Inverse Passive Soil Moisture Retrieval Algorithm
Theory
Dielectric Mixing Model
Assumptions
Correction Procedures
References

If you are interested in a copy of this model,
contact Dr. Charles Laymon at
charles.laymon@msfc.nasa.gov

Theory

The theory behind microwave remote sensing of soil moisture is based on the large contrast between the dielectric properties of liquid water (~ 80) and dry soil (< 4). The dielectric properties of wet soil were studied by several investigators (e.g., Wang and Schmugge, 1980; Dobson et al., 1985; Ulaby et al., 1986). As the moisture increases, the dielectric constant of the soil-water mixture increases and this change is detectable by microwave sensors (Njoku and Kong, 1977). The emission of microwave energy is proportional to the product of the surface temperature and the surface emissivity, which is commonly referred to as the microwave brightness temperature (T_B). For typical soil moisture applications using longer microwave wavelengths at low altitude, temperature contributions from the atmosphere and the sky can be neglected. Thus, the brightness temperature of an emitter of microwave radiation is related to the physical temperature of the source through the emissivity such that

,

(1)

where R is the smooth surface reflectivity, T_sfc is the thermometric temperature of the soil surface, and e=(1-R) is the emissivity, which depends on the dielectric constant of the medium being measured (Schmugge, 1990). Although the relationship between emissivity and T_B is linear, water has a nonlinear dependence on reflectivity because the reflection coefficient of the water, R, is related in a nonlinear manner to the dielectric constant of water. The reflectivity is described by the Fresnel equation that defines the behavior of electromagnetic waves at a smooth dielectric boundary. For horizontally polarized waves (H) at non-nadir incidence (a ), the Fresnel reflection coefficient is given by

,

(2)

where e is the complex dielectric constant of the emitter. Since the angle of incidence is equal to the angle of reflection for a smooth surface, the apparent temperature, the sky temperature and the atmospheric temperature are functions of a only. Through inversion of the Fresnel equation, we obtain an estimate of the effective dielectric constant of the emitting layer as

,

(3)

which reduces to a quadratic equation.

Dielectric Mixing Model

Alternatively, soil dielectric properties have been estimated based on mixing formulae that describe the partitioning of dielectric properties among the various soil components as a function of water content. In our research, the dielectric mixing model of Wang and Schmugge (1980) was inverted to estimate volumetric water content from the dielectric properties of the soil-water-air mixture using e_eff from remote sensing. Wang and Schmugge (1980) recognized that the relationship between volumetric soil moisture and dielectric constant was comprised of two distinct parts separated at a transition moisture value, q _t , defined by

(4)

where q_wp is an empirical approximation of the wilting point moisture based on the sand and clay mass fractions given by

.

(5)

Unique equations describe the relationship between dielectric constant and soil moisture at moisture content less than and greater than the transition moisture. For soil moisture less than q _t ,

(6)

where

,

(7)

,

(8)

and

,

(9)

where e _w , e _i , and e _r are the dielectric constants for water, ice, and rock, respectively, P is the soil porosity, and

.

(10)

The dielectric constants for ice and rock are parameters estimated at 3.2 and 5.5, respectively (Ulaby et al., 1986). For soil moisture greater than q _t ,

,

(11)

where

.

(12)

The real part of the complex dielectric constant for free water is calculated at a given frequency, f, and temperature, T, using a modified Debye equation (after Ulaby et al., 1986)

.

(13)

The magnitude of the high frequency dielectric constant, e _w¥ , was determined by Lane and Saxton (1952) to be 4.9. The static dielectric constant of pure water (dimensionless), e _w0 , is (Klein and Swift, 1977)

.

(14)

The relaxation time of pure water is given by (Stogryn, 1970)

.

(15)

Assumptions

Although it is desirable to use complete and physically based models to estimate soil moisture from microwave remote sensing instruments, it is rarely possible to obtain the ancillary data these require (Jackson, 1993). Consequently, application of the Fresnel equation requires assumptions that the dielectric and temperature properties of the soil are uniform throughout the emitting layer, that emissivity is related principally to the real part of the complex dielectric constant, and that one has knowledge of the soil depth emitting the energy being measured. There are several additional considerations that also should be incorporated in a data interpretation algorithm. Jackson et al. (1997) provides a comprehensive review of many of these issues. Only recently has the validity of the Fresnel equation been demonstrated within this framework (Jackson and Le Vine, 1996; Jackson et al., 1997).

Correction Procedures

The algorithm used in this study follows that of Jackson (1993). From Eq. (1), it is clear that observed microwave emission depends on both soil moisture and temperature. Because soil moisture and temperature are not independent, we must normalize the observed brightness temperature by the effective radiating temperature of the soil if we are to relate T_B to soil moisture. As described by Choudhury et al. (1982), effective temperature, T_eff , may be defined as

,

(16)

where T_sfc is the observed surface temperature, T_d is the deep (e.g. 15 cm) soil temperature, and C is an empirically-defined weighting function based on the relative contribution of individual layers to the microwave emission at the soil surface. Normalizing T_B with respect to T_eff helps to reduce the diurnal hysteresis.

Because the observed microwave emission also depends on the amount of scattering that takes place at the soil surface, compensation is required for rough surfaces. A simple model for correcting for the effects of surface roughness was developed by Choudhury et al. (1979), in which the smooth surface reflectivity is

,

(17)

where R_r is the rough surface reflectivity at look angle a , and h is

,

(18)

where s² is the variance of the height distribution of the surface, and l is the observed wave-length. In practice, however, Eq. (18) tends to exaggerate the effect of roughness and is used mainly as a guideline.

In the presence of vegetation, energy is attenuated requiring a compensating correction. After an extensive review of research on the effects of vegetation on microwave energy transmission, Jackson and Schmugge (1991) proposed a simple model based on the relationship between the optical depth and vegetation water content. In this model, vegetation-corrected emissivity, e_c , becomes (after Shutko, 1986; Jackson and O’Neill, 1990)

,

(19)

where g is the transmissivity of the absorbing layer defined by (Kirdiashev et al., 1979),

,

(20)

where t is the optical depth. Jackson and Schmugge (1991) described optical depth as a function of the vegetation water content, W_v , by

,

(21)

where s is an empirically derived value for the slope of the regression between W_v and t . This correlation is very conducive to remote sensing applications because proxies for vegetation water content can be obtained by other remote sensing methods.

References

Choudhury, B. J., Schmugge, T. J., and Mo, T. (1982), A parameterization of effective soil temperature for microwave emission. J. Geophys. Res., 87, 1301-1304.

Choudhury, B. J., Schmugge, T. J., Newton, R. W., and Chang, A. T. C. (1979), Effect of surface roughness on microwave emission of soils. J. Geophys. Res., 84, 5699-5706.

Dobson, M. C., Ulaby, F. T., Hallikainen, M. T., and El-Rayes, M. A. (1985), Microwave dielectric behavior of wet soil - part II: dielectric mixing models. IEEE Trans. Geosci. Remote Sens., GE-23, 35-46.

Jackson, T. J. (1993), Measuring surface soil moisture using passive microwave remote sensing. Hydrol. Proc., 7, 139-152.

Jackson, T. J., and Le Vine, D. M. (1996), Mapping surface soil moisture using an aircraft-based passive microwave instrument: algorithm and example. J. Hydrol., 184, 85-99.

Jackson, T. J., and O'Neill, P. E. (1990), Attenuation of soil microwave emission by corn and soybeans at 1.4 and 5 GHz. IEEE Trans. Geosci. Remote Sens., 28, 978-980.

Jackson, T. J., O'Neill, P. E., and Swift, C. T. (1997), Passive microwave observation of diurnal surface soil moisture. IEEE Trans. Geosci. Remote Sens., 35, 1210-1222.

Jackson, T. J., O'Neill, P. E., and Swift, C. T. (1997), Passive microwave observation of diurnal surface soil moisture. IEEE Trans. Geosci. Remote Sens., 35, 1210-1222.

Jackson, T. J. and Schmugge, T. J. (1991), Vegetation effects on the microwave emission of soils. Remote Sens. Environ., 36, 203-212.

Kirdiashev, K. P., Chukhlantsev, A. A., and Shutko, A. M. (1979), Microwave radiation of the Earth's surface in the presence of vegetation. Radio Eng. and Elec., 24, 256-264.

Klein, L. A. and Swift, C. T., (1977), An improved model for the dielectric constant of sea water at microwave frequencies. IEEE Trans. Antennas and Propagation, AP-25, 104-111.

Lane, J. and Saxton, J., (1952), Dielectric dispersion in pure polar liquids at very high radio frequencies, III. The effects of electrolytes in solution, Proc. Roy. Soc. London, 214A, 531-545.

Njoku, E. G. and Kong, J. A. (1977), Theory for passive microwave remote sensing of near-surface soil moisture. J. Geophys. Res., 82, 3108-3118.

Schmugge, T. (1990), Measurements of surface soil moisture and temperature, In Remote Sensing of Biosphere Functioning (R.J. Hobbs, and H.A. Mooney, Eds.). Springer-Verlag, New York.

Shutko, A. M. (1986), Remote sensing of the waters and lands via microwave radiometry (the principles of method, problems feasible for solving, economic use). Pontificiae Academiae Scientiarvm Scripta Varia, 68, 413-440.

Stogryn, A. (1970), The brightness temperature of a vertically structured medium. Radio Sci., 5, 1397-1406.

Wang, J. R. and Schmugge, T. J. (1980), An empirical model for the complex dielectric permittivity of soils as a function of water content. IEEE Trans. Geosci. Remote Sens., GE-18, 288-295.

Ulaby, F. T., Moore, R. K., and Fung, A. K. (1986), Microwave Remote Sensing, Active and Passive, vol.III: From Theory to Applications, Artech House, Massachusettes.