def specular_reflection_matrix(self, frequency, eps_1, eps_2, mu1, npol):
"""compute the reflection coefficients for an array of incidence angles (given by their cosine)
in medium 1. Medium 2 is where the beam is transmitted.
:param eps_1: permittivity of the medium where the incident beam is propagating.
:param eps_2: permittivity of the other medium.
:param mu1: array of cosine of incident angles.
:param npol: number of polarization.
:return: the reflection matrix
"""
R01_v, R01_h, R1t_v, R1t_h, exp_kd, exp_2kd, mu_t = self._prepare_computation(frequency, eps_1, eps_2, mu1)
R_v = (R01_v + R1t_v * exp_2kd) / (1 + R01_v * R1t_v * exp_2kd)
R_h = (R01_h + R1t_h * exp_2kd) / (1 + R01_h * R1t_h * exp_2kd)
reflection_coefficients = smrt_matrix.ones((npol, len(mu1)))
reflection_coefficients[0] = abs2(R_v)
reflection_coefficients[1] = abs2(R_h)
if npol >= 3:
reflection_coefficients[2] = (R_v * np.conj(R_h)).real # TsangI Eq 7.2.93
return reflection_coefficients
def coherent_transmission_matrix(self, frequency, eps_1, eps_2, mu1, npol):
"""compute the transmission coefficients for the azimuthal mode m
and for an array of incidence angles (given by their cosine)
in medium 1. Medium 2 is where the beam is transmitted.
:param eps_1: permittivity of the medium where the incident beam is propagating.
:param eps_2: permittivity of the other medium.
:param mu1: array of cosine of incident angles.
:param npol: number of polarization.
:return: the transmission matrix
"""
R01_v, R01_h, R1t_v, R1t_h, exp_kd, exp_2kd, mu_t = self._prepare_computation(frequency, eps_1, eps_2, mu1)
T_v = (1 + R01_v) * (1 + R1t_v) * exp_kd / (1 + R01_v * R1t_v * exp_2kd) # see TsnagI 5.2.10-12
T_h = (1 + R01_h) * (1 + R1t_h) * exp_kd / (1 + R01_h * R1t_h * exp_2kd)
transmission_coefficients = smrt_matrix.ones((npol, len(mu1)))
nt = np.sqrt(eps_2 / eps_1).real
transmission_coefficients[0] = abs2(T_v) * mu_t / mu1 / nt # for the coef see TsangIII 2.1.140b
transmission_coefficients[1] = abs2(T_h) * mu_t / mu1 * nt # for the coef see TsangIII 2.1.140a
if npol >= 3:
# this part is to be confirmed.
R_v = (R01_v + R1t_v * exp_2kd) / (1 + R01_v * R1t_v * exp_2kd)
R_h = (R01_h + R1t_h * exp_2kd) / (1 + R01_h * R1t_h * exp_2kd)
transmission_coefficients[2] = mu_t / mu1 * ((1 + R_v) * np.conj(1 + R_h)).real # TsangI Eq 7.2.95
return transmission_coefficients
def fresnel_reflection_matrix(eps_1, eps_2, mu1, npol):
"""compute the fresnel reflection matrix for/in medium 1 laying above medium 2.
:param npol: number of polarizations to return.
:param eps_1: permittivity of medium 1.
:param eps_2: permittivity of medium 2.
:param mu1: cosine zenith angle in medium 1.
:returns: a matrix or the diagional depending on `return_as_diagonal`
"""
mu1 = np.atleast_1d(mu1)
assert len(mu1.shape) == 1 # 1D array
reflection_coefficients = smrt_matrix.ones((npol, len(mu1)))
rv, rh, _ = fresnel_coefficients(eps_1, eps_2, mu1)
reflection_coefficients[0] = abs2(rv)
reflection_coefficients[1] = abs2(rh)
if npol >= 3:
reflection_coefficients[2] = (rv *
np.conj(rh)).real # TsangI Eq 7.2.93
return reflection_coefficients
def fresnel_transmission_matrix(eps_1, eps_2, mu1, npol):
"""compute the fresnel reflection matrix for/in medium 1 laying above medium 2.
:param npol: number of polarizations to return.
:param eps_1: permittivity of medium 1.
:param eps_2: permittivity of medium 2.
:param mu1: cosine zenith angle in medium 1.
:returns: a matrix or the diagional depending on `return_as_diagonal`
"""
mu1 = np.atleast_1d(mu1)
assert len(mu1.shape) == 1 # 1D array
transmission_coefficients = smrt_matrix.zeros((npol, len(mu1)))
rv, rh, mu2 = fresnel_coefficients(eps_1, eps_2, mu1)
transmission_coefficients[0] = 1 - abs2(rv)
transmission_coefficients[1] = 1 - abs2(rh)
if npol >= 3:
transmission_coefficients[2] = mu2 / mu1 * (
(1 + rv) * np.conj(1 + rh)).real # TsangI Eq 7.2.95
if npol == 4:
raise Exception(
"to be implemented, the matrix is not diagonal anymore")
return transmission_coefficients
def transmission_integrand_for_energy_conservation_test(self, frequency, eps_1, eps_2, mu_t, mu_i, dphi):
"""function relevant to compute energy conservation. See p87 in Tsang_tomeIII.
"""
n_2 = np.sqrt(eps_2)
n_1 = np.sqrt(eps_1)
mu_i = np.atleast_1d(mu_i)[np.newaxis, np.newaxis, :]
mu_t = np.atleast_1d(mu_t)[np.newaxis, :, np.newaxis]
dphi = np.atleast_1d(dphi)[:, np.newaxis, np.newaxis]
sin_i = np.sqrt(1 - mu_i**2)
sin_t = np.sqrt(1 - mu_t**2)
cos_phi = np.cos(dphi)
sin_phi = np.sin(dphi)
ki = vector3.from_xyz(sin_i, 0, -mu_i)
kt = vector3.from_xyz(sin_t * cos_phi, sin_t * sin_phi, -mu_t)
# compute the local transmission Fresnel coefficient
ktd = n_1.real * ki - n_2.real * kt # Eq 2.1.87
n = ktd / (np.sign(ktd.z) * ktd.norm) # Eq 2.1.128
mu_local = -vector3.dot(n, ki)
Rv, Rh, _ = fresnel_coefficients(eps_1, eps_2, mu_local)
# define polarizations
ht = vector3.from_xyz(-sin_phi, cos_phi, 0)
vt = vector3.from_xyz(-mu_t * cos_phi, -mu_t * sin_phi, -sin_t)
#print(vector3.cross(vt, ht), kt)
hi = vector3.from_xyz(0, 1, 0)
vi = vector3.from_xyz(-mu_i, 0, -sin_i)
#print(vector3.cross(vi, hi), ki)
ht_ki = vector3.dot(ht, ki)
vt_ki = vector3.dot(vt, ki)
hi_kt = vector3.dot(hi, kt)
vi_kt = vector3.dot(vi, kt)
Tv = (hi_kt**2) * (1-abs2(Rh)) + (vi_kt**2) * (1-abs2(Rv))
Th = (vi_kt**2) * (1-abs2(Rh)) + (hi_kt**2) * (1-abs2(Rv))
coef = 1/(2 * np.pi * self.mean_square_slope) * eps_2 * ktd.norm2 * vector3.dot(n, kt) * vector3.dot(n, ki) \
/ (mu_i * vector3.cross(ki, kt).norm2 * ktd.z**4 ) * \
np.exp(-(ktd.x**2 + ktd.y**2) / (2 * ktd.z**2 * self.mean_square_slope)) # Eq. 2.1.130 NB: k1^2 -> eps_2
#return Tv, Th #
return coef*Tv, coef*Th
def reflection_integrand_for_energy_conservation_test(self, frequency, eps_1, eps_2, mu_s, mu_i, dphi):
"""function relevant to compute energy conservation. See p87 in Tsang_tomeIII.
"""
mu_i = np.atleast_1d(mu_i)[np.newaxis, np.newaxis, :]
mu_s = np.atleast_1d(mu_s)[np.newaxis, :, np.newaxis]
dphi = np.atleast_1d(dphi)[:, np.newaxis, np.newaxis]
sin_i = np.sqrt(1 - mu_i**2)
sin_s = np.sqrt(1 - mu_s**2)
cos_phi = np.cos(dphi)
sin_phi = np.sin(dphi)
ki = vector3.from_xyz(sin_i, 0, -mu_i)
ks = vector3.from_xyz(sin_s * cos_phi, sin_s * sin_phi, mu_s)
# compute the local reflection Fresnel coefficient
kd = ki - ks # in principe: *sqrt(eps_1), but in the following it appears everywhere as a ratio
n = kd / (np.sign(kd.z) * kd.norm) # EQ 2.1.223 #equivalent to np.vector3(kd_x / kd_z, kd_y / kd_z, 1)
mu_local = -vector3.dot(n, ki)
Rv, Rh, _ = fresnel_coefficients(eps_1, eps_2, mu_local)
# define polarizations
hs = vector3.from_xyz(-sin_phi, cos_phi, 0)
vs = vector3.from_xyz(mu_s * cos_phi, mu_s * sin_phi, -sin_s)
#print(vector3.cross(vs, hs), ks)
hi = vector3.from_xyz(0, 1, 0)
vi = vector3.from_xyz(-mu_i, 0, -sin_i)
#print(vector3.cross(vi, hi), ki)
hs_ki = vector3.dot(hs, ki)
vs_ki = vector3.dot(vs, ki)
hi_ks = vector3.dot(hi, ks)
vi_ks = vector3.dot(vi, ks)
coef = 1/(2 * np.pi * self.mean_square_slope) * kd.norm2**2 / (4 * mu_i * vector3.cross(ki, ks).norm2 * kd.z**4) * \
np.exp( - (kd.x**2 + kd.y**2)/ (2 * kd.z**2 * self.mean_square_slope)) # Eq. 2.1.124
return coef * (hi_ks**2 * abs2(Rh) + vi_ks**2 * abs2(Rv)), \
coef * (vi_ks**2 * abs2(Rh) + hi_ks**2 * abs2(Rv))
def specular_reflection_matrix(self, frequency, eps_1, eps_2, mu1, npol):
"""compute the reflection coefficients for an array of incidence angles (given by their cosine)
in medium 1. Medium 2 is where the beam is transmitted.
:param eps_1: permittivity of the medium where the incident beam is propagating.
:param eps_2: permittivity of the other medium
:param mu1: array of cosine of incident angles
:param npol: number of polarization
:return: the reflection matrix
"""
k2 = (2 * np.pi * frequency / C_SPEED) ** 2 * abs2(eps_1)
return fresnel_reflection_matrix(eps_1, eps_2, mu1, npol) * np.exp(-4 * k2 * self.roughness_rms**2 * mu1**2) # Eq: 2.1.94 in Tsang 2001 Tome I
def diffuse_reflection_matrix(self,
frequency,
eps_1,
eps_2,
mu_s,
mu_i,
dphi,
npol,
debug=False):
"""compute the reflection coefficients for an array of incident, scattered and azimuth angles
in medium 1. Medium 2 is where the beam is transmitted.
:param eps_1: permittivity of the medium where the incident beam is propagating.
:param eps_2: permittivity of the other medium
:param mu1: array of cosine of incident angles
:param npol: number of polarization
:return: the reflection matrix
"""
mu_s = np.atleast_1d(mu_s)
mu_i = np.atleast_1d(mu_i)
if not np.allclose(mu_s, mu_i) or not np.allclose(dphi, np.pi):
raise NotImplementedError(
"Only the backscattering coefficient is implemented at this stage. This is a very preliminary implementation"
)
if len(np.atleast_1d(dphi)) != 1:
raise NotImplementedError(
"Only the backscattering coefficient is implemented at this stage. "
)
mu = mu_i[None, :]
k = vector3.from_angles(
2 * np.pi * frequency / C_SPEED * np.sqrt(eps_1).real, mu, 0)
eps_r = eps_2 / eps_1
# check validity
warning = None
ks = abs(k.norm * self.roughness_rms)
if ks > 3:
warning = "Warning, roughness_rms is too high for the given wavelength. Limit is ks < 3. Here ks=%g" % ks
kskl = abs(ks * k.norm * self.corr_length)
if kskl > np.sqrt(eps_r):
warning = "Warning, roughness_rms or correlation_length are too high for the given wavelength. Limit is ks * kl < sqrt(eps_r). Here ks*kl=%g and sqrt(eps_r)=%g" % (
kskl, np.sqrt(eps_r))
if warning:
if self.warning_handling == "print":
print(warning)
elif self.warning_handling == "nan":
return smrt_matrix.full((npol, len(mu_i)), np.nan)
# chekc validity done
Rv, Rh, _ = fresnel_coefficients(eps_1, eps_2, mu_i)
fvv = 2 * Rv / mu # Eq 44 in Fung et al. 1992
fhh = -2 * Rh / mu # Eq 45 in Fung et al. 1992
# prepare the series
N = self.series_truncation
n = np.arange(1, N + 1, dtype=np.float64)[:, None]
rms2 = self.roughness_rms**2
# Kirchoff term
Iscalar_n = (2 * k.z)**n * np.exp(-rms2 * k.z**2)
Ivv_n = Iscalar_n * fvv # Eq 82 in Fung et al. 1992
Ihh_n = Iscalar_n * fhh
# Complementary term
mu2 = mu**2
sin2 = 1 - mu2
tan2 = sin2 / mu2
Ivv_n += k.z**n * (
sin2 / mu * (1 + Rv)**2 * (1 - 1 / eps_r) * (1 + tan2 / eps_r)
) # part of Eq 91. We don't use all the simplification because we want validity for n>1, especially not np.exp(-rms2*k.z**2)=1
Ihh_n += -k.z**n * (sin2 / mu * (1 + Rh)**2 *
(eps_r - 1) / mu2) # part of Eq 95.
# compute the series
rms2_over_fractorial = np.cumprod(rms2 / n)[:, None]
# Eq 82 in Fung et al. 1992
coef = k.norm2 / 2 * np.exp(-2 * rms2 * k.z**2)
coef_n = rms2_over_fractorial * self.W_n(n, -2 * k.x)
sigma_vv = coef * np.sum(coef_n * abs2(Ivv_n), axis=0)
sigma_hh = coef * np.sum(coef_n * abs2(Ihh_n), axis=0)
#if debug:
# self.sigma_vv_1 = ( 8*k.norm2**2*rms2*abs2(Rv*mu2 + (1-mu2)*(1+Rv)**2 / 2 * (1 - 1 / eps_r)) * self.W_n(1, -2 * k.x) ).flat
# self.sigma_hh_1 = ( 8*k.norm2**2*rms2*abs2(Rh*mu2) * self.W_n(1, -2 * k.x) ).flat
reflection_coefficients = smrt_matrix.zeros((npol, len(mu_i)))
reflection_coefficients[0] = sigma_vv / (4 * np.pi * mu_i)
reflection_coefficients[1] = sigma_hh / (4 * np.pi * mu_i)
return reflection_coefficients
def diffuse_reflection_matrix(self, frequency, eps_1, eps_2, mu_s, mu_i, dphi, npol):
"""compute the reflection coefficients for an array of incident, scattered and azimuth angles
in medium 1. Medium 2 is where the beam is transmitted.
:param eps_1: permittivity of the medium where the incident beam is propagating.
:param eps_2: permittivity of the other medium
:param mu1: array of cosine of incident angles
:param npol: number of polarization
:return: the reflection matrix
"""
mu_i = np.atleast_1d(mu_i)[np.newaxis, np.newaxis, :]
mu_s = np.atleast_1d(mu_s)[np.newaxis, :, np.newaxis]
dphi = np.atleast_1d(dphi)[:, np.newaxis, np.newaxis]
sin_i = np.sqrt(1 - mu_i**2)
sin_s = np.sqrt(1 - mu_s**2)
cos_phi = np.cos(dphi)
sin_phi = np.sin(dphi)
ki = vector3.from_xyz(sin_i, 0, -mu_i)
ks = vector3.from_xyz(sin_s * cos_phi, sin_s * sin_phi, mu_s)
# compute the local reflection Fresnel coefficient
kd = ki - ks # in principe: *sqrt(eps_1), but in the following it appears everywhere as a ratio
n = kd / (np.sign(kd.z) * kd.norm) # EQ 2.1.223 #equivalent to np.vector3(kd_x / kd_z, kd_y / kd_z, 1)
mu_local = -vector3.dot(n, ki)
Rv, Rh, _ = fresnel_coefficients(eps_1, eps_2, mu_local)
# define polarizations
hs = vector3.from_xyz(-sin_phi, cos_phi, 0)
vs = vector3.from_xyz(mu_s * cos_phi, mu_s * sin_phi, -sin_s)
hi = vector3.from_xyz(0, 1, 0)
vi = vector3.from_xyz(-mu_i, 0, -sin_i)
# compute the dot products
hs_ki = vector3.dot(hs, ki)
vs_ki = vector3.dot(vs, ki)
hi_ks = vector3.dot(hi, ks)
vi_ks = vector3.dot(vi, ks)
fvv = abs2( hs_ki * hi_ks * Rh + vs_ki* vi_ks * Rv) # Eqs 2.1.122 in Tsang_tomeIII
fhh = abs2( vs_ki * vi_ks * Rh + hs_ki* hi_ks * Rv)
fhv = abs2( vs_ki * hi_ks * Rh - hs_ki* vi_ks * Rv)
fvh = abs2( hs_ki * vi_ks * Rh + vs_ki* hi_ks * Rv)
reflection_coefficients = smrt_matrix.zeros((npol, npol, dphi.shape[0], mu_s.shape[1], mu_i.shape[2]))
reflection_coefficients[0, 0] = fvv
reflection_coefficients[0, 1] = fvh
reflection_coefficients[1, 0] = fhv
reflection_coefficients[1, 1] = fhh
smrt_norm = 1 / (4 * np.pi) # divide by 4*pi because this is the norm for SMRT
coef = smrt_norm / (2 * self.mean_square_slope) / mu_i * kd.norm2**2 / (vector3.cross(ki, ks).norm2**2 * kd.z**4) * \
np.exp( - (kd.x**2 + kd.y**2)/ (2 * kd.z**2 * self.mean_square_slope)) # Eq. 2.1.124
if self.shadow_correction:
backward = dphi == np.pi
higher_thetas = mu_s <= mu_i
zero_i = backward & higher_thetas
zero_s = backward & ~higher_thetas
s = 1 / (1 + zero_i * shadow_function(self.mean_square_slope, mu_i/sin_i) + zero_s * shadow_function(self.mean_square_slope, mu_s/sin_s))
coef *= s
return reflection_coefficients * coef
def diffuse_transmission_matrix(self, frequency, eps_1, eps_2, mu_t, mu_i, dphi, npol):
"""compute the transmission coefficients for an array of incident, scattered and azimuth angles
in medium 1. Medium 2 is where the beam is transmitted.
:param eps_1: permittivity of the medium where the incident beam is propagating.
:param eps_2: permittivity of the other medium
:param mu_i: array of cosine of incident angles
:param mu_t: array of cosine of transmitted wave angles
:param npol: number of polarization
:return: the transmission matrix
"""
n_2 = np.sqrt(eps_2)
n_1 = np.sqrt(eps_1)
eta1_eta = n_1 / n_2 # eta1 is the impedance in medium 2 and eta in medium 1. Impedance is sqrt(permeability/permittivity)
mu_i = np.atleast_1d(mu_i)[np.newaxis, np.newaxis, :]
mu_t = np.atleast_1d(mu_t)[np.newaxis, :, np.newaxis]
dphi = np.atleast_1d(dphi)[:, np.newaxis, np.newaxis]
sin_i = np.sqrt(1 - mu_i**2)
sin_t = np.sqrt(1 - mu_t**2)
cos_phi = np.cos(dphi)
sin_phi = np.sin(dphi)
ki = vector3.from_xyz(sin_i, 0, -mu_i)
kt = vector3.from_xyz(sin_t * cos_phi, sin_t * sin_phi, -mu_t)
# compute the local transmission Fresnel coefficient
ktd = n_1.real * ki - n_2.real * kt # Eq 2.1.87
n = ktd / (np.sign(ktd.z) * ktd.norm) # Eq 2.1.128
mu_local = -vector3.dot(n, ki)
Rv, Rh, _ = fresnel_coefficients(eps_1, eps_2, mu_local)
n_kt = vector3.dot(n, kt)
n_kt[mu_local < 0] = 0
# define polarizations
ht = vector3.from_xyz(-sin_phi, cos_phi, 0)
vt = vector3.from_xyz(-mu_t * cos_phi, -mu_t * sin_phi, -sin_t)
hi = vector3.from_xyz(0, 1, 0)
vi = vector3.from_xyz(-mu_i, 0, -sin_i)
# compute the cosines
ht_ki = vector3.dot(ht, ki)
vt_ki = vector3.dot(vt, ki)
hi_kt = vector3.dot(hi, kt)
vi_kt = vector3.dot(vi, kt)
Wvv = abs2( ht_ki * hi_kt * (1 + Rh) + vt_ki * vi_kt * (1 + Rv) * eta1_eta) # Eqs 2.1.130 in Tsang_tomeIII
Whh = abs2( vt_ki * vi_kt * (1 + Rh) + ht_ki * hi_kt * (1 + Rv) * eta1_eta)
Whv = abs2(-vt_ki * hi_kt * (1 + Rh) + ht_ki * vi_kt * (1 + Rv) * eta1_eta)
Wvh = abs2( ht_ki * vi_kt * (1 + Rh) - vt_ki * hi_kt * (1 + Rv) * eta1_eta)
transmission_coefficients = smrt_matrix.zeros((npol, npol, dphi.shape[0], mu_t.shape[1], mu_i.shape[2]))
transmission_coefficients[0, 0] = Wvv
transmission_coefficients[0, 1] = Wvh
transmission_coefficients[1, 0] = Whv
transmission_coefficients[1, 1] = Whh
smrt_norm = 1 / (4 * np.pi) # SMRT requires scattering coefficient / 4 * pi
coef = smrt_norm * 2 * eps_2 / (eta1_eta * self.mean_square_slope) / mu_i * \
ktd.norm2 * n_kt**2 / (vector3.cross(ki, kt).norm2**2 * ktd.z**4) * \
np.exp(-(ktd.x**2 + ktd.y**2) / (2 * ktd.z**2 * self.mean_square_slope)) # Eq. 2.1.130 NB: k1^2 -> eps_2
if self.shadow_correction:
s = 1 / (1 + shadow_function(self.mean_square_slope, mu_i/sin_i) + shadow_function(self.mean_square_slope, mu_t/sin_t))
coef *= s
return transmission_coefficients * coef.real
def diffuse_reflection_matrix(self, frequency, eps_1, eps_2, mu_s, mu_i,
dphi, npol):
"""compute the reflection coefficients for an array of incident, scattered and azimuth angles
in medium 1. Medium 2 is where the beam is transmitted.
:param eps_1: permittivity of the medium where the incident beam is propagating.
:param eps_2: permittivity of the other medium
:param mu1: array of cosine of incident angles
:param npol: number of polarization
:return: the reflection matrix
"""
mu_i = np.atleast_1d(self.clip_mu(mu_i))[np.newaxis, np.newaxis, :]
mu_s = np.atleast_1d(self.clip_mu(mu_s))[np.newaxis, :, np.newaxis]
dphi = np.atleast_1d(dphi)[:, np.newaxis, np.newaxis]
sin_i = np.sqrt(1 - mu_i**2)
sin_s = np.sqrt(1 - mu_s**2)
cos_phi = np.cos(dphi)
sin_phi = np.sin(dphi)
ki = vector3.from_xyz(sin_i, 0, -mu_i)
ks = vector3.from_xyz(sin_s * cos_phi, sin_s * sin_phi, mu_s)
# compute the local reflection Fresnel coefficient
kd = ki - ks # in principe: *sqrt(eps_1), but in the following it appears everywhere as a ratio
n = kd / (
np.sign(kd.z) * kd.norm
) # EQ 2.1.123 #equivalent to np.vector3(kd_x / kd_z, kd_y / kd_z, 1)
mu_local = -vector3.dot(n, ki)
assert (np.all(mu_local >= 0))
assert (np.all(mu_local <= 1.0001)) # compare with some tolerance
Rv, Rh, _ = fresnel_coefficients(eps_1, eps_2, self.clip_mu(mu_local))
# define polarizations
hs = vector3.from_xyz(-sin_phi, cos_phi, 0)
vs = vector3.from_xyz(mu_s * cos_phi, mu_s * sin_phi, -sin_s)
hi = vector3.from_xyz(0, 1, 0)
vi = vector3.from_xyz(-mu_i, 0, -sin_i)
# compute the dot products
cross_ki_ks_norm = vector3.cross(ki, ks).norm
colinear = cross_ki_ks_norm < 1e-4
# avoid warning due to divide error, the colinear case is solved independantly:
cross_ki_ks_norm[colinear] = 1
hs_ki = vector3.dot(hs, ki) / cross_ki_ks_norm
hs_ki[colinear] = -1
vs_ki = vector3.dot(vs, ki) / cross_ki_ks_norm
vs_ki[colinear] = 0
hi_ks = vector3.dot(hi, ks) / cross_ki_ks_norm
hi_ks[colinear] = 1
vi_ks = vector3.dot(vi, ks) / cross_ki_ks_norm
vi_ks[colinear] = 0
fvv = abs2(hs_ki * hi_ks * Rh +
vs_ki * vi_ks * Rv) # Eqs 2.1.122 in Tsang_tomeIII
fhh = abs2(vs_ki * vi_ks * Rh + hs_ki * hi_ks * Rv)
fhv = abs2(vs_ki * hi_ks * Rh - hs_ki * vi_ks * Rv)
fvh = abs2(hs_ki * vi_ks * Rh - vs_ki * hi_ks * Rv)
reflection_coefficients = smrt_matrix.zeros(
(npol, npol, dphi.shape[0], mu_s.shape[1], mu_i.shape[2]))
reflection_coefficients[0, 0] = fvv
reflection_coefficients[0, 1] = fvh
reflection_coefficients[1, 0] = fhv
reflection_coefficients[1, 1] = fhh
smrt_norm = 1 / (4 * np.pi
) # divide by 4*pi because this is the norm for SMRT
coef = smrt_norm / (2 * self.mean_square_slope) / mu_i * kd.norm2**2 / kd.z**4 * \
np.exp(-(kd.x**2 + kd.y**2) / (2 * kd.z**2 * self.mean_square_slope)) # Eq. 2.1.124
if self.shadow_correction:
backward = dphi == np.pi
higher_thetas = mu_s <= mu_i
zero_i = backward & higher_thetas
zero_s = backward & ~higher_thetas
# this hack to avoid division-by-zero is safe, because the shadow_function is only important for large angles
sin_i[sin_i < 1e-3] = 1e-3
sin_s[sin_s < 1e-3] = 1e-3
s = 1 / (1 + (~zero_i) *
shadow_function(self.mean_square_slope, mu_i / sin_i) +
(~zero_s) *
shadow_function(self.mean_square_slope, mu_s / sin_s))
coef *= s
return reflection_coefficients * coef
def transmission_integrand_for_energy_conservation_test(
self, frequency, eps_1, eps_2, mu_t, mu_i, dphi):
"""function relevant to compute energy conservation. See p87 in Tsang_tomeIII.
"""
n_2 = np.sqrt(eps_2)
n_1 = np.sqrt(eps_1)
mu_i = np.atleast_1d(self.clip_mu(mu_i))[np.newaxis, np.newaxis, :]
mu_t = np.atleast_1d(self.clip_mu(mu_t))[np.newaxis, :, np.newaxis]
dphi = np.atleast_1d(dphi)[:, np.newaxis, np.newaxis]
sin_i = np.sqrt(1 - mu_i**2)
sin_t = np.sqrt(1 - mu_t**2)
cos_phi = np.cos(dphi)
sin_phi = np.sin(dphi)
ki = vector3.from_xyz(sin_i, 0, -mu_i)
kt = vector3.from_xyz(sin_t * cos_phi, sin_t * sin_phi, -mu_t)
# compute the local transmission Fresnel coefficient
ktd = n_1.real * ki - n_2.real * kt # Eq 2.1.87
n = ktd / (np.sign(ktd.z) * ktd.norm) # Eq 2.1.128
# compute Fresnel coefficients at stationary point
n_kt = -vector3.dot(n, kt)
n_ki = -vector3.dot(n, ki)
Rh = (n_1.real * n_ki - n_2.real * n_kt) / (
n_1.real * n_ki + n_2.real * n_kt) # Eq. 2.1.132a
Rv = (n_2.real * n_ki - n_1.real * n_kt) / (
n_2.real * n_ki + n_1.real * n_kt) # Eq. 2.1.132b
no_compatible_slopes = (n_kt < 0) | (n_ki < 0)
Rh[no_compatible_slopes] = -1
Rv[no_compatible_slopes] = -1
# define polarizations
hi = vector3.from_xyz(0, 1, 0)
vi = vector3.from_xyz(-mu_i, 0, -sin_i)
hi_kt = vector3.dot(hi, kt)
vi_kt = vector3.dot(vi, kt)
Tv = (hi_kt**2) * (1 - abs2(Rh)) + (vi_kt**2) * (1 - abs2(Rv))
Th = (vi_kt**2) * (1 - abs2(Rh)) + (hi_kt**2) * (1 - abs2(Rv))
coef = eps_2 / (2 * np.pi * self.mean_square_slope) * ktd.norm2 * n_kt * n_ki \
/ (mu_i * vector3.cross(ki, kt).norm2 * ktd.z**4) * \
np.exp(- (ktd.x**2 + ktd.y**2) / (2 * ktd.z**2 * self.mean_square_slope)) # Eq. 2.1.130 NB: k1^2 -> eps_2
if self.shadow_correction:
sin_i[sin_i < 1e-3] = 1e-3
sin_t[sin_t < 1e-3] = 1e-3
s = 1 / (1 + shadow_function(self.mean_square_slope, mu_i / sin_i)
+ shadow_function(self.mean_square_slope, mu_t / sin_t))
coef *= s
return coef * Tv, coef * Th
def diffuse_transmission_matrix(self, frequency, eps_1, eps_2, mu_t, mu_i,
dphi, npol):
"""compute the transmission coefficients for an array of incident, scattered and azimuth angles
in medium 1. Medium 2 is where the beam is transmitted.
:param eps_1: permittivity of the medium where the incident beam is propagating.
:param eps_2: permittivity of the other medium
:param mu_i: array of cosine of incident angles
:param mu_t: array of cosine of transmitted wave angles
:param npol: number of polarization
:return: the transmission matrix
"""
n_2 = np.sqrt(eps_2)
n_1 = np.sqrt(eps_1)
eta1_eta = n_1 / n_2 # eta1 is the impedance in medium 2 and eta in medium 1. Impedance is sqrt(permeability/permittivity)
if abs(eta1_eta - 1) < 1e-6:
raise NotImplementedError(
"the case of successive layers with identical index (%g) is not implemented"
% n_2)
return 1 / (
4 * np.pi
) # return the identity matrix. The coef is to be checked.
mu_i = np.atleast_1d(self.clip_mu(mu_i))[np.newaxis, np.newaxis, :]
mu_t = np.atleast_1d(self.clip_mu(mu_t))[np.newaxis, :, np.newaxis]
dphi = np.atleast_1d(dphi)[:, np.newaxis, np.newaxis]
sin_i = np.sqrt(1 - mu_i**2)
sin_t = np.sqrt(1 - mu_t**2)
cos_phi = np.cos(dphi)
sin_phi = np.sin(dphi)
ki = vector3.from_xyz(sin_i, 0, -mu_i)
kt = vector3.from_xyz(sin_t * cos_phi, sin_t * sin_phi, -mu_t)
# compute the local transmission Fresnel coefficient
ktd = ki * n_1.real - kt * n_2.real # Eq 2.1.87
n = ktd / (np.sign(ktd.z) * ktd.norm) # Eq 2.1.128
n_kt = -vector3.dot(n, kt)
n_ki = -vector3.dot(n, ki)
# compute Fresnel coefficients at stationary point
Rh = (n_1.real * n_ki - n_2.real * n_kt) / (
n_1.real * n_ki + n_2.real * n_kt) # Eq. 2.1.132a
Rv = (n_2.real * n_ki - n_1.real * n_kt) / (
n_2.real * n_ki + n_1.real * n_kt) # Eq. 2.1.132b
no_compatible_slopes = (n_kt < 0) | (n_ki < 0)
Rh[no_compatible_slopes] = -1
Rv[no_compatible_slopes] = -1
# define polarizations
ht = vector3.from_xyz(-sin_phi, cos_phi, 0)
vt = vector3.from_xyz(-mu_t * cos_phi, -mu_t * sin_phi, -sin_t)
hi = vector3.from_xyz(0, 1, 0)
vi = vector3.from_xyz(-mu_i, 0, -sin_i)
# compute the cosines
cross_ki_kt_norm = vector3.cross(ki, kt).norm
colinear = cross_ki_kt_norm < 1e-4
# avoid warning due to divide error, the colinear case is solved independantly:
cross_ki_kt_norm[colinear] = 1
ht_ki = vector3.dot(ht, ki) / cross_ki_kt_norm
ht_ki[colinear] = -1 # checked with sympy
vt_ki = vector3.dot(vt, ki) / cross_ki_kt_norm
vt_ki[colinear] = 0 # checked with sympy
hi_kt = vector3.dot(hi, kt) / cross_ki_kt_norm
hi_kt[colinear] = 1 # checked with sympy
vi_kt = vector3.dot(vi, kt) / cross_ki_kt_norm
vi_kt[colinear] = 0 # checked with sympy
Wvv = abs2(ht_ki * hi_kt * (1 + Rh) + vt_ki * vi_kt *
(1 + Rv) * eta1_eta) # Eqs 2.1.130 in Tsang_tomeIII
Whh = abs2(vt_ki * vi_kt * (1 + Rh) + ht_ki * hi_kt *
(1 + Rv) * eta1_eta)
Whv = abs2(-vt_ki * hi_kt * (1 + Rh) + ht_ki * vi_kt *
(1 + Rv) * eta1_eta)
Wvh = abs2(ht_ki * vi_kt * (1 + Rh) - vt_ki * hi_kt *
(1 + Rv) * eta1_eta)
transmission_coefficients = smrt_matrix.zeros(
(npol, npol, dphi.shape[0], mu_t.shape[1], mu_i.shape[2]))
transmission_coefficients[0, 0] = Wvv
transmission_coefficients[0, 1] = Wvh
transmission_coefficients[1, 0] = Whv
transmission_coefficients[1, 1] = Whh
smrt_norm = 1 / (4 * np.pi
) # SMRT requires scattering coefficient / 4 * pi
coef = smrt_norm * 2 * eps_2 * ktd.norm2 * n_kt**2 / (eta1_eta * self.mean_square_slope * mu_i * ktd.z**4) * \
np.exp(-(ktd.x**2 + ktd.y**2) / (2 * ktd.z**2 * self.mean_square_slope)) # Eq. 2.1.130 NB: k1^2 -> eps_2
if self.shadow_correction:
# this hack to avoid division-by-zero is safe, because the shadow_function is only important for large angles
sin_i[sin_i < 1e-3] = 1e-3
sin_t[sin_t < 1e-3] = 1e-3
s = 1 / (1 + shadow_function(self.mean_square_slope, mu_i / sin_i)
+ shadow_function(self.mean_square_slope, mu_t / sin_t))
coef *= s
return transmission_coefficients * coef.real
