def fktrafo(stream, normalize=True): """ Calculates the f,k - transformation of the data in stream. Returns the trafo as an array. :param st: Stream :type st: obspy.core.stream.Stream :param inv: inventory :type inv: obspy.station.inventory.Inventory :param event: Event :type event: obspy.core.event.Event returns :param fkdata: f,k - transformation of data in stream :type fkdata: numpyndarray """ st_tmp = stream.copy() ArrayData = stream2array(st_tmp, normalize) ix = ArrayData.shape[0] iK = int(math.pow(2,nextpow2(ix))) it = ArrayData.shape[1] iF = int(math.pow(2,nextpow2(it))) fkdata = np.fft.fft2(ArrayData, s=(iK,iF)) return fkdata
def ifktrafo(fkdata, stream, normalize=True): """ Calculates the inverse f,k - transformation of the data in fkdata. Returns the trafo as an array. """ StreamData= stream2array(stream) ix = StreamData.shape[0] iK = int(math.pow(2,nextpow2(ix))) it = StreamData.shape[1] iF = int(math.pow(2,nextpow2(it))) fk_tmp = fkdata.copy() ArrayData = np.fft.ifft2(fkdata, s=(iK,iF)) ArrayData = ArrayData[0:ix, 0:it] return ArrayData
def pocs(data, maxiter, noft, alpha=0.9, beta=None, method='linear', dmethod='denoise', peaks=None, maskshape=None, dt=None, p=None, flow=None, fhigh=None, slidingwindow=False, overlap=0.5):
"""
This functions reconstructs missing signals in the f-k domain, using the original data,
including gaps, filled with zeros. It applies the projection onto convex sets (pocs) algorithm in
2D.
Reference: 3D interpolation of irregular data with a POCS algorithm, Abma & Kabir, 2006
:param data:
:type data:
:param maxiter:
:type maxiter:
:param nol: Number of loops
:type nol:
:param alpha: Factor of threshold decrease after each iteration
:type alpha: float
:param method: Method to be used for the pocs algorithm
-'linear', 'exp', 'mask' or 'ssa'
:param peaks: Slope values for the mask
:param maskshape: Shape of the corners of mask, see makemask
returns:
:param datap:
:type datap:
"""
#if not decrease in ('linear', 'exp', None):
# msg='No decrease method chosen'
# raise IOError(msg)
ArrayData = data.copy()
ix = ArrayData.shape[0]
iK = int(math.pow(2,nextpow2(ix)))
it = ArrayData.shape[1]
iF = int(math.pow(2,nextpow2(it)))
fkdata = np.fft.fft2(ArrayData, s=(iK,iF))
threshold = abs(fkdata.max())
ADold = ArrayData.copy()
ADnew = ArrayData.copy()
ADfinal = np.zeros(ArrayData.shape).astype('complex')
if method in ('linear', 'exp'):
if slidingwindow:
if dmethod in ('reconstruct'):
w_length = int(data.shape[1] / 3.)
swh = np.hanning(w_length)
loc = 0.
inside = True
while inside:
curr_win = int(loc)
ADtemp = ArrayData[:,curr_win:curr_win+w_length].copy()
for i in range(maxiter):
data_tmp = ADtemp.copy()
fkdata = np.fft.fft2(data_tmp, s=(iK,iF))
fkdata[ np.where(abs(fkdata) < threshold)] = 0. + 0j
if method in ('linear'):
threshold = threshold * alpha
elif method in ('exp'):
threshold = threshold * sp.exp(-(i+1) * alpha)
data_tmp = np.fft.ifft2(fkdata, s=(iK,iF)).real[0:ix, 0:it].copy()
ADtemp[noft] = data_tmp[noft][:,curr_win:curr_win+w_length].copy()
if loc == 0.:
ADfinal[:,curr_win:curr_win+w_length] = ADtemp.copy()
else:
ADfinal[:,curr_win:curr_win+w_length] = ADfinal[:,curr_win:curr_win+w_length] + ADtemp
ADfinal[:,curr_win-int(overlap*w_length):int(curr_win)] = ( ADold[:,int((1-overlap)*w_length):] + ADtemp[:,:int(overlap*w_length)] ) / 2.
ADold = ADtemp.copy()
threshold = abs(np.fft.fft2(ADold, s=(iK,iF)).max())
loc += overlap * w_length
print(loc)
if loc >= data.shape[1]: inside=False
else:
if dmethod in ('reconstruct'):
ADtemp = ArrayData.copy()
for i in range(maxiter):
data_tmp = ADtemp.copy()
fkdata = np.fft.fft2(data_tmp, s=(iK,iF))
fkdata[ np.where(abs(fkdata) < threshold)] = 0. + 0j
if method in ('linear'):
threshold = threshold * alpha
elif method in ('exp'):
threshold = threshold * sp.exp(-(i+1) * alpha)
data_tmp = np.fft.ifft2(fkdata, s=(iK,iF)).real[0:ix, 0:it].copy()
ADtemp[noft] = data_tmp[noft]
name = str(i) + '.png'
# plt.ion()
# plotfk(fkdata)
# fig = plt.gcf()
# fig.set_size_inches(7,8)
# fig.savefig(name)
# plt.ioff()
ADfinal = ADtemp.copy()
threshold = abs(np.fft.fft2(ADfinal, s=(iK,iF)).max())
elif dmethod in ('denoise', 'de-noise'):
for n in noft:
ADtemp = ArrayData.copy()
for i in range(maxiter):
data_tmp = ADtemp.copy()
fkdata = np.fft.fft2(data_tmp, s=(iK,iF))
fkdata[ np.where(abs(fkdata) < threshold)] = 0. + 0j
if method in ('linear'):
threshold = threshold * alpha
elif method in ('exp'):
threshold = threshold * sp.exp(-(i+1) * alpha)
data_tmp = np.fft.ifft2(fkdata, s=(iK,iF)).real[0:ix, 0:it].copy()
ADtemp[n] = data_tmp[n]
#save = 'pocsdata' + str(i) + '.png'
#plot(data_tmp, ylabel='Distance(m)', xlabel='Time(s)', fs=22, yinfo=2, savefig=save)
ADfinal = ADtemp.copy()
threshold = abs(np.fft.fft2(ADfinal, s=(iK,iF)).max())
elif method in ('mask'):
W = makeMask(fkdata, peaks[0], maskshape)
ADfinal = ArrayData.copy()
for n in noft:
ADtemp = ArrayData.copy()
threshold = abs(W*np.fft.fft2(ADfinal, s=(iK,iF))).max()
for i in range(maxiter):
data_tmp =ADtemp.copy()
fkdata = W * np.fft.fft2(data_tmp, s=(iK,iF))
fkdata[ np.where(abs(fkdata) < threshold)] = 0. + 0j
threshold = threshold * alpha
data_tmp = np.fft.ifft2(fkdata, s=(iK,iF)).real[0:ix, 0:it].copy()
ADtemp[n] = data_tmp[n]
ADfinal[n] = ADtemp[n].copy()
elif method in ('ssa'):
for n in noft:
for i in range(maxiter):
data_tmp = ArrayData.copy()
data_ssa = fx_ssa(data_tmp,dt,p,flow,fhigh)
ArrayData = alpha * ArrayData
ArrayData[n] = (1. - alpha) * data_ssa[n]
ADfinal[n] = ArrayData[n].copy()
elif method in ('average'):
threshold = beta * abs(np.fft.fft2(ArrayData, s=(iK,iF)).max())
ADtemp = ArrayData.copy()
for n in noft:
for i in range(maxiter):
data_tmp = ADtemp.copy()
fkdata = np.fft.fft2(data_tmp, s=(iK,iF))
fkdata[ np.where(abs(fkdata) < threshold)] = 0. + 0j
ADtemp = alpha*data_tmp + (1. - alpha) * np.fft.ifft2(fkdata, s=(iK,iF)).real[0:ix, 0:it]
ADtemp[n] = (1. - alpha) * np.fft.ifft2(fkdata, s=(iK,iF)).real[0:ix, 0:it][n]
ADfinal = ADtemp.copy()
elif method == 'maskvary':
ADfinal = ArrayData.copy()
ADtemp = ArrayData.copy()
for n in noft:
for i in range(maxiter):
W = makeMask(fkdata, peaks, shape=maskshape, expl_cutoff=i)
data_tmp = ADtemp.copy()
fkdata = W * np.fft.fft2(data_tmp, s=(iK,iF))
data_tmp = np.fft.ifft2(fkdata, s=(iK,iF)).real[0:ix, 0:it].copy()
ADtemp[n] = alpha * ArrayData[n].copy()
ADtemp[n] += (1. - alpha) * data_tmp[n]
ADfinal[n] = ArrayData[n].copy()
else:
print('no method specified')
return
datap = ADfinal.copy()
return datap
def fk_filter(st, inv=None, event=None, ftype='eliminate', fshape=['butterworth', 4, 4], phase=None, polygon=4, normalize=True, stack=False,
slopes=[-3,3], deltaslope=0.05, slopepicking=False, smoothpicks=False, dist=0.5, maskshape=['boxcar',None],
order=4., peakinput=False, eval_mean=1, fs=25):
"""
Import stream, the function applies an 2D FFT, removes a certain window around the
desired phase to surpress a slownessvalue corresponding to a wavenumber and applies an 2d iFFT.
To fill the gap between uneven distributed stations use array_util.gaps_fill_zeros(). A method to interpolate the signals in the
fk-domain is beeing build, also a method using a norm minimization method.
Alternative is an nonequidistant 2D Lombard-Scargle transformation.
param st: Stream
type st: obspy.core.stream.Stream
param inv: inventory
type inv: obspy.station.inventory.Inventory
param event: Event
type event: obspy.core.event.Event
param ftype: type of method, default is 'eliminate-polygon', possible inputs are:
-eliminate
-extract
-eliminate-polygon
-extract-polygon
-mask
-fk
type ftype: string
param fshape: fshape[0] describes the shape of the fk-filter in case of ftype is 'eliminate' or 'extract'. Possible inputs are:
-spike (default)
-boxcar
-taper
-butterworth
fshape[1] is an additional attribute to the shape of taper and butterworth, for:
-taper: fshape[1] = slope of sides
-butterworth: fshape[1] = number of poles
fshape[3] describes the length of the filter shape, respectivly wavenumber corner points around k=0,
e.g.: fshape['taper', 2, 4] produces a symmetric taper with slope of side = 2, where the signal is reduced about 50% at k=+-2
type fshape: list
param phase: name of the phase to be investigated
type phase: string
param polygon: number of vertices of polygon for fk filter, only needed
if ftype is set to eliminate-polygon or extract-polygon.
Default is 12.
type polygon: int
param normalize: normalize data to 1
type normalize: bool
param SSA: Force SSA algorithm or let it check, default:False
type SSA: bool
param eval_mean: number of linear events used to calculate the average of the area in the fk domain.
returns: stream_filtered, the filtered stream.
References: Yilmaz, Thomas
Author: S. Schneider 2016
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published
by the Free Software Foundation, either version 3 of the License, or
any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details: http://www.gnu.org/licenses/
"""
# Convert format and prepare Variables.
# Check for Data type of variables.
if not type(st ) == Stream:
print( "Wrong input type of stream, must be obspy.core.stream.Stream" )
raise TypeError
if len(fshape) == 1:
fshape = [fshape[0], None, None]
st_tmp = st.copy()
ArrayData = stream2array(st_tmp, normalize)
ix = ArrayData.shape[0]
iK = int(math.pow(2,nextpow2(ix)))
try:
yinfo = epidist2nparray(attach_epidist2coords(inv, event, st_tmp))
dx = (yinfo.max() - yinfo.min() + 1) / yinfo.size
k_axis = np.fft.fftfreq(iK, dx)
except:
try:
ymax = st_tmp[0].stats.distance
ymin = st_tmp[0].stats.distance
for trace in st_tmp:
if trace.stats.distance > ymax: ymax = trace.stats.distance
if trace.stats.distance < ymin: ymin = trace.stats.distance
dx = (ymax - ymin + 1) / len(st_tmp)
k_axis = np.fft.fftfreq(iK, dx)
except:
print("\nNo inventory or event-information found. \nContinue without specific distance and wavenumber information.")
yinfo=None
dx=None
k_axis=None
it = ArrayData.shape[1]
iF = int(math.pow(2,nextpow2(it)))
dt = st_tmp[0].stats.delta
f_axis = np.fft.fftfreq(iF,dt)
# Calc mean diff of each epidist entry if it is reasonable
# do a partial stack and apply filter.
"""
2D Frequency-Space / Wavenumber-Frequency Filter #########################################################
"""
# 2D f-k Transformation
# Note array_fk has f on the x-axis and k on the y-axis!!!
# For interaction the conj.-transposed Array is shown!!!
# Decide when to use SSA to fill the gaps, calc mean distance of each epidist entry
# if it differs too much --> SSA
if ftype in ("eliminate"):
if phase:
if not isinstance(event, Event) and not isinstance(inv, Inventory):
msg='For alignment on phase calculation inventory and event information is needed, not found.'
raise IOError(msg)
st_al = alignon(st_tmp, inv, event, phase)
ArrayData = stream2array(st_al, normalize)
array_fk = np.fft.fft2(ArrayData, s=(iK,iF))
array_filtered_fk = line_set_zero(array_fk, shape=fshape)
else:
array_fk = np.fft.fft2(ArrayData, s=(iK,iF))
array_filtered_fk = line_set_zero(array_fk, shape=fshape)
elif ftype in ("extract"):
if phase:
if not isinstance(event, Event) and not isinstance(inv, Inventory):
msg='For alignment on phase calculation inventory and event information is needed, not found.'
raise IOError(msg)
st_al = alignon(st_tmp, inv, event, phase)
ArrayData = stream2array(st_al, normalize)
array_fk = np.fft.fft2(ArrayData, s=(iK,iF))
array_filtered_fk = line_cut(array_fk, shape=fshape)
else:
array_fk = np.fft.fft2(ArrayData, s=(iK,iF))
array_filtered_fk = line_cut(array_fk, shape=fshape)
elif ftype in ("eliminate-polygon"):
array_fk = np.fft.fft2(ArrayData, s=(iK,iF))
if phase:
if not isinstance(event, Event) and not isinstance(inv, Inventory):
msg='For alignment on phase calculation inventory and event information is needed, not found.'
raise IOError(msg)
st_al = alignon(st_tmp, inv, event, phase)
ArrayData = stream2array(st_al, normalize)
array_fk = np.fft.fft2(ArrayData, s=(iK,iF))
array_filtered_fk = _fk_eliminate_polygon(array_fk, polygon, ylabel=r'frequency domain f in Hz', \
yticks=f_axis, xlabel=r'wavenumber domain k in $\frac{1}{^{\circ}}$', xticks=k_axis, eval_mean=eval_mean, fs=fs)
else:
array_filtered_fk = _fk_eliminate_polygon(array_fk, polygon, ylabel=r'frequency domain f in Hz', \
yticks=f_axis, xlabel=r'wavenumber domain k in $\frac{1}{^{\circ}}$', xticks=k_axis, eval_mean=eval_mean, fs=fs)
elif ftype in ("extract-polygon"):
array_fk = np.fft.fft2(ArrayData, s=(iK,iF))
if phase:
if not isinstance(event, Event) and not isinstance(inv, Inventory):
msg='For alignment on phase calculation inventory and event information is needed, not found.'
raise IOError(msg)
st_al = alignon(st_tmp, inv, event, phase)
ArrayData = stream2array(st_al, normalize)
array_fk = np.fft.fft2(ArrayData, s=(iK,iF))
array_filtered_fk = _fk_extract_polygon(array_fk, polygon, ylabel=r'frequency domain f in Hz', \
yticks=f_axis, xlabel=r'wavenumber domain k in $\frac{1}{^{\circ}}$', xticks=k_axis, eval_mean=eval_mean, fs=fs)
else:
array_filtered_fk = _fk_extract_polygon(array_fk, polygon, ylabel=r'frequency domain f in Hz', \
yticks=f_axis, xlabel=r'wavenumber domain k in $\frac{1}{^{\circ}}$', xticks=k_axis, eval_mean=eval_mean, fs=fs)
elif ftype in ("mask"):
array_fk = np.fft.fft2(ArrayData)
M, prange, peaks = slope_distribution(array_fk, slopes, deltaslope, peakpick=None, mindist=dist, smoothing=smoothpicks, interactive=slopepicking)
W = makeMask(array_fk, peaks[0], maskshape)
array_filtered_fk = array_fk * W
array_filtered = np.fft.ifft2(array_filtered_fk)
stream_filtered = array2stream(array_filtered, st_original=st.copy())
return stream_filtered, array_fk, W
elif ftype in ("fk"):
if phase:
if not isinstance(event, Event) and not isinstance(inv, Inventory):
msg='For alignment on phase calculation inventory and event information is needed, not found.'
raise IOError(msg)
st_al = alignon(st_tmp, inv, event, phase)
ArrayData = stream2array(st_al, normalize)
array_fk = np.fft.fft2(ArrayData, s=(iK,iF))
### BUILD DOUBLE TAPER ###
#array_filtered_fk =
else:
array_fk = np.fft.fft2(ArrayData, s=(iK,iF))
### BUILD DOUBLE TAPER ###
#array_filtered_fk =
else:
print("No type of filter specified")
raise TypeError
array_filtered = np.fft.ifft2(array_filtered_fk, s=(iK,iF)).real
# Convert to Stream object.
array_filtered = array_filtered[0:ix, 0:it]
stream_filtered = array2stream(array_filtered, st_original=st.copy())
return stream_filtered
def radon_inverse(st, inv, event, p, weights, line_model, inversion_model, hyperparameters):
"""
This function inverts move-out data to the Radon domain given the inputs:
:param st:
:param inv:
:param event:
:param p: -- vector of slowness axis you would like to invert to.
:param weights: -- weighting vector that determines importance of each trace.
set vector to ones for no preference.
:param line_model: select one of the following options for path integration:
'linear' - linear paths in the spatial domain (default)
'parabolic' - parabolic paths in the spatial domain.
:param inversion model: select one of the following options for regularization schema:
'L2' - Regularized on the L2 norm of the Radon domain (default)
'L1' - Non-linear regularization based on L1 norm and iterative
reweighted least sqaures (IRLS) see Sacchi 1997.
'Cauchy' - Non-linear regularization see Sacchi & Ulrych 1995
:param hyperparameters: trades-off between fitting the data and chosen damping.
returns: radon domain is ordered size(R)==[length(p),length(t)], time-axis and distance-axis.
Known limitations:
- Assumes evenly sampled time axis.
- Assumes move-out data isn't complex.
References: Schultz, R., Gu, Y. J., 2012. Flexible Matlab implementation
of the Radon Transform. Computers and Geosciences.
An, Y., Gu, Y. J., Sacchi, M., 2007. Imaging mantle
discontinuities using least-squares Radon transform.
Journal of Geophysical Research 112, B10303.
Author: R. Schultz, 2012
Translated to Python by: S. Schneider, 2016
"""
# Check for Data type of variables.
if not isinstance(st, Stream) or not isinstance(inv, Inventory) or not isinstance(event, Event):
msg = "Wrong input type must be obspy Stream, Inventory and Event"
raise TypeError
if not isinstance(hyperparameters,list):
msg = "Wrong input type of mu, must be list"
raise TypeError
# Define some array/matrices lengths.
st_tmp = st.copy()
M = stream2array(st_tmp)
epi = epidist2nparray(attach_epidist2coords(inv, event, st_tmp))
delta = np.array([ epi.copy() ])
ref_dist = np.mean(delta)
if not weights:
weights = np.ones(delta.size)
t = np.linspace(0,st_tmp[0].stats.delta * st_tmp[0].stats.npts, st_tmp[0].stats.npts)
it=t.size
print(it)
iF=int(math.pow(2,nextpow2(it)+1)) # Double length
iDelta=delta.size
ip=len(p)
iw=len(weights)
#Exit if inconsistent data is input.
if M.shape != (iDelta, it):
print("Dimensions inconsistent!\nShape of M is not equal to (len(delta),len(t)) \nShape of M = (%i , %i)\n(len(delta),len(t)) = (%i, %i) \n" % (M.shape[0], M.shape[1], iDelta, it) )
R=0
return(R)
if iw != iDelta:
print("Dimensions inconsistent!\nlen(delta) ~= len(weights)\nlen(delta) = %i\nlen(weights) = %i\n" % (iDelta, iw))
R=0
return(R)
#Exit if improper hyperparameters are entered.
if inversion_model in ["L1", "Cauchy"]:
if not len(hyperparameters == 2):
print("Improper number of trade-off parameters\n")
R=0
return(R)
else: #The code's default is L2 inversion.
if not len(hyperparameters) == 1:
print("Improper number of trade-off parameters\n")
R=0
return(R)
#Preallocate space in memory.
R=np.zeros((ip,it))
Rfft=np.zeros((ip,iF)) + 0j
A=np.zeros((iDelta,ip)) + 0j
Tshift=np.zeros((iDelta,ip)) + 0j
AtA=np.zeros((ip,ip)) + 0j
AtM=np.zeros((ip,1)) + 0j
Ident=np.identity(ip)
#Define some values
Dist_array=delta-ref_dist
dF=1./(t[0]-t[1])
Mfft=np.fft.fft(M,iF,1)
W=sparse.spdiags(weights.conj().transpose(), 0, iDelta, iDelta).A
dCOST=0.
COST_curv=0.
COST_prev=0.
#Populate ray parameter then distance data in time shift matrix.
for j in range(iDelta):
if line_model == "parabolic":
Tshift[j]=p
else: #Linear is default
Tshift[j]=p
for k in range(ip):
if line_model == 'parabolic':
Tshift[:,k]=(2. * ref_dist * Tshift[:,k] * Dist_array.conj().transpose()) + (Tshift[:,k] * (Dist_array**2).conj().transpose())
else: #Linear is default
Tshift[:,k]=Tshift[:,k] * Dist_array[0].conj().transpose()
# Loop through each frequency.
for i in range( int(math.floor((iF+1)/2)) ):
print('Step %i of %i' % (i, int(math.floor((iF+1)/2))) )
# Make time-shift matrix, A.
f = ((float(i)/float(iF))*dF)
A = np.exp( (0.+1j)*2*pi*f * Tshift )
# M = A R ---> AtM = AtA R
# Solve the weighted, L2 least-squares problem for an initial solution.
AtA = dot( dot(A.conj().transpose(), W), A )
AtM = dot( A.conj().transpose(), dot( W, Mfft[:,i] ) )
mu = abs(np.trace(AtA)) * hyperparameters[0]
Rfft[:,i] = sp.linalg.solve((AtA + mu*Ident), AtM)
#Non-linear methods use IRLS to solve, iterate until convergence to solution.
if inversion_model in ("Cauchy", "L1"):
#Initialize hyperparameters.
b=hyperparameters[1]
lam=mu*b
#Initialize cost functions.
dCOST = float("Inf")
if inversion_model == "Cauchy":
COST_prev = np.linalg.norm( Mfft[:,i] - dot(A,Rfft[:,i]), 2 ) + lam*sum( np.log( abs(Rfft[:,i]**2 + b) ) )
elif inversion_model == "L1":
COST_prev = np.linalg.norm( Mfft[:,i] - dot(A,Rfft[:,i]), 2 ) + lam*np.linalg.norm( abs(Rfft[:,i]+1), 1 )
itercount=1
#Iterate until negligible change to cost function.
while dCost > 0.001 and itercount < 10:
#Setup inverse problem.
if inversion_model == "Cauchy":
Q = sparse.spdiags( 1./( abs(Rfft[:,i]**2) + b), 0, ip, ip).A
elif inversion_model == "L1":
Q = sparse.spdiags( 1./( abs(Rfft[:,i]) + b), 0, ip, ip).A
Rfft[:,i]=sp.linalg.solve( ( lam * Q + AtA ), AtM )
#Determine change to cost function.
if inversion_model == "Cauchy":
COST_cur = np.linalg.norm( Mfft[:,i]-A*Rfft[:,i], 2 ) + lam*sum( np.log( abs(Rfft[:,i]**2 + b )-np.log(b) ) )
elif inversion_model == "L1":
COST_cur = np.linalg.norm( Mfft[:,i]-A*Rfft[:,i], 2 ) + lam*np.linalg.norm( abs(Rfft[:,i]+1) + b, 1 )
dCOST = 2*abs(COST_cur - COST_prev)/(abs(COST_cur) + abs(COST_prev))
COST_prev = COST_cur
itercount += 1
#Assuming Hermitian symmetry of the fft make negative frequencies the complex conjugate of current solution.
if i != 0:
Rfft[:,iF-i] = Rfft[:,i].conjugate()
R = np.fft.ifft(Rfft, iF)
R = R[:,0:it]
return R, t, epi
def radon_forward(t,p,R,delta,ref_dist,line_model):
"""
This function applies the time-shift Radon operator A, to the Radon
domain. Will calculate the move-out data, given the inputs:
-t -- vector of time axis.
-p -- vector of slowness axis you would like to invert to.
-R -- matrix of Radon data, ordered size(R)==[length(p),length(t)].
-delta -- vector of distance axis.
-ref_dist -- reference distance the path-function will shift about.
-line_model, select one of the following options for path integration:
'linear' - linear paths in the spatial domain (default)
'parabolic' - parabolic paths in the spatial domain.
Output spatial domain is ordered size(M)==[length(delta),length(t)].
Known limitations:
- Assumes evenly sampled time axis.
- Assumes Radon data isn't complex.
References: Schultz, R., Gu, Y. J., 2012. Flexible Matlab implementation
of the Radon Transform. Computers and Geosciences [In Preparation]
An, Y., Gu, Y. J., Sacchi, M., 2007. Imaging mantle
discontinuities using least-squares Radon transform.
Journal of Geophysical Research 112, B10303.
Author: R. Schultz, 2012
Translated to Python by: S. Schneider, 2016
"""
# Check for Data type of variables.
if not isinstance(t, numpy.ndarray) or not isinstance(delta, numpy.ndarray):
print( "Wrong input type of t or delta, must be numpy.ndarray" )
raise TypeError
it=t.size
iF=int(math.pow(2,nextpow2(it)+1)) # Double length
iDelta=delta.size
ip=len(p)
#Exit if inconsistent data is input.
if R.shape != (ip, it):
print("Dimensions inconsistent!\nShape of M is not equal to (len(delta),len(t)) \nShape of M = (%i , %i)\n(len(delta),len(t)) = (%i, %i) \n" % (M.shape[0], M.shape[1], iDelta, it) )
M=0
return(M)
#Preallocate space in memory.
Mfft = np.zeros((iDelta, iF)) + 0j
A = np.zeros((iDelta, ip)) + 0j
Tshift = np.zeros((iDelta, ip)) + 0j
#Define some values.
Dist_array=delta-ref_dist
dF=1./(t[0]-t[1])
Rfft=np.fft.fft(R,iF,1)
#Populate ray parameter then distance data in time shift matrix.
for j in range(iDelta):
if line_model == "parabolic":
Tshift[j,:]=p
else: #Linear is default
Tshift[j,:]=p
for k in range(ip):
if line_model == 'parabolic':
Tshift[:,k]=(2. * ref_dist * Tshift[:,k] * Dist_array.conj().transpose()) + (Tshift[:,k] * (Dist_array**2).conj().transpose())
else: #Linear is default
Tshift[:,k]=Tshift[:,k] * Dist_array.conj().transpose()
# Loop through each frequency.
for i in range( int(math.floor((iF+1)/2))-1 ):
# Make time-shift matrix, A.
f = ((float(i)/float(iF))*dF)
A = np.exp( (0.+1j)*2*pi*f * Tshift )
# Apply Radon operator.
Mfft[:,i]=dot(A, Rfft[:,i])
# Assuming Hermitian symmetry of the fft make negative frequencies the complex conjugate of current solution.
if i != 0:
Mfft[:,iF-i] = Mfft[:,i].conjugate()
M = np.fft.ifft(Mfft, iF)
M = M[:,0:it]
return(M)
