smrt.permittivity package 

Provide many formulations for the permittivity of various materials (ice, water, etc.) or for mixing formulae.

The former are to be used as input of the functions in smrt.inputs in order to prescribe the scatterers and background permittivity, while the latter are to be used in smrt.emmodels to reformulate how the effective permittivity is calculated. This latter usage is very specific and should not concern most users. See smrt.emmodel.symsce_torquato21.derived_SymSCETK21 and smrt.emmodel.iba.derived_IBA.

For developers

To add a new permittivity function proceed as follows:

1. To add a new permittivity formulation add a function either in an existing file or in a new file (recommended for testing). E.g. for salty ice permittivity formulations should be in saltyice.py and so on.

2. Any function defining a permittivity model must declare the mapping between the layer properties and the arguments of the function (see ice.py for examples). It means that the arguments of the function must be listed (in order) in the @required_layer_properties decorator. In most cases, the name of the arguments should be the same as a properties, but this is not strictly necessary, only the order matters. For example:

@required_layer_properties("temperature", "salinity")
def permittivity_something(frequency, t, s):

maps the layer property “temperature” to the argument “t” of the function (and “salinity” to s) However, it is recommended to change t into temperature for sake of clarity.

For curious ones, this declaration is required because the function can be called either with its arguments (normal case) or with only two arguments like this (frequency, layer). In this latter case, the arguments required by the original function are automatically extracted from the layer attributes (=properties) based on the declaration in @required_layer_properties. This complication is necessary because there is no way in Python to inspect the name of the arguments of a function, so the need for explicit declaration.

3. To use the new function, import the module (e.g. from smrt.permittivity.ice import permittivity_something) and pass this function to smrt.core.snowpack.make_snowpack or smrt.core.layer:make_snow_layer.

smrt.permittivity.ice module 

Contain functions to compute various permittivity of ice.

ice_permittivity_maetzler06(frequency, temperature)

Calculate the complex ice dielectric constant depending on the frequency and temperature based on Mätzler (2006).

This is the default model used in smrt.inputs.make_medium.make_snow_layer().

Parameters:

frequency – frequency in Hz.
temperature – temperature in K.

Returns:

complex permittivity of pure ice.

Usage:

from smrt.permittivity.ice import ice_permittivity_maetzler06
eps_ice = ice_permittivity_maetzler06(frequency=18e9, temperature=270)

Note

Ice permittivity is automatically calculated in smrt.inputs.make_medium.make_snow_layer() and is not set by the electromagnetic model module. An alternative to ice_permittivity_maetzler06 may be specified as an argument to the make_snow_layerfunction. The usage example is provided for external reference or testing purposes.

References

Mätzler, C. (2006). Thermal Microwave Radiation: Applications for Remote Sensing p456-461, https://doi.org/10.1049/PBEW052E

ice_permittivity_maetzler98(frequency, temperature)

Compute permittivity of ice (accounting for ionic impurities in ice?), equations from Hufford (1991) as given in Mätzler (1998).

Parameters:

temperature – ice temperature in K.
frequency – Frequency in Hz.

Returns:

complex permittivity of pure ice.

References

Hufford, G. A model for the complex permittivity of ice at frequencies below 1 THz. Int J Infrared Milli Waves 12, 677–682 (1991). https://doi.org/10.1007/BF01008898

Mätzler, C. (1998). Microwave Properties of Ice and Snow. In: Schmitt, B., De Bergh, C., Festou, M. (eds) Solar System Ices. Astrophysics and Space Science Library, vol 227. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5252-5_10

ice_permittivity_maetzler87(frequency, temperature)

Calculate the complex ice dielectric constant depending on the frequency and temperature based on Mätzler, C. and Wegmüller (1987).

Parameters:

frequency – frequency in Hz.
temperature – temperature in K.

Returns:

complex permittivity of pure ice.

Usage:

from smrt.permittivity.ice import ice_permittivity_maetzler87
eps_ice = ice_permittivity_maetzler87(frequency=18e9, temperature=270)

Note

This is only suitable for testing at -5 deg C and -15 deg C. If used at other temperatures a warning will be displayed.

References

Mätzler, C. and Wegmüller (1987). Dielectric properties of fresh-water ice at microwave frequencies. J. Phys. D: Appl. Phys. 20, 1623-1630. https://doi.org/10.1088/0022-3727/20/12/013

ice_permittivity_tiuri84(frequency, temperature)

Calculate the complex ice dielectric constant depending on the frequency and temperature based on Tiuri et al. (1984).

Parameters:

frequency – frequency in Hz.
temperature – temperature in K.

Returns:

complex permittivity of pure ice.

Usage:

from smrt.permittivity.ice import ice_permittivity_tiuri84
eps_ice = ice_permittivity_tiuri84(frequency=1.9e9, temperature=250)

References

Tiuri et al. (1984). The Complex Dielectric Constant of Snow at Microwave Frequencies. IEEE Journal of Oceanic Engineering, vol. 9, no. 5., pp. 377-382. https://doi.org/10.1109/JOE.1984.1145645.

ice_permittivity_hufford91_maetzler87(frequency, temperature)

Calculate the complex ice dielectric constant depending on the frequency and temperature with the real part of permittivity follows Mätzler and Wegmuller (1987) and the imaginary part is based on Hufford 1991.

Note

The Hufford model is derived for frequencies up to 1000 GHz and temperatures from -40°C to 0°C. This gives exact agreement with the MEMLS_ice model version used in Rückert et al. (2023).

Parameters:

frequency – Frequency in Hz.
temperature – ice temperature in K.

Returns:

complex permittivity of pure ice.

References

Hufford, G. A model for the complex permittivity of ice at frequencies below 1 THz. Int J Infrared Milli Waves 12, 677–682 (1991). https://doi.org/10.1007/BF01008898

Mätzler, C. and Wegmüller (1987). Dielectric properties of fresh-water ice at microwave frequencies. J. Phys. D: Appl. Phys. 20, 1623-1630. https://doi.org/10.1088/0022-3727/20/12/013

Rückert, J., Huntemann, M., Tonboe, RT., and Spreen, G., (2023). Modeling Snow and Ice Microwave Emissions in the Arctic for a Multi-Parameter Retrieval of Surface and Atmospheric Variables From Microwave Radiometer Satellite Data, Earth and Space Scienc, 10(10), https://doi.org/10.1029/2023EA003177

smrt.permittivity.water module 

Provide equations to compute the effective permittivity of water.

water_permittivity_maetzler87(frequency, temperature)

Calculate the complex water dielectric constant depending on the frequency and temperature based on Mätzler & Wegmuller (1987).

Parameters:

frequency – Frequency in Hz.
temperature – Temperature in K.

Raises:

Exception – If liquid water > 0 or salinity > 0 (model unsuitable).

Returns:

Complex permittivity of pure ice.

Return type:

complex

References

Matzler, C., Wegmuller, U., (1987) Dielectric properties of freshwater ice at microwave frequencies. J. Phys. D: Appl. Phys, 20, 1623–1630, https://doi.org/10.1088/0022-3727/20/12/013.

water_permittivity(frequency, temperature)

Calculate the complex water dielectric constant depending on the frequency and temperature based on Mätzler & Wegmuller (1987).

Parameters:

frequency – Frequency in Hz.
temperature – Temperature in K.

Raises:

Exception – If liquid water > 0 or salinity > 0 (model unsuitable).

Returns:

Complex permittivity of pure ice.

Return type:

complex

References

Matzler, C., Wegmuller, U., (1987) Dielectric properties of freshwater ice at microwave frequencies. J. Phys. D: Appl. Phys, 20, 1623–1630, https://doi.org/10.1088/0022-3727/20/12/013.

water_permittivity_tiuri80(frequency, temperature)

Calculate the complex water dielectric constant reported by Tiuri and Schultz (1980).

Parameters:

frequency – Frequency in Hz.
temperature – Temperature in K.

Raises:

SMRTError – If the water temperature is below the freezing point.

Returns:

Complex permittivity of water.

Return type:

complex

References

Tiuri, M. and Schultz, H., Theoretical and experimental studies of microwave radiation from a natural snow field. In Rango, A. , ed. Microwave remote sensing of snowpack properties. Proceedings of a workshop … Fort Collins, Colorado, May 20-22, 1980. Washington, DC, National Aeronautics and Space Center, 225-234. (Conference Publication 2153.) https://ntrs.nasa.gov/api/citations/19810010984/downloads/19810010984.pdf

water_permittivity_turner16(frequency, temperature)

Calculate the complex water dielectric constant depending on the frequency and temperature based on Turner et al. (2016).

Parameters:: frequency – Frequency in Hz. temperature: Temperature in K.

References:

Turner, D. D., Kneifel, S., & Cadeddu, M. P. (2016). An Improved Liquid Water Absorption Model at Microwave Frequencies for Supercooled Liquid Water Clouds. Journal of Atmospheric and Oceanic Technology, 33(1), 33–44. https://doi.org/10.1175/jtech-d-15-0074.1

debye_delta_i(a_i, b_i, tempC): Compute Delta_i as a function of temperature (T in °C). Eq. 9 in Turner 2016

debye_tau_i(c_i, d_i, tempC, t_c): Compute Tau_i as a function of temperature (T in °C). Eq. 10 in Turner 2016

debye_A_i(tau_i, delta_i, frequency): Compute A_i relaxation term. Eq. 7 in Turner 2016

debye_B_i(tau_i, delta_i, frequency): Compute B_i relaxation term. Eq. 8 in Turner 2016

smrt.permittivity.wetice module 

Provide equations to compute the effective permittivity of wet ice.

wetice_permittivity_bohren83(frequency, temperature, liquid_water)

Calculate the dielectric constant of wet particules of ice using Maxwell Garnet equation using water as the background and ice as the inclusions. As reported by Bohren and Huffman 1983 according to Ya Qi Jin (1993), eq 8-69, p282.

Note

smrt.permittivity.saline_ice module 

Provide permittivity formulations for use with saline ice, and possibly in some cases with saline snow.

See also smrt.permittivity.saline_snow.py in the latter case.

impure_ice_permittivity_maetzler06(frequency, temperature, salinity)

Compute permittivity of impure ice from Maetzler (2006).

Note

Model developed for salinity around 0.013 PSU. The extrapolation is based on linear assumption to salinity, so it is not recommended for much higher salinity.

Parameters:

temperature – ice temperature in K.
salinity – salinity of ice in kg/kg (see PSU constant in smrt module).

Returns:

complex permittivity of saline ice.

Usage:

from smrt.permittivity.saline_ice import impure_ice_permittivity_maetzler06
eps_ice = impure_ice_permittivity_maetzler06(frequency=18e9, temperature=270, salinity=0.013)

References

Mätzler, C. (2006). Thermal Microwave Radiation: Applications for Remote Sensing p456-461, https://doi.org/10.1049/PBEW052E

saline_ice_permittivity_pvs_mixing(frequency, temperature, brine_volume_fraction, brine_inclusion_shape='spheres', brine_mixing_ratio: float | None = None, ice_permittivity_model=None, brine_permittivity_model=None)

Compute effective permittivity of saline ice using the Polder Van Santen mixing formulaes.

Parameters:

frequency – frequency in Hz.
temperature – ice temperature in K.
brine_volume_fraction – brine / liquid water fraction in sea ice.
brine_inclusion_shape – Assumption for shape(s) of brine inclusions. Can be a string for single shape, or a list/tuple/dict of strings for mixture of shapes. So far, we have the following shapes: “spheres” and “random_needles” (i.e. randomly-oriented elongated ellipsoidal inclusions). If the argument is a dict, the keys are the shapes and the values are the mixing ratio. If it is a list, the mixing_ratio argument is required.
brine_mixing_ratio – The mixing ratio of the shapes. This is only relevant when inclusion_shape is a list/tuple. Mixing ratio must be a sequence with length len(inclusion_shape)-1. The mixing ratio of the last shapes is deduced as the sum of the ratios must equal to 1.
ice_permittivity_model – pure ice permittivity formulation (default is ice_permittivity_matzler87).
brine_permittivity_model – brine permittivity formulation (default is brine_permittivity_stogryn85).

Returns:

complex permittivity of saline ice.

smrt.permittivity.brine module 

Contain functions to compute various properties of brines.

brine_conductivity_stogryn85(temperature)

Compute ionic conductivity of dissolved salts, Stogryn and Desargant, 1985.

Parameters:: temperature – thermometric temperature in K.
Returns:: Brine conductivity.

References

Stogryn, A., & Desargant, G. (1985). The dielectric properties of brine in sea ice at microwave frequencies. IEEE Transactions on Antennas and Propagation, 33(5), 523–532. https://doi.org/10.1109/tap.1985.1143610

brine_relaxation_time_stogryn85(temperature)

Compute relaxation time of brine, Stogryn and Desargant, 1985.

Parameters:: temperature – thermometric temperature in K.
Returns:: Brine relaxation time in second.

References

Stogryn, A., & Desargant, G. (1985). The dielectric properties of brine in sea ice at microwave frequencies. IEEE Transactions on Antennas and Propagation, 33(5), 523–532. https://doi.org/10.1109/tap.1985.1143610

brine_salinity(temperature)

Compute the salinity of brine (in ppt) for a given temperature.

Note

The origin of this code is unknown, to be investigate. It was attributed to (Cox and Weeks, 1975) in earlier commits but their Eq 15 is different.

Parameters:: temperature – snow temperature in K.
Returns:: Brine salinity in ppt.

brine_salinity_coxandweeks75(temperature)

Compute the salinity of brine (in ppt) for a given temperature (Cox and Weeks, 1975, equation 15).

Parameters:: temperature – snow temperature in K.
Returns:: Brine salinity in ppt.

References

Cox, G. F. N., and Weeks, W-f., and Cold Regions Research and Engineering Laboratory (U.S.). 1975. Brine Drainage and Initial Salt Entrapment in Sodium Chloride Ice. Hanover, N.H.: U.S. Dept. of Defense, Dept. of the Army, Corps of Engineers, Cold Regions Research and Engineering Laboratory.

brine_salinity_assur60poe72(temperature)

Compute the salinity of brine (in psu) for a given temperature (Assur, 1960; Poe et al., 1972; as cited in Ulaby and Long (2014) (equation 4.46)).

Validity range:: -43.2°C <= temperature in K <= -2°C

Parameters:: temperature – snow temperature in K
Returns:: Brine salinity in psu.

References

Ulaby, Fawwaz, and David Long. Microwave Radar and Radiometric Remote Sensing. University of Michigan Press, 2014. https://doi.org/10.3998/0472119356.

static_brine_permittivity_stogryn85(temperature)

Compute static dielectric constant of brine, after Stogryn and Desargant, 1985.

Parameters:: temperature – thermometric temperature in K.

References

Stogryn, A., & Desargant, G. (1985). The dielectric properties of brine in sea ice at microwave frequencies. IEEE Transactions on Antennas and Propagation, 33(5), 523–532. https://doi.org/10.1109/tap.1985.1143610

permittivity_high_frequency_limit_stogryn85(temperature)

Compute permittivity, after Stogryn and Desargant, 1985.

Parameters:: temperature – ice or snow temperature in K.

References

Stogryn, A., & Desargant, G. (1985). The dielectric properties of brine in sea ice at microwave frequencies. IEEE Transactions on Antennas and Propagation, 33(5), 523–532. https://doi.org/10.1109/tap.1985.1143610

water_freezing_temperature(salinity)

Calculate temperature at which saline water freezes.

Note

Use polynomial fits of the Gibbs function given in TEOS-10: The international thermodynamic equation of seawater - 2010 (http://www.teos-10.org/pubs/TEOS-10_Manual.pdf). The error of this fit ranges between -5e-4 K and 6e-4 K when compared with the temperature calculated from the exact in-situ freezing temperature, which is found by a Newton-Raphson iteration of the equality of the chemical potentials of water in seawater and in ice.

Parameters:: salinity – salinity of ice in kg/kg (see PSU constant in smrt module)

brine_volume_cox83_lepparanta88(temperature, salinity, porosity=0, bulk_density=None)

Compute brine volume fraction using coefficients from Cox and Weeks (1983). If ice temperature is below -2 deg C, coefficients determined by Lepparanta and Manninen (1988).

Parameters:

temperature – ice temperature in K.
salinity – salinity of ice in kg/kg (see PSU constant in smrt module).
porosity – fractional air volume in ice (0..1). Default is 0.
bulk_density – density of bulk ice in kg m ^-3.

References

Cox, G. F. N., and Weeks, W-f.. Equations for Determining the Gas and Brine Volumes in Sea-Ice Samples. Journal of Glaciology. 1983;29(102):306-316. https://doi.org/10.3189/S0022143000008364

Leppäranta, M., & Manninen, T. (1988). The brine and gas content of sea ice with attention to low salinities and high temperatures. Finnish Institute of Marine Research. http://hdl.handle.net/1834/23905

brine_volume(*args, **kwargs)

brine_volume_frankenstein67(temperature, salinity)

Compute brine volume fraction using the simpliest equation of Frankenstein and Garner (1967).

Parameters:

temperature – ice temperature in K.
salinity – salinity of ice in kg/kg (see PSU constant in smrt module).

References

Frankenstein G, and Garner R. Equations for Determining the Brine Volume of Sea Ice from −0.5° to −22.9°C. Journal of Glaciology.1967;6(48):943-944. https://doi.org:/10.3189/S0022143000020244

brine_volume_function_stogryn_1987(temperature, salinity)

Compute brine volume fraction using coefficients from Stogryn (1987).

Parameters:

temperature – ice temperature in K.
salinity – salinity of ice in kg/kg (see PSU constant in smrt module).

Usage:

salinity = 33*PSU
temperature = 270
make_ice_column('firstyear',
                thickness=[1.0],
                microstructure_model='exponential',
                temperature=temperature,
                salinity=salinity,
                corr_length=[0.2e-3],
                water_salinity=34,
                brine_volume_fraction=brine_volume_function_stogryn_1987(temperature,salinity))

Note: example usage might change

References

A. Stogryn (1987). An Analysis of the Tensor Dielectnc Constant of Sea Ice at Microwave Frequencies, IEEE Transactions on Geoscience and Remote Sensing, vol. GE-25, no. 2, pp. 147-158 https://doi.org:/10.1109/TGRS.1987.289814.

smrt.permittivity.saline_water module 

Provide equations to compute the effective permittivity of saline water.

seawater_permittivity_klein76(frequency, temperature, salinity)

Calculate permittivity (dielectric constant) of water using an empirical relationship described by Klein and Swift (1976).

Parameters:

frequency – frequency in Hz.
temperature – water temperature in K.
salinity – water salinity in kg/kg (see PSU constant in smrt module).

Returns:

complex water permittivity for a frequency f.

Raises:

SMRTError – If the water temperature is lower than the freezing point at the given salinity.

References

Klein, A.L., and Swift, C.T. (1975) An Improved Model for-the Dielectric Constant of Sea Water at Microwave. Frequencies. https://aquarius.oceansciences.org/docs/ed_klien_swift_1978.pdf

seawater_permittivity_stogryn71(frequency, temperature)

Compute dielectric constant of brine, complex_b (Stogryn, 1971 approach)

Note

From polynomial fit in Stogryn and Desargent, 1985

Parameters:

frequency – frequency in Hz.
temperature – water temperature in K.

Returns:

complex water permittivity.

Source:: Matlab code, Ludovic Brucker.

References

A. Stogryn. (1971) Equations for Calculating the Dielectric Constant of Saline Water (Correspondence), IEEE Transactions on Microwave Theory and Techniques, vol. 19, no. 8, pp. 733-736, https://doi.org/10.1109/TMTT.1971.1127617

brine_permittivity_stogryn85(frequency, temperature)

Compute permittivity and loss of brine using equations given in Stogryn and Desargant (1985).

Parameters:

frequency – frequency in Hz.
temperature – temperature in K.

Returns:

complex water permittivity for a frequency f.

References

A. Stogryn and G. Desargant. (1985) The dielectric properties of brine in sea ice at microwave frequencies, IEEE Transactions on Antennas and Propagation, vol. 33, no. 5, pp. 523-532, https://doi.org/10.1109/TAP.1985.1143610

seawater_permittivity_stogryn95(frequency, temperature, salinity)

Compute seawater dielectric constant using Stogryn 1995.

Parameters:

frequency – frequency in Hz.
temperature – water temperature in K.
salinity – water salinity in kg/kg (see PSU constant in smrt module).

Returns:

complex water permittivity for a frequency f.

Source:: Stogryn 1995 + http://rime.aos.wisc.edu/MW/models/src/eps_sea_stogryn.f90; Matlab code, Ludovic Brucker

References

Stogryn, A. P., Bull, H. T., Rubayi, K., & Iravanchy, S. (1995). The microwave dielectric properties of sea and fresh water. GenCorp Aerojet.

seawwater_permittivity_boutin23_2function(frequency, temperature, salinity)

Compute the permittivity using BVZ 2 from Equations (7) and (8) in Boutin et al. (2023).

BVZ 2functions’ has been derived from L-Band GW2020 measurements, following the assumptions of Somaraju and Trumpf (2006). It reasonnably fits P-Band dielectric constants laboratory measurements over the 0-150 pss range (Le vine et al. 2025, IEEE TGRS). With respect to the BV (Boutin et al. 2021) model, the following changes have been performed:

Conductivity=pss78 conductivity-salinity relationship from TEOS10.

tau = tauMW(1+gSST) instead of tau=tauMW.

alpha = (par(1)-par(2).SST).

Note

This function requires the Gibbs SeaWater Oceanographic Toolbox package (gsw): https://github.com/TEOS-10/GSW-python

Parameters:

frequency – frequency in Hz.
temperature – ice temperature in K.

Returns:

complex water permittivity for a frequency f.

References

Boutin, J., Vergely, J-L., Bonjean, F., Perrot, X., Zhou, Y., and Dinnat, E.P., (2023). New Seawater Dielectric Constant Parametrization and Application to SMOS Retrieved Salinity, IEEE Transactions on Geoscience and Remote Sensing, vol. 61, pp. 1-13, 2023, Art no. 2000813, https://doi.org/10.1109/TGRS.2023.3257923.

Boutin, J., Vergely, J. L., Perrot, X., Zhou, Y., Dinnat, E., and Sabia, R., (2021). Seawater Dielectric Constant At L-Band: How Consistent Are New Parametrisations Inferred from Smos and Laboratory Measurements?, IEEE International Geoscience and Remote Sensing Symposium IGARSS, Brussels, Belgium, 2021, pp. 7465-7467, https://doi.org/10.1109/IGARSS47720.2021.9554909.

Le Vine, D. M., Lang, R. H., Li, M., Dinnat, E. P., Boutin, J., and Zhou, Y., (2025). The Dielectric Constant of Sea Water at P-Band for Salinity From 0 to 150 pss, IEEE Transactions on Geoscience and Remote Sensing, vol. 63, pp. 1-11, 2025, Art no. 4202011, https://doi.org/10.1109/TGRS.2025.3532180.

Somaraju, R., and Trumpf, J., (2006). Frequency, Temperature and Salinity Variation of the Permittivity of Seawater, IEEE Transactions on Antennas and Propagation, vol. 54, no. 11, pp. 3441-3448, https://doi.org/10.1109/TAP.2006.884290.

seawwater_permittivity_boutin23_3function(frequency, temperature, salinity)

Compute the permittivity using BVZ 2 from Equations (9,10,11) Boutin et al. (2023).

Model derived from L-Band GW2020 measurements and validated wit SMOS SSS retrievals. In order to better fit GW2020 measurements, and with respect to Somaraju and Trumpf (2006) assumptions. An additional dependency of alpha with S has been introduced. This parametization is not valid outside the 0-38pss range. With respect to the BV (Boutin et al. 2020) model, the following changes have been performed:

Conductivity=pss78 conductivity-salinity relationship (TEOS10).

tau=tauMW(1+gSST) instead of tau=tauMW.

alpha=(par(1)-par(2).SST)(1+hSSS).

gSST and hSSS are polynomial functions of SST and SSS respectively.

par(1) and par(2) : linear fit parameters of alpha.

Notations and fresh parameters as in Meissner and Wentz (2004).

Note

This function requires the Gibbs SeaWater Oceanographic Toolbox package (gsw): https://github.com/TEOS-10/GSW-python

Parameters:

frequency – frequency in Hz.
temperature – ice temperature in K.

Returns:

complex water permittivity for a frequency f.

References

Boutin, J., Vergely, J-L., Bonjean, F., Perrot, X., Zhou, Y., and Dinnat, E.P., (2023). New Seawater Dielectric Constant Parametrization and Application to SMOS Retrieved Salinity, IEEE Transactions on Geoscience and Remote Sensing, vol. 61, pp. 1-13, 2023, Art no. 2000813, https://doi.org/10.1109/TGRS.2023.3257923.

Boutin, J., Vergely, J. L., Perrot, X., Zhou, Y., Dinnat, E., and Sabia, R., (2021). Seawater Dielectric Constant At L-Band: How Consistent Are New Parametrisations Inferred from Smos and Laboratory Measurements?, IEEE International Geoscience and Remote Sensing Symposium IGARSS, Brussels, Belgium, 2021, pp. 7465-7467, https://doi.org/10.1109/IGARSS47720.2021.9554909.

Meissner, T., and Wentz, F. J., (2004). The complex dielectric constant of pure and sea water from microwave satellite observations, IEEE Transactions on Geoscience and Remote Sensing, vol. 42, no. 9, pp. 1836-1849, https://doi.org/10.1109/TGRS.2004.831888

Somaraju, R., and Trumpf, J., (2006). Frequency, Temperature and Salinity Variation of the Permittivity of Seawater, IEEE Transactions on Antennas and Propagation, vol. 54, no. 11, pp. 3441-3448, https://doi.org/10.1109/TAP.2006.884290.

smrt.permittivity.generic_mixing_formula module 

Provide functions that are not tied to a particular electromagnetic model and are available to be imported by any electromagnetic model.

Note

It is the responsibility of the developer to ensure these functions, if used, are appropriate and consistent with the physics of the electromagnetic model.

Calculate the effective permittivity of snow by solution of quadratic Polder Van Santen equation for spherical or random needle inclusions only, potentially a mixing of these shapes (Sihvola, 1999).

Note

See general_polder_van_santen() for an alternative function (more generic, but slower).

Parameters:

frac_volume – Fractional volume of inclusions.
e0 – Permittivity of background (default is 1).
eps – Permittivity of scattering material (default is 3.185 to compare with MEMLS)
depolarization_factors – [Optional] Depolarization factors, spherical isotropy is default. It is not taken into account here.
length_ratio – Length_ratio. Used to estimate depolarization factors when they are not given.
inclusion_shape – Assumption for shape(s) of brine inclusions. Can be a string for single shape, or a list/tuple/dict of strings for mixture of shapes. So far, we have the following shapes: “spheres” and “random_needles” (i.e. randomly-oriented elongated ellipsoidal inclusions). If the argument is a dict, the keys are the shapes and the values are the mixing ratio. If it is a list, the mixing_ratio argument is required.
mixing_ratio – The mixing ratio of the shapes. This is only relevant when inclusion_shape is a list/tuple. Mixing ratio must be a sequence with length len(inclusion_shape)-1. The mixing ratio of the last shapes is deduced as the sum of the ratios must equal to 1.

Returns:

Effective permittivity

Usage:

>>> from smrt.permittivity.generic_mixing_formula import polder_van_santen
>>> effective_permittivity = polder_van_santen(frac_volume, e0, eps)

for a mixture of 30% spheres and 70% needles >>> effective_permittivity = polder_van_santen(frac_volume, e0, eps, inclusion_shape={“spheres”: 0.3, “random_needles”: 0.7})

or >>> effective_permittivity = polder_van_santen(frac_volume, e0, eps, inclusion_shape=(“spheres”, “random_needles”), mixing_ratio=0.3)

Todo

Extend Polder Van Santen model to account for ellipsoidal inclusions

References

Sihvola, A.: Electromagnetic Mixing Formulas and Applications, 1999, INSTITUTION OF ENGINEERING & T http://www.ebook.de/de/product/21470462/a_sihvola_electromagnetic_mixing_formulas_and_applications.html

general_polder_van_santen(frac_volume: float, e0: complex = 1, eps: complex = 3.185, depolarization_factors=None, length_ratio: float | None = None, inclusion_shape: str | None = None)

Calculate the effective permittivity of snow by solution of quadratic Polder Van Santen equation.

Note

For spherical or random needle inclusions only, potentially a mixing of these shapes (Sihvola, 1999). See polder_van_santen() for an alternative function (faster but specific to some shapes).

Parameters:

frac_volume – Fractional volume of inclusions.
e0 – Permittivity of background (default is 1).
eps – Permittivity of scattering material (default is 3.185 to compare with MEMLS).
depolarization_factors – [Optional] Depolarization factors, spherical isotropy is default. It is not taken into account here.
length_ratio – Length_ratio. Used to estimate depolarization factors when they are not given.
inclusion_shape – Assumption for shape(s) of brine inclusions. Can be a string for single shape, or a list/tuple/dict of strings for mixture of shapes. So far, we have the following shapes: “spheres” and “random_needles” (i.e. randomly-oriented elongated ellipsoidal inclusions). If the argument is a dict, the keys are the shapes and the values are the mixing ratio. If it is a list, the mixing_ratio argument is required.
mixing_ratio – The mixing ratio of the shapes. This is only relevant when inclusion_shape is a list/tuple. Mixing ratio must be a sequence with length len(inclusion_shape)-1. The mixing ratio of the last shapes is deduced as the sum of the ratios must equal to 1.

Returns:

Effective permittivity.

References: Sihvola, A.: Electromagnetic Mixing Formulas and Applications, 1999, INSTITUTION OF ENGINEERING & T http://www.ebook.de/de/product/21470462/a_sihvola_electromagnetic_mixing_formulas_and_applications.html

Calculate the effective permittivity of snow by solution of quadratic Polder Van Santen equation for spherical or random needle inclusions only, potentially a mixing of these shapes (Sihvola, 1999).

Note

See general_polder_van_santen() for an alternative function (more generic, but slower).

Parameters:

frac_volume – Fractional volume of inclusions.
e0 – Permittivity of background (default is 1).
eps – Permittivity of scattering material (default is 3.185 to compare with MEMLS)
depolarization_factors – [Optional] Depolarization factors, spherical isotropy is default. It is not taken into account here.
length_ratio – Length_ratio. Used to estimate depolarization factors when they are not given.
inclusion_shape – Assumption for shape(s) of brine inclusions. Can be a string for single shape, or a list/tuple/dict of strings for mixture of shapes. So far, we have the following shapes: “spheres” and “random_needles” (i.e. randomly-oriented elongated ellipsoidal inclusions). If the argument is a dict, the keys are the shapes and the values are the mixing ratio. If it is a list, the mixing_ratio argument is required.
mixing_ratio – The mixing ratio of the shapes. This is only relevant when inclusion_shape is a list/tuple. Mixing ratio must be a sequence with length len(inclusion_shape)-1. The mixing ratio of the last shapes is deduced as the sum of the ratios must equal to 1.

Returns:

Effective permittivity

Usage:

>>> from smrt.permittivity.generic_mixing_formula import polder_van_santen
>>> effective_permittivity = polder_van_santen(frac_volume, e0, eps)

for a mixture of 30% spheres and 70% needles >>> effective_permittivity = polder_van_santen(frac_volume, e0, eps, inclusion_shape={“spheres”: 0.3, “random_needles”: 0.7})

or >>> effective_permittivity = polder_van_santen(frac_volume, e0, eps, inclusion_shape=(“spheres”, “random_needles”), mixing_ratio=0.3)

Todo

Extend Polder Van Santen model to account for ellipsoidal inclusions

References

Sihvola, A.: Electromagnetic Mixing Formulas and Applications, 1999, INSTITUTION OF ENGINEERING & T http://www.ebook.de/de/product/21470462/a_sihvola_electromagnetic_mixing_formulas_and_applications.html

polder_van_santen_three_spherical_components(f1, f2, eps0, eps1, eps2)

Calculate effective permittivity using Polder and van Santen with three components assuming spherical inclusions.

Parameters:

f1 – fractional volume of component 1.
f2 – fractional volume of component 2.
eps0 – permittivity of material 0.
eps1 – permittivity of material 1.
eps2 – permittivity of material 2.

Returns:

Effective permittivity.

References: Sihvola, A.: Electromagnetic Mixing Formulas and Applications, 1999, INSTITUTION OF ENGINEERING & T http://www.ebook.de/de/product/21470462/a_sihvola_electromagnetic_mixing_formulas_and_applications.html

polder_van_santen_three_components(f1, f2, eps0, eps1, eps2, A1, A2)

Calculate effective permittivity using Polder and van Santen with three components

Parameters:

f1 – fractional volume of component 1.
f2 – fractional volume of component 2.
eps0 – permittivity of material 0.
eps1 – permittivity of material 1.
eps2 – permittivity of material 2.
A1 – depolarization factor for material 1.
A2 – depolarization factor for material 2.

Returns:

Effective permittivity.

References: Sihvola, A.: Electromagnetic Mixing Formulas and Applications, 1999, INSTITUTION OF ENGINEERING & T http://www.ebook.de/de/product/21470462/a_sihvola_electromagnetic_mixing_formulas_and_applications.html

maxwell_garnett(frac_volume, e0, eps, depolarization_factors=None, inclusion_shape=None, length_ratio=None)

Calculate effective permittivity using Maxwell-Garnett equation.

Parameters:

frac_volume – Fractional volume of snow.
e0 – Permittivity of background (no default, must be provided).
eps – Permittivity of scattering material (no default, must be provided).
depolarization_factors – [Optional] Depolarization factors, spherical isotropy is default.
length_ratio – Length_ratio. Used if depolarization factors are not given to estimate depolarization factors of spheroids when they are not given.
inclusion_shape – Assumption for shape(s) of brine inclusions. Can be a string for single shape, or a list/tuple/dict of strings for mixture of shapes. So far, we have the following shapes: “spheres” and “random_needles” (i.e. randomly-oriented elongated ellipsoidal inclusions). If the argument is a dict, the keys are the shapes and the values are the mixing ratio. If it is a list, the mixing_ratio argument is required.

Returns:

random orientation effective permittivity.

Usage:

# If used by electromagnetic model module:
from .commonfunc import maxwell_garnett
effective_permittivity = maxwell_garnett(frac_volume=0.2, e0=1, eps=3.185, depol_xyz=[0.3, 0.3, 0.4])

#If accessed from elsewhere, use absolute import
from smrt.emmodel.commonfunc import maxwell_garnett

maxwell_garnett_for_spheres(frac_volume, e0, eps)

Calculate effective permittivity using Maxwell-Garnett equation assuming spherical inclusion.

Note

This function is essentially an optimized version of py:func:maxwell_garnett.

Parameters:

frac_volume – Fractional volume of snow.
e0 – Permittivity of background (no default, must be provided).
eps – Permittivity of scattering material (no default, must be provided).

Returns:

Effective permittivity.

smrt.permittivity.depolarization_factors module 

Provide formaluations for the depolarization factors used with the Polden van Santen or Maxwell Garnett mixing formulations.

depolarization_factors_spheroids(length_ratio: float | None = None, **kwargs)

Calculate depolarization factors for use in effective permittivity models. These are a measure of the anisotropy of the snow. Default is spherical isotropy.

Parameters:: length_ratio – [Optional] ratio of microstructure length measurement in x/y direction to z-direction [unitless].
Returns:: [x, y, z] depolarization factor array.

Usage:

from smrt.permittivity.generic_mixing_formula import depolarization_factors
#default depol of 1
depol_xyz = depolarization_factors()
#provide a length ratio
depol_xyz = depolarization_factors(length_ratio=1.2)

References

Mätzler, C. (1996). Microwave permittivity of dry snow. Geoscience and Remote Sensing, EEE Transactions on Geoscience and Remote Sensing,, 34(2), 573–581. https://doi.org/10.1109/36.485133

Löwe, H., Riche, F., and Schneebeli, M.: A general treatment of snow microstructure exemplified by an improved relation for thermal conductivity, The Cryosphere, 7, 1473–1480, https://doi.org/10.5194/tc-7-1473-2013, 2013.

depolarization_factors_matzler96(frac_volume: float, **kwargs)

Calculate depolarization factors with Mätzler (1996) for the Polden van Santen (Sihvola, 1999) permittivity model.

Parameters:: frac_volume – fractional volume of ice.
Returns:: [x, y, z] depolarization factor array.

References

Mätzler, C. (1996). Microwave permittivity of dry snow. Geoscience and Remote Sensing, EEE Transactions on Geoscience and Remote Sensing,, 34(2), 573–581. https://doi.org/10.1109/36.485133

Sihvola, A.: Electromagnetic Mixing Formulas and Applications, 1999, INSTITUTION OF ENGINEERING & T http://www.ebook.de/de/product/21470462/a_sihvola_electromagnetic_mixing_formulas_and_applications.html

depolarization_factors_oblate_matzler98(frac_volume: float, **kwargs)

Calculate depolarization factors with Mätzler (1998) for the Polden van Santen (Sihvola, 1999) permittivity model.

Parameters:: frac_volume – fractional volume of ice.
Returns:: [x, y, z] depolarization factor array.

References

Mätzler, C. (1998). Improved Born approximation for scattering of radiation in a granular medium. J. Appl. Phys., 83(11), 6111–6117. https://doi.org/10.1063/1.367496

Sihvola, A.: Electromagnetic Mixing Formulas and Applications, 1999, INSTITUTION OF ENGINEERING & T http://www.ebook.de/de/product/21470462/a_sihvola_electromagnetic_mixing_formulas_and_applications.html

smrt.permittivity.saline_snow module 

Provide mixing formulae relevant to saline snow and equations to compute the effective permittivity of saline snow.

These functions are to be used with smrt.emmodel.iba.derived_IBA or smrt.emmodel.symsce_torquato21.derived_SymSCETK21 to change the default of most emmodels (IBA, DMRT, SFT Rayleigh, SCE) using the generic mixing formula Polder van Staten that automatically mixes the permittivities of the background (e.g.) and the scatterer materials (e.g. saline ice) to compute the effective permittivity of snow in a proportion determined by frac_volume.

Note

They should not be used to set the material permittivities as input of smrt.inputs.make_medium.make_snowpack() and similar functions (because the emmodel would re-mix the already mixed materials with the background material).

saline_snow_permittivity_geldsetzer09(frequency, density, temperature, salinity)

Compute permittivity of saline snow using the frequency dispersion model published by Geldsetzer et al. (2009).

Note

In-situ measurements collected had salinity concentration between 0.1e-3 and 12e3 kg/kg, temperatures ranging between 257 and 273 K, and a mean snow density of 352 kg/m3. Validity between 10 MHz and 40 GHz.

Parameters:

frequency – frequency in Hz.
density – snow density in kg m-3.
temperature – ice temperature in K.
salinity – salinity of ice in kg/kg (see PSU constant in smrt module).

Returns:

complex permittivity of saline snow.

Source:: Matlab code, Ludovic Brucker.

References

Geldsetzer, T., Langlois, A., Yackel, J., (2009) Dielectric properties of brine-wetted snow on first-year sea ice, Cold Regions Science and Technology, Volume 58, Issues 1–2,pp 47-56. https://doi.org/10.1016/j.coldregions.2009.03.009.

saline_snow_permittivity_scharien_with_stogryn71(frequency, density, temperature, salinity)

Compute permittivity of saline snow with seawater_permittivity_stogryn71.

See saline_snow_permittivity_scharien documentation.

saline_snow_permittivity_scharien_with_stogryn95(frequency, density, temperature, salinity)

Compute permittivity of saline snow with seawater_permittivity_stogryn95.

See saline_snow_permittivity_scharien documentation.

saline_snow_permittivity_scharien(density, temperature, salinity, brine_permittivity)

Compute permittivity of saline snow using the Denoth / Matzler Mixture Model - Dielectric Constant of Saline Snow.

Note

Assumptions:

Brine inclusion geometry as oblate spheroids. Depolarization factor, A0 = 0.053 (Denoth, 1980).
Brine inclusions are isotropically oriented. Coupling factor, X = 2/3 (Drinkwater and Crocker, 1988).

Validity ranges:

Temperature, Ts, down to - 22.9 degrees Celcius;
Brine salinity, Sb, up to 157ppt; i.e.up to a Normality of 3 for NaCl. Not valid for wet snow.

Parameters:

density – snow density in kg m-3.
temperature – snow temperature in K.
salinity – snow salinity in kg/kg (see PSU constant in smrt module).
brine_permittivity – brine_permittivity.

Returns:

complex permittivity of saline snow.

Source:: Matlab code, Randall Scharien.

References

Denoth A. The Pendular-Funicular Liquid Transition in Snow. Journal of Glaciology. 1980;25(91):93-98. https://doi.org/10.3189/S0022143000010315

Drinkwater MR, Crocker GB. Modelling Changes in Scattering Properties of the Dielectric and Young Snow-Covered Sea Ice at GHz Requencies. Journal of Glaciology. 1988;34(118):274-282. https://doi.org/10.3189/S0022143000007012

smrt.permittivity.snow_mixing_formula module 

Provide mixing formulae relevant to snow and equations to compute the effective permittivity of snow.

These functions are to be used with smrt.emmodel.iba.derived_IBA or smrt.emmodel.symsce_torquato21.derived_SymSCETK21 to change the default of most emmodels (IBA, DMRT, SFT Rayleigh, SCE) using the generic mixing formula Polder van Staten that automatically mixes the permittivities of the background (e.g.) and the scatterer materials (e.g. ice) to compute the effective permittivity of snow in a proportion determined by frac_volume.

Note

They should not be used to set the material permittivities as input of smrt.inputs.make_medium.make_snowpack() and similar functions (because the emmodel would re-mix the already mixed materials with the background material).

wetsnow_permittivity_tinga73(frequency, temperature, density, liquid_water, ice_permittivity_model=None, water_permittivity_model=None)

Compute the effective permittivity proposed by Tinga et al. 1(973) for three-component mixing.

The component 1 is the background (“a” here), the compoment 2 (“w” here) is a spherical shell surrounding the component 3 (“i” here). It was used by Tiuri as well as T. Mote to compute wet snolw permittivity.

Parameters:

temperature – temperature in K.
density – snow density in kg m-3.
liquid_water – fractional volume of water with respect to ice+water volume.
ice_permittivity_model – [optional] permittivity model of ice use for mixing, default is water_permittivity_tiuri80
water_permittivity_model – [optional] permittivity model of water use for mixing, default is ice_permittivity_tiuri84

Returns:

Complex permittivity of snow.

References

Tinga, W.R., Voss, W.A.G. and Blossey, D. F.: General approach to multiphase dielectric mixture theory. Journal of Applied Physics, Vol.44(1973) No.9,pp.3897-3902. https://doi.org/10.1063/1.1662868

Tiuri, M. and Schultz, H., Theoretical and experimental studies of microwave radiation from a natural snow field. In Rango, A. , ed. Microwave remote sensing of snowpack properties. Proceedings of a workshop … Fort Collins, Colorado, May 20-22, 1980. Washington, DC, National Aeronautics and Space Center, 225-234. (Conference Publication 2153.)

compute_frac_volumes(density, liquid_water)

Compute the fractional volume of ice+water, the fractional volume of ice, and the fractional volume of water from the (wet) snow density and the liquid_water which is the volume fraction of liquid with respect to ice + liquid (but no air).

Parameters:

density – snow density in kg m-3.
liquid_water – fractional volume of water with respect to ice+water volume.

Returns:

frac_volume, fi, fw

wetsnow_permittivity_colbeck80_caseI(frequency, temperature, density, liquid_water, ice_permittivity_model=None, water_permittivity_model=None)

Compute the effective permittivity proposed by Colbeck, 1980 for the pendular regime.

Parameters:

temperature – temperature in K.
density – snow density in kg m-3.
liquid_water – fractional volume of water with respect to ice+water volume.
ice_permittivity_model – [optional] permittivity model of ice use for mixing, default is ice_permittivity_maetzler06
water_permittivity_model – [optional] permittivity model of water use for mixing, default is water_permittivity_maetzler87

Returns:

Complex permittivity of snow.

References

Colbeck, S. C. (1980). Liquid distribution and the dielectric constant of wet snow. Goddard Space Flight Center Microwave Remote Sensing of Snowpack Properties, 21–40. https://apps.dtic.mil/sti/tr/pdf/ADP000148.pdf

wetsnow_permittivity_colbeck80_caseII(frequency, temperature, density, liquid_water, ice_permittivity_model=None, water_permittivity_model=None)

Compute the effective permittivity proposed by Colbeck, 1980 for the funicular regime and low dry snow density.

Parameters:

temperature – temperature in K.
density – snow density in kg m-3.
liquid_water – fractional volume of water with respect to ice+water volume.
ice_permittivity_model – [optional] permittivity model of ice use for mixing, default is ice_permittivity_maetzler06
water_permittivity_model – [optional] permittivity model of water use for mixing, default is water_permittivity_maetzler87

Returns:

Complex permittivity of snow.

References

Colbeck, S. C. (1980). Liquid distribution and the dielectric constant of wet snow. Goddard Space Flight Center Microwave Remote Sensing of Snowpack Properties, 21–40. https://apps.dtic.mil/sti/tr/pdf/ADP000148.pdf

wetsnow_permittivity_colbeck80_caseIII(frequency, temperature, density, liquid_water, ice_permittivity_model=None, water_permittivity_model=None)

Compute the effective permittivity proposed by Colbeck, 1980 for the low porosity.

Parameters:

temperature – temperature in K.
density – snow density in kg m-3.
liquid_water – fractional volume of water with respect to ice+water volume.
ice_permittivity_model – [optional] permittivity model of ice use for mixing, default is ice_permittivity_maetzler06
water_permittivity_model – [optional] permittivity model of water use for mixing, default is water_permittivity_maetzler87

Returns:

Complex permittivity of snow.

References

Colbeck, S. C. (1980). Liquid distribution and the dielectric constant of wet snow. Goddard Space Flight Center Microwave Remote Sensing of Snowpack Properties, 21–40. https://apps.dtic.mil/sti/tr/pdf/ADP000148.pdf

wetsnow_permittivity_hallikainen86(frequency, density, liquid_water)

Compute the effective permittivity of a snow mixture calculated with the Modified Debye model by Hallikainen 1986

The implementation of this function follows the equations formulation of the original paper (Hallikainen et al. 1986). Anyway this formulation does not allow the reproduction of the results as reported in the paper. A new formulation of eq. 12a have been presented in the book Microwave Radar and Radiometric Remote Sensing by Ulaby et al. 2014 from which the SMRT function wetsnow_permittivity_hallikainen86_ulaby14 have been implemented. The users are pointed to that definition.

Note

The implemented equation are 10, 11 and 13a-c. The validity of the model is: frequency between 3 and 37GHz; mv between 1% and 12%; dry_snow_density between 0.09 and 0.38g/cm3.

Parameters:

density – snow density in kg m-3.
liquid_water – fractional volume of water with respect to ice+water volume.

Returns:

Complex permittivity of snow.

References

Hallikainen, M., F. Ulaby, and M. Abdelrazik, Dielectric properties of snow in 3 to 37 GHz range, IEEE Trans. on Antennasand Propagation,Vol. 34, No. 11, 1329–1340, 1986. https://doi.org/10.1109/TAP.1986.1143757

Ulaby, F. T., et al. (2014). Microwave radar and radiometric remote sensing.

wetsnow_permittivity_hallikainen86_ulaby14(frequency, density, liquid_water)

Compute the effective permittivity of a snow mixture calculated with the Modified Debye model by Hallikainen 1986 and revised in Microwave Radar and Radiometric Remote Sensing by Ulaby et al. 2014.

Note

Equations implemented are ch 4 pp 143-15 4.60a - 4.61h. The validity of the model is: frequency between 3 and 37GHz; mv between 1% and 12%; dry_snow_density between 0.09 and 0.38g/cm3. Same formulation can be reproduced by the book code https://mrs.eecs.umich.edu/codes/Module4_6/Module4_6.html

Parameters:

density – snow density in kg m-3.
liquid_water – fractional volume of water with respect to ice+water volume.

Returns:

Complex permittivity of snow.

References

Hallikainen, M., F. Ulaby, and M. Abdelrazik, Dielectric properties of snow in 3 to 37 GHz range, IEEE Trans. on Antennasand Propagation,Vol. 34, No. 11, 1329–1340, 1986. https://doi.org/10.1109/TAP.1986.1143757

Ulaby, F. T., et al. (2014). Microwave radar and radiometric remote sensing. Chapter 4

wetsnow_permittivity_wiesmann99(frequency, temperature, density, liquid_water, ice_permittivity_model=None)

Compute the effective permittivity of a snow mixture as presented in MEMLS by Wiesmann and Matzler, 1999.

Note

The version implemented in MEMLS v3 is different.

Parameters:

temperature – temperature in K.
density – snow density in kg m-3.
liquid_water – fractional volume of water with respect to ice+water volume.
ice_permittivity_model – [optional] permittivity model of ice use for mixing, default is ice_permittivity_maetzler06

Returns:

Complex permittivity of snow.

References

Wiesmann, A., Mätzler, C., (1999). Microwave Emission Model of Layered Snowpacks, Remote Sensing of Environment, Volume 70, Issue 3,Pages 307-316, ISSN 0034-4257, https://doi.org/10.1016/S0034-4257(99)00046-2.

wetsnow_permittivity_memls(frequency, temperature, density, liquid_water, ice_permittivity_model=None, water_permittivity_model=None)

Compute the effective permittivity of a snow mixture as calculated in MEMLS using Maxwell-Garnett Mixing rule of water in dry snow for prolate spheroidal water with experimentally determined. Dry snow permittivity is here determined with Polder van Santen.

Parameters:

temperature – temperature in K.
density – snow density in kg m-3.
liquid_water – fractional volume of water with respect to ice+water volume.
ice_permittivity_model – [optional] permittivity model of ice use for mixing, default is ice_permittivity_maetzler06
water_permittivity_model – [optional] permittivity model of water use for mixing, default is water_permittivity_maetzler87

Returns:

Complex permittivity of snow.

References

Mätzler, C., Wiesmann, A., (2009). Extension of the Microwave Emission Model of Layered Snowpacks to Coarse-Grained Snow, Remote Sensing of Environment, Volume 70, Issue 3. https://doi.org/10.1016/S0034-4257(99)00047-4

wetsnow_permittivity_three_component_polder_van_santen(frequency, temperature, density, liquid_water, ice_permittivity_model=None, water_permittivity_model=None)

Compute the effective permittivity of a snow mixture using the three components polder_van_santen, assuming spherical inclusions.

Parameters:

temperature – temperature in K.
density – snow density in kg m-3.
liquid_water – fractional volume of water with respect to ice+water volume.
ice_permittivity_model – [optional] permittivity model of ice use for mixing, default is ice_permittivity_maetzler06
water_permittivity_model – [optional] permittivity model of water use for mixing, default is water_permittivity_maetzler87

Returns:

Complex permittivity of snow.

References

Sihvola, A.: Electromagnetic Mixing Formulas and Applications, 1999, INSTITUTION OF ENGINEERING & T http://www.ebook.de/de/product/21470462/a_sihvola_electromagnetic_mixing_formulas_and_applications.html

depolarization_factors_maetzler96(density)

Compute the empirical depolarization factors of snow estimated by Mäzler 1996.

Note

It is supposed to provide more accurate permittivity=f(density) than using constant depolarization factors in Polder van Santen (e.g. spheres).

Parameters:: density – snow density in kg m-3.
Returns:: [x, y, z] depolarization factor array.

References

Matzler, C., (1996). Microwave permittivity of dry snow. IEEE Transactions on Geoscience and Remote Sensing, vol. 34, no. 2, pp. 573-581, March 1996, https://doi.org/10.1109/36.485133

drysnow_permittivity_maetzler96(density, e0=1, eps=3.185)

default_ice_water_permittivity(ice_permittivity_model, water_permittivity_model)

smrt.permittivity.wetsnow module 

Provide equations to compute the effective permittivity of wet ice.

The wetsnow module is to be removed in a future version. It has been renamed wetice because the naming was ambiguous.

wetsnow_permittivity(frequency, temperature, liquid_water)

Calculate the dielectric constant of wet particles of ice using Bohren and Huffman (1983) according to Ya Qi Jin (1996), eq 8-69, p282.

Note