def calc_L(ustar, T_A_K, rho, c_p, H, LE):
'''Calculates the Monin-Obukhov length.
Parameters
----------
ustar : float
friction velocity (m s-1).
T_A_K : float
air temperature (Kelvin).
rho : float
air density (kg m-3).
c_p : float
Heat capacity of air at constant pressure (J kg-1 K-1).
H : float
sensible heat flux (W m-2).
LE : float
latent heat flux (W m-2).
Returns
-------
L : float
Obukhov stability length (m).
References
----------
.. [Brutsaert2005] Brutsaert, W. (2005). Hydrology: an introduction (Vol. 61, No. 8).
Cambridge: Cambridge University Press.'''
# Convert input scalars to numpy arrays
ustar, T_A_K, rho, c_p, H, LE = map(
np.asarray, (ustar, T_A_K, rho, c_p, H, LE))
# first convert latent heat into rate of surface evaporation (kg m-2 s-1)
Lambda = met.calc_lambda(T_A_K) # in J kg-1
E = LE / Lambda
del LE, Lambda
# Virtual sensible heat flux
Hv = H + (0.61 * T_A_K * c_p * E)
del H, E
L = np.asarray(np.ones(ustar.shape) * float('inf'))
i = Hv != 0
L_const = np.asarray(k * gravity / T_A_K)
L[i] = -ustar[i]**3 / (L_const[i] * (Hv[i] / (rho[i] * c_p[i])))
return np.asarray(L)
def calc_stomatal_resistance_TSEB(
LE_C,
LE,
R_A,
R_x,
e_a,
T_A,
T_C,
F,
p=1013.0,
leaf_type=1,
f_g=1,
f_dry=1):
''' TSEB Stomatal conductace
Estimates the effective Stomatal conductace by inverting the
resistance-based canopy latent heat flux from a Two source perspective
Parameters
----------
LE_C : float
Canopy latent heat flux (W m-2).
LE : float
Surface (bulk) latent heat flux (W m-2).
R_A : float
Aerodynamic resistance to heat transport (s m-1).
R_x : float
Bulk aerodynamic resistance to heat transport at the canopy boundary layer (s m-1).
e_a : float
Water vapour pressure at the reference height (mb).
T_A : float
Air temperature at the reference height (K).
T_C : float
Canopy (leaf) temperature (K).
F : float
local Leaf Area Index.
p : float, optional
Atmospheric pressure (mb) use 1013.0 as default.
leaf_type : int, optional
type of leaf regarding stomata distribution.
1=HYPOSTOMATOUS stomata in the lower surface of the leaf (default).
2=AMPHISTOMATOUS, stomata in both surfaces of the leaf.
f_g : float, optional
Fraction of green leaves.
f_dry : float, optional
Fraction of dry (non-wet) leaves.
Returns
-------
G_s : float
effective leaf stomata conductance (m s-1).
References
----------
.. [Anderson2000] M.C. Anderson, J.M. Norman, T.P. Meyers, G.R. Diak, An analytical
model for estimating canopy transpiration and carbon assimilation fluxes based on
canopy light-use efficiency, Agricultural and Forest Meteorology, Volume 101,
Issue 4, 12 April 2000, Pages 265-289, ISSN 0168-1923,
http://dx.doi.org/10.1016/S0168-1923(99)00170-7.'''
# Convert input scalars to numpy arrays
LE_C, LE, R_A, R_x, e_a, T_A, T_C, F, p, leaf_type, f_g, f_dry = map(
np.asarray, (LE_C, LE, R_A, R_x, e_a, T_A, T_C, F, p, leaf_type, f_g, f_dry))
# Invert the bulk SW to obtain eb (vapor pressure at the canopy interface)
rho = met.calc_rho(p, e_a, T_A)
Cp = met.calc_c_p(p, e_a)
Lambda = met.calc_lambda(T_A)
psicr = met.calc_psicr(Cp, p, Lambda)
e_ac = e_a + LE * R_A * psicr / (rho * Cp)
# Calculate the saturation vapour pressure in the leaf in mb
e_star = met.calc_vapor_pressure(T_C)
# Calculate the boundary layer canopy resisitance to water vapour (Anderson et al. 2000)
# Invert the SW LE_S equation to calculate the bulk stomatal resistance
R_c = np.asarray((rho * Cp * (e_star - e_ac) / (LE_C * psicr)) - R_x)
K_c = np.asarray(f_dry * f_g * leaf_type)
# Get the mean stomatal resistance (here LAI comes in as stomatal resistances
# are in parallel: 1/Rc=sum(1/R_st)=LAI/Rst
r_st = R_c * K_c * F
return np.asarray(r_st)
def ESVEP(Tr_K,
vza,
T_A_K,
u,
ea,
p,
Sn_C,
Sn_S,
L_dn,
LAI,
emis_C,
emis_S,
z_0M,
d_0,
z_u,
z_T,
z0_soil=0.001,
x_LAD=1,
f_c=1.0,
f_g=1.0,
w_C=1.0,
calcG_params=[[1], 0.35],
UseL=False):
'''ESVEP
Calculates soil and vegetation energy fluxes using Soil and Vegetation Energy Patrtitioning
`ESVEP` model and a single observation of composite radiometric temperature.
Parameters
----------
Tr_K : float
Radiometric composite temperature (Kelvin).
vza : float
View Zenith Angle (degrees).
T_A_K : float
Air temperature (Kelvin).
u : float
Wind speed above the canopy (m s-1).
ea : float
Water vapour pressure above the canopy (mb).
p : float
Atmospheric pressure (mb), use 1013 mb by default.
Sn_C : float
Canopy net shortwave radiation (W m-2).
Sn_S : float
Soil net shortwave radiation (W m-2).
L_dn : float
Downwelling longwave radiation (W m-2).
LAI : float
Effective Leaf Area Index (m2 m-2).
emis_C : float
Leaf emissivity.
emis_S : flaot
Soil emissivity.
z_0M : float
Aerodynamic surface roughness length for momentum transfer (m).
d_0 : float
Zero-plane displacement height (m).
z_u : float
Height of measurement of windspeed (m).
z_T : float
Height of measurement of air temperature (m).
z0_soil : float, optional
bare soil aerodynamic roughness length (m).
x_LAD : float, optional
Campbell 1990 leaf inclination distribution function chi parameter.
f_c : float, optional
Fractional cover.
f_g : float, optional
Fraction of vegetation that is green.
w_C : float, optional
Canopy width to height ratio.
calcG_params : list[list,float or array], optional
Method to calculate soil heat flux,parameters.
* [[1],G_ratio]: default, estimate G as a ratio of Rn_S, default Gratio=0.35.
* [[0],G_constant] : Use a constant G, usually use 0 to ignore the computation of G.
* [[2,Amplitude,phase_shift,shape],time] : estimate G from Santanello and Friedl with G_param list of parameters (see :func:`~TSEB.calc_G_time_diff`).
UseL : float or None, optional
If included, its value will be used to force the Moning-Obukhov stability length.
Returns
-------
flag : int
Quality flag, see Appendix for description.
T_S : float
Soil temperature (Kelvin).
T_C : float
Canopy temperature (Kelvin).
T_sd: float
End-member temperature of dry soil (Kelvin)
T_vd: float
End-member temperature of dry vegetation (Kelvin)
T_sw: float
End-member temperature of saturated soil (Kelvin)
T_vw: float
End-member temperature of well-watered vegetation (Kelvin)
T_star: float
Critical surface temperature (Kelvin)
L_nS : float
Soil net longwave radiation (W m-2)
L_nC : float
Canopy net longwave radiation (W m-2)
LE_C : float
Canopy latent heat flux (W m-2).
H_C : float
Canopy sensible heat flux (W m-2).
LE_S : float
Soil latent heat flux (W m-2).
H_S : float
Soil sensible heat flux (W m-2).
G : float
Soil heat flux (W m-2).
r_vw: float
Canopy resistance to heat transport of well-watered vegetation (s m-1)
r_vd: float
Canopy resistance to heat transport of vegetation with zero soil water avaiability (s m-1)
r_av: float
Aerodynamic resistance to heat transport of the vegetation (s m-1)
r_as: float
Aerodynamic resistance to heat transport of the soil (s m-1)
L : float
Monin-Obuhkov length (m).
n_iterations : int
number of iterations until convergence of L.
References
----------
.. [Tang2017] Tang, R., and Z. L. Li. An End-Member-Based Two-Source Approach for Estimating
Land Surface Evapotranspiration From Remote Sensing Data. IEEE Transactions on Geoscience
and Remote Sensing 55, no. 10 (October 2017): 5818–32.
https://doi.org/10.1109/TGRS.2017.2715361.
'''
# Convert input float scalars to arrays and check parameters size
Tr_K = np.asarray(Tr_K)
(vza, T_A_K, u, ea, p, Sn_C, Sn_S, L_dn, LAI, emis_C, emis_S, z_0M, d_0,
z_u, z_T, z0_soil, x_LAD, f_c, f_g, w_C,
calcG_array) = map(tseb._check_default_parameter_size, [
vza, T_A_K, u, ea, p, Sn_C, Sn_S, L_dn, LAI, emis_C, emis_S, z_0M,
d_0, z_u, z_T, z0_soil, x_LAD, f_c, f_g, w_C, calcG_params[1]
], [Tr_K] * 21)
# Create the output variables
[
flag, T_S, T_C, T_sd, T_vd, T_sw, T_vw, T_star, Ln_S, Ln_C, LE_C, H_C,
LE_S, H_S, G, r_vw, r_vd, r_av, r_as, iterations
] = [np.zeros(Tr_K.shape) + np.NaN for i in range(20)]
# iteration of the Monin-Obukhov length
if isinstance(UseL, bool):
# Initially assume stable atmospheric conditions and set variables for
L = np.asarray(np.zeros(T_S.shape) + np.inf)
max_iterations = ITERATIONS
else: # We force Monin-Obukhov lenght to the provided array/value
L = np.asarray(np.ones(T_S.shape) * UseL)
max_iterations = 1 # No iteration
# Calculate the general parameters
rho = met.calc_rho(p, ea, T_A_K) # Air density
c_p = met.calc_c_p(p, ea) # Heat capacity of air
z_0H = res.calc_z_0H(z_0M, kB=kB) # Roughness length for heat transport
z_0H_soil = res.calc_z_0H(z0_soil,
kB=kB) # Roughness length for heat transport
s = met.calc_delta_vapor_pressure(
T_A_K) * 10 # slope of the saturation pressure curve (mb C-1)
lbd = met.calc_lambda(T_A_K) # latent heat of vaporisation (MJ./kg)
gama = met.calc_psicr(p, lbd) # psychrometric constant (mb C-1)
vpd = met.calc_vapor_pressure(T_A_K) - ea # vapor pressure deficit (mb)
# Calculate LAI dependent parameters for dataset where LAI > 0
omega0 = CI.calc_omega0_Kustas(LAI, f_c, x_LAD=x_LAD, isLAIeff=True)
F = np.asarray(LAI / f_c) # Real LAI
# Fraction of vegetation observed by the sensor
f_theta = tseb.calc_F_theta_campbell(vza,
F,
w_C=w_C,
Omega0=omega0,
x_LAD=x_LAD)
# Initially assume stable atmospheric conditions and set variables for
# iteration of the Monin-Obukhov length
u_friction = MO.calc_u_star(u, z_u, L, d_0, z_0M)
u_friction = np.asarray(np.maximum(u_friction_min, u_friction))
u_friction_s = MO.calc_u_star(u, z_u, L, np.zeros(d_0.shape), z0_soil)
u_friction_s = np.asarray(np.maximum(u_friction_min, u_friction_s))
L_old = np.ones(Tr_K.shape)
L_diff = np.asarray(np.ones(Tr_K.shape) * float('inf'))
# First assume that canopy temperature equals the minumum of Air or
# radiometric T
T_C = np.asarray(np.minimum(Tr_K, T_A_K))
flag, T_S = tseb.calc_T_S(Tr_K, T_C, f_theta)
# Loop for estimating stability.
# Stops when difference in consecutives L is below a given threshold
for n_iterations in range(max_iterations):
i = flag != 255
if np.all(L_diff[i] < L_thres):
if L_diff[i].size == 0:
print("Finished iterations with no valid solution")
else:
print("Finished interations with a max. L diff: " +
str(np.max(L_diff[i])))
break
i = np.logical_and(L_diff >= L_thres, flag != 255)
print("Iteration " + str(n_iterations) + ", max. L diff: " +
str(np.max(L_diff[i])))
iterations[i] = n_iterations
# Calculate net longwave radiation with current values of T_C and T_S
Ln_C[i], Ln_S[i] = rad.calc_L_n_Kustas(T_C[i], T_S[i], L_dn[i], LAI[i],
emis_C[i], emis_S[i])
Rn_C = Sn_C + Ln_C
Rn_S = Sn_S + Ln_S
# Compute Soil Heat Flux
G[i] = tseb.calc_G([calcG_params[0], calcG_array], Rn_S, i)
# Calculate aerodynamic resistances
r_vw[i] = 100.0 / LAI[i]
r_vd[i] = np.zeros(LAI[i].shape) + 2000.0
r_av_params = {
"z_T": z_T[i],
"u_friction": u_friction[i],
"L": L[i],
"d_0": d_0[i],
"z_0H": z_0H[i]
}
r_av[i] = tseb.calc_resistances(tseb.KUSTAS_NORMAN_1999,
{"R_A": r_av_params})[0]
r_as_params = {
"z_T": z_T[i],
"u_friction": u_friction_s[i],
"L": L[i],
"d_0": np.zeros(d_0[i].shape),
"z_0H": z_0H_soil[i]
}
r_as[i] = tseb.calc_resistances(tseb.KUSTAS_NORMAN_1999,
{"R_A": r_as_params})[0]
# Estimate the surface temperatures of the end-members
# Eq 8a
T_sd[i] = r_as[i] * (Rn_S[i] - G[i]) / (rho[i] * c_p[i]) + T_A_K[i]
# Eq 8b
T_vd[i] = r_av[i] * Rn_C[i] / (rho[i] * c_p[i]) *\
gama[i] * (1 + r_vd[i] / r_av[i]) / (s[i] + gama[i] * (1 + r_vd[i] / r_av[i])) -\
vpd[i] / (s[i] + gama[i] * (1 + r_vd[i] / r_av[i])) + T_A_K[i]
# Eq 8c
T_sw[i] = r_as[i] * (Rn_S[i] - G[i]) / (rho[i] + c_p[i]) *\
gama[i] / (s[i] + gama[i]) - vpd[i] / (s[i] + gama[i]) + T_A_K[i]
# Eq 8d
T_vw[i] = r_av[i] * Rn_C[i] / (rho[i] * c_p[i]) *\
gama[i] * (1 + r_vw[i] / r_av[i]) / (s[i] + gama[i] * (1 + r_vw[i] / r_av[i])) -\
vpd[i] / (s[i] + gama[i] * (1 + r_vw[i] / r_av[i])) + T_A_K[i]
# Estimate critical surface temperature - eq 10
T_star[i] = (T_sd[i]**4 * (1 - f_theta[i]) +
T_vw[i]**4 * f_theta[i])**0.25
# Estimate latent heat fluxes when water in the top-soil is avaiable for evaporation
j = np.logical_and(Tr_K <= T_star, i)
# Eq 12a
T_C[j] = T_vw[j]
# Eq 12b
T_S[j] = ((Tr_K[j]**4 - f_theta[j] * T_C[j]**4) /
(1 - f_theta[j]))**0.25
# Eq 13a
LE_C[j] = (s[j] * Rn_C[j] + rho[j] * c_p[j] * vpd[j] / r_av[j]) /\
(s[j] + gama[j] * (1 + r_vw[j] / r_av[j]))
# Eq 13b
LE_S[j] = (T_sd[j] - T_S[j]) / (T_sd[j] - T_sw[j]) *\
((s[j] * (Rn_S[j] - G[j]) + rho[j] * c_p[j] * vpd[j] / r_as[j]) /
(s[j] + gama[j]))
# Estimate latent heat fluxes when no water in the top-soil is avaiable for evaporation
j = np.logical_and(Tr_K > T_star, i)
# Eq 14a
T_S[j] = T_sd[j]
# Eq 14b
T_C[j] = ((Tr_K[j]**4 - (1 - f_theta[j]) * T_S[j]**4) /
f_theta[j])**0.25
# Eq 15a
LE_C[j] = (T_vd[j] - T_C[j]) / (T_vd[j] - T_vw[j]) *\
((s[j] * Rn_C[j] + rho[j] * c_p[j] * vpd [j] / r_av[j]) /
(s[j] + gama[j] * (1 + r_vw[j] / r_av[j])))
# Eq 15b
LE_S[j] = 0
# Estimate sensible heat fluxes as residuals
H_C[i] = Rn_C[i] - LE_C[i]
H_S[i] = Rn_S[i] - G[i] - LE_S[i]
# Calculate total fluxes
H = np.asarray(H_C + H_S)
LE = np.asarray(LE_C + LE_S)
# Now L can be recalculated and the difference between iterations
# derived
if isinstance(UseL, bool):
L[i] = MO.calc_L(u_friction[i], T_A_K[i], rho[i], c_p[i], H[i],
LE[i])
L_diff = np.asarray(np.fabs(L - L_old) / np.fabs(L_old))
L_diff[np.isnan(L_diff)] = float('inf')
L_old = np.array(L)
L_old[L_old == 0] = 1e-36
# Calculate again the friction velocity with the new stability
# correctios
u_friction[i] = MO.calc_u_star(u[i], z_u[i], L[i], d_0[i], z_0M[i])
u_friction = np.asarray(np.maximum(u_friction_min, u_friction))
u_friction_s[i] = MO.calc_u_star(u[i], z_u[i], L[i],
np.zeros(d_0[i].shape),
np.zeros(z_0M[i].shape) + 0.005)
u_friction_s = np.asarray(np.maximum(u_friction_min, u_friction_s))
return [
flag, T_S, T_C, T_sd, T_vd, T_sw, T_vw, T_star, Ln_S, Ln_C, LE_C, H_C,
LE_S, H_S, G, r_vw, r_vd, r_av, r_as, L, n_iterations
]
def pet_fao56(T_A_K,
u,
ea,
p,
Sdn,
z_u,
z_T,
f_cd=1,
reference=SHORT_REFERENCE):
'''Calcultaes the latent heat flux for well irrigated and cold pixel using
FAO56 potential ET from a short (grass) crop
Parameters
----------
T_A_K : float or array
Air temperature (Kelvin).
u : float or array
Wind speed above the canopy (m s-1).
ea : float or array
Water vapour pressure above the canopy (mb).
p : float or array
Atmospheric pressure (mb), use 1013 mb by default.
Sdn : float or array
Solar irradiance (W m-2).
z_u : float or array
Height of measurement of windspeed (m).
z_T : float or array
Height of measurement of air temperature (m).
f_cd : float or array
cloudiness factor, default = 1
reference : bool
If true, reference ET is for a tall canopy (i.e. alfalfa)
'''
# Atmospheric constants
delta = 10. * met.calc_delta_vapor_pressure(
T_A_K) # slope of saturation water vapour pressure in mb K-1
lambda_ = met.calc_lambda(T_A_K) # latent heat of vaporization MJ kg-1
c_p = met.calc_c_p(p, ea) # Heat capacity of air
psicr = met.calc_psicr(c_p, p, lambda_) # Psicrometric constant (mb K-1)
es = met.calc_vapor_pressure(
T_A_K) # saturation water vapour pressure in mb
rho = met.calc_rho(p, ea, T_A_K)
# Net shortwave radiation
# Sdn = Sdn * 3600 / 1e6 # W m-2 to MJ m-2 h-1
albedo = 0.23
Sn = Sdn * (1.0 - albedo)
# Net longwave radiation
Ln = calc_Ln(T_A_K, ea, f_cd=f_cd)
# Net radiation
Rn = Sn + Ln
if reference == TALL_REFERENCE:
G_ratio = 0.04
h_c = 0.5
else:
G_ratio = 0.1
h_c = 0.12
R_c = 70.0
# Soil heat flux
G = G_ratio * Rn
# Windspeed at 2m height
z_0M = h_c * 0.123
d = h_c * 2. / 3.
u_2 = wind_profile(u, z_u, z_0M, d, 2.0)
R_a = 208. / u_2
LE = (delta * (Rn - G) + rho * c_p * (es - ea) / R_a) / (delta + psicr *
(1.0 + R_c / R_a))
return LE