np.sum(P*S))
cct=cct+np.real(filt[idf]*ccf[idf]*np.exp(1j*omega[idf]*t))
cct=cct*p.dx*p.dy*p.df
#==============================================================================
#- Plot result.
#==============================================================================
if (plot==1):
#- Frequency domain.
plt.semilogy(f,np.abs(ccf),'k')
plt.semilogy(f,np.real(ccf),'b')
plt.title('frequency-domain correlation function (black=abs, blue=real)')
plt.xlabel('frequency [Hz]')
plt.ylabel('correlation [N^2 s^2/m^4]')
plt.show()
#- Time domain.
tt=np.sqrt((p.x[rec0]-p.x[rec1])**2+(p.y[rec0]-p.y[rec1])**2)/p.v
cct_max=np.max(np.abs(cct))
plt.plot(t,cct,'k',linewidth=2.0)
plt.plot([tt,tt],[-1.1*cct_max,1.1*cct_max],'--',color=(0.5,0.5,0.5),linewidth=1.5)
plt.plot([-tt,-tt],[-1.1*cct_max,1.1*cct_max],'--',color=(0.5,0.5,0.5),linewidth=1.5)
plt.ylim((-1.1*cct_max,1.1*cct_max))
plt.title('correlation function')
plt.xlabel('time [s]')
plt.ylabel('correlation [N^2 s/m^4]')
plt.show()
#==============================================================================
#- Return.
#==============================================================================
return cct, t, ccf, f
def precompute(rec=0, verbose=0):
"""
precompute(verbose=0)
Compute correlation wavefield in the frequency domain and store for in /OUTPUT for re-use in kernel computation.
INPUT:
------
rec: index of reference receiver.
verbose: give screen output when 1.
OUTPUT:
-------
Frequency-domain interferometric wavefield stored in /OUTPUT.
Last updated: 16 May 2016.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
p = parameters.Parameters()
#- Spatial grid.
x_line = np.arange(p.xmin, p.xmax, p.dx)
y_line = np.arange(p.ymin, p.ymax, p.dy)
x, y = np.meshgrid(x_line, y_line)
#- Frequency line.
f = np.arange(p.fmin - p.fwidth, p.fmax + p.fwidth, p.df)
omega = 2.0 * np.pi * f
#- Power-spectral density.
S, indeces = s.space_distribution()
instrument, natural = s.frequency_distribution(f)
filt = natural * instrument * instrument
#- Check if the indeces are actually available. If not, interrupt.
if len(indeces) == 0:
print 'Correlation field cannot be computed because source index array is empty.'
return
C = np.zeros((len(y_line), len(x_line), len(omega)), dtype=complex)
#==============================================================================
#- Compute correlation field for all frequencies and store.
#==============================================================================
#- March through the spatial grid. --------------------------------------------
for idx in range(len(x_line)):
if (verbose == 1):
print str(100 * float(idx) / float(len(x_line))) + ' %'
t1 = time.time()
for idy in range(len(y_line)):
#- March through all sources.
for k in indeces:
C[idy, idx, :] += S[k] * filt * g.conjG1_times_G2(
x[idy, idx], y[idy, idx], p.x[rec], p.y[rec], x[k], y[k],
omega, p.dx, p.dy, p.rho, p.v, p.Q)
C[idy, idx, :] = np.conj(C[idy, idx, :])
t2 = time.time()
print t2 - t1
#- Normalisation.
C = C * p.dx * p.dy
#==============================================================================
#- Save interferometric wavefield.
#==============================================================================
fn = 'OUTPUT/cf_' + str(rec)
fid = open(fn, 'w')
np.save(fid, C)
fid.close()
def correlation_function(rec0=0,
rec1=1,
effective=False,
plot=False,
save=False):
"""
cct, t, ccf, f = correlation_function(rec0=0,rec1=1,effective=False,plot=False,save=False)
Compute time- and frequency-domain correlation functions.
INPUT:
------
rec0, rec1: indeces of the receivers used in the correlation.
plot: When plot=True, the source distribution, and the time- and frequency domain correlation functions are plotted.
save: Save pdf figures to OUTPUT when True.
effective: When effective==True, effective correlations are computed using the propagation correctors stored in OUTPUT/correctors.
The source power-spectral density is then interpreted as the effective one.
OUTPUT:
-------
cct, t: Time-domain correlation function and time axis [N^2 s / m^4],[s].
ccf, f: Frequency-domain correlation function and frequency axis [N^2 s^2 / m^4],[1/s].
Last updated: 16 July 2019.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
p = parameters.Parameters()
x_line = np.arange(p.xmin, p.xmax, p.dx)
y_line = np.arange(p.ymin, p.ymax, p.dy)
x, y = np.meshgrid(x_line, y_line)
f = np.arange(p.fmin - p.fwidth, p.fmax + p.fwidth, p.df)
omega = 2.0 * np.pi * f
t = np.arange(p.tmin, p.tmax, p.dt)
#- Frequency- and space distribution of the source. ---------------------------
S, indices = s.space_distribution(plot=plot, save=save)
instrument, natural = s.frequency_distribution(f)
filt = natural * instrument * instrument
#- Read propagation corrector if needed. --------------------------------------
if effective:
gf = gpc.get_propagation_corrector(rec0, rec1, plot=False)
else:
gf = np.ones(len(f), dtype=complex)
#==============================================================================
#- Compute inter-station correlation function.
#==============================================================================
cct = np.zeros(np.shape(t), dtype=float)
ccf = np.zeros(np.shape(f), dtype=complex)
for idf in range(len(omega)):
P = g.conjG1_times_G2(p.x[rec0], p.y[rec0], p.x[rec1], p.y[rec1], x, y,
omega[idf], p.dx, p.dy, p.rho, p.v, p.Q)
ccf[idf] = gf[idf] * np.conj(np.sum(P * S))
cct = cct + np.real(filt[idf] * ccf[idf] * np.exp(1j * omega[idf] * t))
cct = cct * p.dx * p.dy * p.df
#==============================================================================
#- Plot result.
#==============================================================================
if (plot or save):
plt.rcParams["font.family"] = "serif"
plt.rcParams.update({'font.size': 10})
#- Frequency domain.
plt.semilogy(f, np.abs(ccf), 'k')
plt.semilogy(f, np.real(ccf), 'b')
plt.title(
'frequency-domain correlation function (black=abs, blue=real)')
plt.xlabel('frequency [Hz]')
plt.ylabel(r'correlation [N$^2$ s$^2$/m$^4$]')
if plot:
plt.show()
else:
fn = 'OUTPUT/correlations_computed/c_frequency_domain_' + str(
rec0) + '-' + str(rec1) + '.pdf'
plt.savefig(fn, format='pdf')
plt.clf()
#- Time domain.
tt = np.sqrt((p.x[rec0] - p.x[rec1])**2 +
(p.y[rec0] - p.y[rec1])**2) / p.v
cct_max = np.max(np.abs(cct))
plt.plot(t, cct, 'k', linewidth=2.0)
plt.plot([tt, tt], [-1.1 * cct_max, 1.1 * cct_max],
'--',
color=(0.5, 0.5, 0.5),
linewidth=1.5)
plt.plot([-tt, -tt], [-1.1 * cct_max, 1.1 * cct_max],
'--',
color=(0.5, 0.5, 0.5),
linewidth=1.5)
plt.ylim((-1.1 * cct_max, 1.1 * cct_max))
plt.title('correlation function')
plt.xlabel('time [s]')
plt.ylabel(r'correlation [N$^2$ s/m$^4$]')
if plot:
plt.show()
else:
fn = 'OUTPUT/correlations_computed/c_time_domain_' + str(
rec0) + '-' + str(rec1) + '.pdf'
plt.savefig(fn, format='pdf')
plt.clf()
#==============================================================================
#- Save results if wanted.
#==============================================================================
if save == 1:
#- Store frequency and time axes.
fn = 'OUTPUT/correlations_computed/t'
np.save(fn, t)
fn = 'OUTPUT/correlations_computed/t'
np.save(fn, t)
#- Store computed correlations in the time and frequency domain.
fn = 'OUTPUT/correlations_computed/cct_' + str(rec0) + '-' + str(rec1)
np.save(fn, cct)
fn = 'OUTPUT/correlations_computed/ccf_' + str(rec0) + '-' + str(rec1)
np.save(fn, ccf)
#==============================================================================
#- Return.
#==============================================================================
return cct, t, ccf, f
def structure_kernel(cct, t, rec0=0, rec1=1, measurement='cctime', dir_forward='OUTPUT/', effective=0, plot=0):
"""
x,y,K_kappa = structure_kernel(cct, t, rec0=0, rec1=1, measurement='cctime', dir_forward='OUTPUT/', effective=0, plot=0):
Compute structure kernel K_kappa for a frequency-independent source power-spectral density.
INPUT:
------
cct, t: Time-domain correlation function and time axis as obtained from correlation_function.py.
rec0, rec1: Indeces of the receivers used in the correlation.
measurement: Type of measurement used to compute the adjoint source. See adsrc.py for options.
dir_forward: Location of the forward interferometric fields from rec0 and rec1. Must exist.
plot: When plot=1, plot structure kernel.
effective: When effective==1, effective correlations are computed using the propagation correctors stored in OUTPUT/correctors.
The source power-spectral density is then interpreted as the effective one.
OUTPUT:
-------
x,y: Space coordinates.
K: Structure kernel [unit of measurement * 1/N].
Last updated: 11 July 2016.
"""
#==============================================================================
#- Initialisation.
#==============================================================================
p=parameters.Parameters()
x_line=np.arange(p.xmin,p.xmax,p.dx)
y_line=np.arange(p.ymin,p.ymax,p.dy)
x,y=np.meshgrid(x_line,y_line)
f=np.arange(p.fmin-p.fwidth,p.fmax+p.fwidth,p.df)
df=f[1]-f[0]
omega=2.0*np.pi*f
K_kappa=np.zeros(np.shape(x))
C1=np.zeros((len(y_line),len(x_line),len(omega)),dtype=complex)
C2=np.zeros((len(y_line),len(x_line),len(omega)),dtype=complex)
nx=len(x_line)
ny=len(y_line)
kappa=p.rho*(p.v**2)
#- Frequency- and space distribution of the source. ---------------------------
S,indeces=s.space_distribution(plot=0)
instrument,natural=s.frequency_distribution(f)
filt=natural*instrument*instrument
#- Compute the adjoint source. ------------------------------------------------
a=adsrc.adsrc(cct, t, measurement, plot)
#- Compute number of grid points corresponding to the minimum wavelength. -----
L=int(np.ceil(p.v/(p.fmax*p.dx)))
#- Read propagation corrector if needed. --------------------------------------
if (effective==1):
gf=gpc.get_propagation_corrector(rec0,rec1,plot=0)
else:
gf=np.ones(len(f),dtype=complex)
#==============================================================================
#- Load forward interferometric wavefields.
#==============================================================================
fn=dir_forward+'/cf_'+str(rec0)
fid=open(fn,'r')
C1=np.load(fid)
fid.close()
fn=dir_forward+'/cf_'+str(rec1)
fid=open(fn,'r')
C2=np.load(fid)
fid.close()
#==============================================================================
#- Loop over frequencies.
#==============================================================================
for k in range(len(omega)):
w=omega[k]
#- Adjoint fields. --------------------------------------------------------
G1=-w**2*g.green_input(x,y,p.x[rec0],p.y[rec0],w,p.dx,p.dy,p.rho,p.v,p.Q)*gf[k]
G2=-w**2*g.green_input(x,y,p.x[rec1],p.y[rec1],w,p.dx,p.dy,p.rho,p.v,p.Q)*gf[k]
#- Multiplication with adjoint fields. ------------------------------------
K_kappa=K_kappa-2.0*np.real(G2*C1[:,:,k]*np.conj(a[k])+G1*C2[:,:,k]*a[k])
K_kappa=K_kappa/kappa
#==============================================================================
#- Smooth over minimum wavelength.
#==============================================================================
for k in range(L):
K_kappa[1:ny-2,:]=(K_kappa[1:ny-2,:]+K_kappa[0:ny-3,:]+K_kappa[2:ny-1,:])/3.0
for k in range(L):
K_kappa[:,1:nx-2]=(K_kappa[:,1:nx-2]+K_kappa[:,0:nx-3]+K_kappa[:,2:nx-1])/3.0
#==============================================================================
#- Visualise if wanted.
#==============================================================================
if plot==1:
cmap = plt.get_cmap('RdBu')
plt.pcolormesh(x,y,K_kappa,cmap=cmap,shading='interp')
plt.clim(-np.max(np.abs(K_kappa))*0.25,np.max(np.abs(K_kappa))*0.25)
plt.axis('image')
plt.colorbar()
plt.title('Structure (kappa) kernel [unit of measurement / N]')
plt.xlabel('x [km]')
plt.ylabel('y [km]')
plt.plot(p.x[rec0],p.y[rec0],'ro')
plt.plot(p.x[rec1],p.y[rec1],'ro')
plt.show()
return x,y,K_kappa