A Method for Surface Roughness Parameter Estimation in Passive Microwave Remote Sensing

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1 Chin. Geogra. Sci (4) DOI: /s A Method for Surface Roughness Parameter Estimation in Passive Microwave Remote Sensing ZHENG Xingming 1,, ZHAO Kai 1 (1. Northeast Institute of Geography and Agroecology, Chinese Academy of Sciences, Changchun 13001, China;. Graduate University of the Chinese Academy of Sciences, Beijing , China) Abstract: Surface roughness parameter is an important factor and obstacle for retrieving soil moisture in passive microwave remote sensing. Two statistical parameters, root mean square (RMS) height (s) and correlation length (l), are designed for describing the roughness of a randomly rough surface. The roughness parameter measured by traditional way is independence of frequency, soil moisture and soil heterogeneity and just the geometric roughness of random surface. This geometric roughness can not fully explain the scattered thermal radiation by the earth s surface. The relationship between geometric roughness and integrated roughness (contain both geometric roughness and dielectric roughness) is linked by empirical coefficient. In view of this problem, this paper presents a method for estimating integrated surface roughness from radiometer sampling data at different frequencies, which mainly based on the flourier relationship between power spectral density distribution and spatial autocorrelation function. We can obtain integrated surface roughness at different frequencies by this method. Besides "geometric" roughness, this integrated surface roughness not only contains "dielectric" roughness but also includes frequency dependence. Combined with Q/H model the polarization coupling coefficient can also be obtained for both H and V polarization. Meanwhile, the simulated numerical results show that radiometer with a sensitivity of 0.1 K can distinguish the different surface roughness and the change of roughness with frequency for the same rough surface. This confirms the feasibility of radiometer sampling method for estimating the surface roughness theoretically. This method overcomes the problem of dielectric roughness measurement to some extent and can achieve the integrated surface roughness within a microwave pixel which can serve soil moisture inversion better than the geometric roughness. Keywords: surface roughness; passive microwave remote sensing; statistical parameter estimation; soil moisture; radiometer 1 Introduction Water and energy fluxes at the surface/atmosphere interface are strongly dependent upon soil moisture. Surface soil moisture (SSM) not only influences the surface evaporation, infiltration and surface runoff, but also governs the rate of water uptake by vegetation in the vadose zone. The spatio-temporal evolution of soil moisture fields is an important factor for numerical weather and climate models, and should be accounted for in hydrology and vegetation monitoring (Kerr et al., 001). Microwave measurements have the benefit of being largely unaffected by cloud cover and variable surface solar illumination, and can provide a unique capability for obtaining frequent observations of soil moisture at global and regional scales (Song et al., 007). The emissivity of the bare soil is the function of wavelength of incidence electromagnetic wave, polarization, incidence angel, soil texture, soil temperature, soil moisture and soil surface roughness. Under a given wavelength, polarization and incidence angle of electromagnetic wave, the surface roughness is the key parameter for retrieving soil moisture accurately by microwave radiation. So far, the solutions for rough surface microwave radiation can be divided into two general categories of Received date: ; accepted date: Foundation item: Under the auspices of the Key Direction in Knowledge Innovation Programs of Chinese Academy of Sciences (No. KZCX-YW-340) Corresponding author: ZHAO Kai. zhaokai@neigae.ac.cn Science Press, Northeast Institute of Geography and Agroecology, CAS and Springer-Verlag Berlin Heidelberg 010

2 346 ZHENG Xingming, ZHAO Kai physical modeling and semi-empirical approaches (Shi et al., 00). The former contains Kirchhoff approximation (KA) (Ulaby et al., 1986), small perturbation method (SPM) (Ulaby et al., 1986), integral equation model (IEM) (Fung et al., 199), advanced integral equation model (AIEM) (Chen et al., 003) and bi-spectrum model (BSM) (Liu and Li, 003). The KA is applicable for a rough surface with a large surface curvature, while the SPM is valid for slightly rough surfaces. There are a few methods that overlap these two methods such as IEM, AIEM and BSM. IEM is in agreement with SPM for slightly rough surface and with KA for greatly rough surface (Chen et al., 000). In the development of the IEM, in order to obtain an explicit mathematical formula that is easy to compute, several assumptions were made. While AIEM is the modification of IEM by removing the simplifying assumption in the spectral representation of Green s function (Chen et al., 000). BSM considers that a perfectly conducting randomly rough surface is a composite of a Kirchhoff surface and a small perturbation surface. Physical modeling can describe the interaction of electromagnetic waves and rough surface analytically, and its bias mainly results from the difference between physical structure of real rough surface and theoretical physical model. The semi-empirical model that is most commonly used to describe the emissivity of bare surface is the socalled Q/H model (Wang and Choudhury, 1981). The parameter Q describes the energy emitted in orthogonal polarizations due to surface roughness effects. The parameter H is a measure of the surface roughness effect to increase surface emission. In all forms of Q/H model, the form of Q has no obvious change, and oppositely, H has a variety of different forms (Wigneron et al., 001). Those simplified Q/H models have generally focused on evaluating the strength of the relationship between the emission signal and the surface roughness. Q/P model is a specific form of Q/H by setting H = 1 and making Q dependent on polarization, proposed by Shi et al. (005). As polarization coupling coefficient varies with polarization, instead of Q in Q/H model, Q p is used for Q/P model. Q p can be simply described as a single-surface roughness property, i.e., the ratio of the surface root mean square (RMS) height (s) to the correlation length (l). The computational complexity of Q/P model is the same as Q/H in modeling the effective reflectivity or emission signals. In Q/H model, Q and H are often with the values between 0 and 1 to justify the surface roughness effect on the reflectivity and usually determined empirically from the experimental data for a given frequency and incidence angle (Wegmüller and Mätzler, 1999). Similarly, Q p in Q/P model is empirically obtained from experimental data (Shi et al., 005). The key point for rough surface reflectivity model is the quantitative measurement of Q, and this will be one of the problems discussed in this paper. Another problem need to be solved is the estimation of surface roughness statistical parameter. In general, in the study on soil moisture inversion by passive microwave remote sensing, microwave emissivity described by no matter physical model or semi-empirical model is closely related with the surface roughness. The commonly used instruments for measuring soil surface roughness are grid plate, profile meter and laser imager (Moreno et al., 008). The main advantage of grid plate is its simplicity and ease of handling under extreme field conditions. So grid plate is frequently used to measure surface height distribution, and RMS height and correlation length can be calculated by the surface height distribution. Considering the heterogeneity of surface height distribution, RMS height and correlation length need to be measured in multiple directions, and in order to obtain representative surface roughness parameter within a microwave pixel, the measured results of all directions should be averaged statistically. This approach is feasible to ground-based remote sensing experiment, but can not be applied in space remote sensing. Moreover, because surface roughness varies with frequency, it is difficult to solve the surface roughness problem using multi-frequency remote sensor. So how to detect the integrated surface roughness and "dielectric" roughness accurately is the key for soil moisture retrieval by passive microwave remote sensing. This paper presents a spatial sampling estimation method to obtain statistical parameters of integrated surface roughness, and in combination with microwave data of ground-based measurement, the value of the polarization coupling coefficient (Q or Q p ) can be obtained. By this method, the numerical simulation is implemented. Based on this spatial sampling method by pencil beam radiometer, the regional surface roughness database can be built, which can lay the foundation for soil moisture inversion from passive microwave remote sensing data.

3 A Method for Surface Roughness Parameter Estimation in Passive Microwave Remote Sensing 347 Methods.1 Rough surface microwave reflectivity For a smooth surface and a medium with uniform dielectric constant, the expression for Fresnel reflectivity at vertical and horizontal polarizations, derived from electromagnetic theory, is as follows (Ulaby et al., 1986): cosθ ε sin θ rh ( θ ) = cosθ + ε sin θ r ( θ) = v εcosθ ε sin θ εcosθ + ε sin θ (1) () where r h (θ) and r v (θ) are reflectivity of smooth surface for horizontal and vertical polarization, θ is the view angle (measured from the surface normal), and ε is the complex relative dielectric constant (relative permitivity) of the medium. The complex relative dielectric constant ε is mainly affected by soil moisture, soil texture, electromagnetic wave length, soil bulk density and soil temperature. The semi-empirical dielectric constant model proposed by Dobson et al., (1985) is used to compute complex relative dielectric constant of soil because it is applicable to a wide range of electromagnetic wave length (4 18 GHz) and no longer depends on soil type. The soil layer can be considered to be infinite medium in hemisphere space, and the relation between emissivity and reflectivity in microwave is described as (Njoku and Entekhabi, 1996): e = 1 Γ (3) p where e p and Γ p is emissivity and reflectivity of rough surface in the polarization p (horizontal polarization (H) or vertical polarization (V)). Theoretically, the reflectivity of rough surface after roughness correction can be expressed as (Peake, 1959): Γ (, ) 1 4π (, ;, ) (, ;, ) dω (4) ' ' ' ' ' p θφ = π γpp θφθ φ + γpq θφθ φ where Γ p (θ, φ) is the effective reflectivity of rough surface in the polarization p at direction (θ, φ); γ pp and γ pq is the co-polarization and cross-polarization scattering co- ' efficient from direction (θ, φ) to direction (θ, φ ); Ω is the solid angle. The integration in the solid angle only contains the hemisphere above the surface. Though this p theoretical formula is accurate for reflectivity, it is too complicated and not suitable for mass computing of remote sensing data. In combination of Q p and Fresnel reflectivity of smooth surface (r p or r q ), the effective reflectivity of rough surface (Г p (θ)) can be obtained by: where Q p is expressed as: Γ ( θ) = Qr( θ) + (1 Q ) r ( θ) (5) p p q p p log[ Q ( f)] = a ( f) + b ( f) log( s/ l) + c ( f) ( s/ l) p p p p (6) where a p, b p and c p are regression coefficients depending on view angle, frequency and polarization; f (GHz) denotes frequency; s and l are RMS height and correlation length of randomly rough surface.. Surface roughness parameter (s and l) estimation For simplifying analysis process, surface height distribution is assumed to be one-dimension stationary random process ξ (u),u is spatial coordinates (x or y), R(τ) and S(v) are respectively the correlation function and power spectral density function of the stationary random process, v is spatial frequency (1/m), τ is the coordinate x difference of two sample point. A Fourier transform relationship exists between the power spectral density and correlation function as follows: R() τ = Sv ()cos(π vτ)dv (7) The value of S(v) can be obtained from pencil beam antenna radiometer. τ represents the correlation length (l) when the amplitude of R(τ) decreases to its 1/e. R(τ) derived from Equation (7) can be explained by frequency component and high frequency component is eliminated as disturbance. The sensitivity of microwave radiometer (ΔT rms ) represents the minimum detectable signal and controls the precision of measurement. Under a given wavelength of electromagnetic wave and soil conditions, penetration depth of electromagnetic wave is considered as a fixed value, and in this case surface height distribution is considered to be stationary random process. The RMS height (s) can be obtained from R(τ) (Shen et al., 001): s = R(0) (8) In the process of surface roughness measurement for calculating spatial autocorrelation function, the test for

4 348 ZHENG Xingming, ZHAO Kai device parameter is very important. Assuming that the 3dB beam angle and view angle of microwave radiometer are α and θ, device height is h, surface brightness temperature of microwave is T B, based on the sensitivity of microwave radiometer and λ/ (λ is band length) spatial sampling intervals, the relation between test parameters is as follows: ( λ /) [ h tan( α/ ) / cos( θ) ] T B 3ΔT A large number of remote-sensing experiments and theoretical analysis show that L-band is the best for retrieving soil moisture in passive microwave remote sensing. Taking the working frequency of microwave o radiometer being 1.4 GHz, α = 50 and θ = 15 o, T B = 73 K for non-frozen soil and ΔT rms = 0.1 K, the result from Equation (9) indicates that the device height is equal or less than 14.7 m. At far-field conditions, the device height of h =14 m is adopted in the actual measurement. rms (9).3 Polarization coupling coefficient estimation In previous studies, the polarization coupling coefficient (Q or Q p ) is obtained by empirical regression, which is estimated by a new proposed method in this paper. In Q/H model, the effective reflectivity for rough surface (Г p (θ, p)) could be derived by (Shi et al., 005; Wegmüller and Mätzler, 1999): ( θ p) ( Q) r ( θ) Qr ( θ) ' cos Γ, = 1 e h θ p + q (10) where h = 4k s, k = π / λ, Q and h are independent of each other; and r p (θ) and r q (θ) denote the reflectivity of smooth surface in the p and q polarization. The best view angle (θ) range for Q/H model is (Barré et al., 008). At the view angle of θ, the brightness temperature for the vertical and horizontal polarizations (T BV and T BH ) are respectively measured by microwave radiometer, and the effective reflectivity ((Г(θ, V), (Г(θ, V)) can be calculated from T BV and T BH and soil equivalent temperature T sm (K): ( θ ) ( TBV Tsm ) ( θ ) ( T T ) Γ,V = 1 / (11) Γ,H = 1 / (1) BH sm Once the smooth surface reflectivity is computed from Fresnel reflectivity equations (1) (), the polarization coupling coefficient can be obtained from equations (10) (1). This method for calculating polarization coupling coefficient is also suitable for both Q/P and Q/H models. By this way, we can have access to achieve the quantitative polarization coupling coefficient Q at a given roughness and frequency, and get rid of the way of using empirical coefficient, hoping to improve rough surface reflectivity calculation accuracy. The value of polarization coupling coefficient computed here is frequency-dependent. 3 Numerical Simulations To prove the feasibility of surface roughness measurement by the different frequency microwave radiometer, firstly the difference of brightness temperature between rough surface and smooth surface is simulated by the Q/P model, and scatter plot between the brightness temperature difference and s/l is established, then the changes of brightness temperature and roughness parameter Q p with frequency are simulated, last one-dimension randomly rough surface is generated according to Wiener-Khintchine theorem (Tsang et al., 000) and random surface simulation method (Jin, 008). 3.1 Variation of brightness temperature difference with s/l Soil is four-phase mixed media composed of soil solid particles, air, free water and bound water. Assuming that soil effective temperature is 93 K, sand weight percent of soil is 70%, clay weight percent of soil 10%, soil volumetric water content 4%, soil bulk density 1.4 g/cm 3 and frequency GHz, the modified Debye formula (Lane and Saxton, 195) and Dobson mixed media dielectric model (Dobson et al., 1985) are firstly used to compute the relative complex permittivity of free water and soil, respectively; secondly, the reflectivity of smooth surface is derived by equations (1) and () according to soil relative complex permittivity at a given incidence angle; lastly, the reflectivity of smooth and rough surface is linked by Q/P model. Based on the above conditions, brightness temperature of smooth surface and rough surface with s/l change range of [0.01, 0.30] can be computed by equations (3), (11) and (1). The brightness temperature difference of smooth and rough surfaces is showed in Fig. 1. While surface roughness parameter (s/l) varies from 0.01 to 0.30, the

5 A Method for Surface Roughness Parameter Estimation in Passive Microwave Remote Sensing 349 maximal brightness temperature difference between smooth and rough surface is 4.7 K for horizontal polarization (H), and 11 K for vertical polarization (V). For radiometer with 0.1 K sensitivity, the minimum s/l which can be distinguished is for H polarization, and for V polarization. This indicates that the variation of surface roughness can be identified effectively. 3. Variation of Q p and brightness temperature with frequency In order to understand the variation of polarization coupling coefficient Q p with frequency, the Q/P model was used to simulate brightness temperature and Q p variation in 6.95 GHz and GHz with the roughness parameter s/l (Fig. ). Under the same random surface, the difference of brightness temperature and polarization coupling coefficient between 6.95 GHz and GHz have a little variation for V polarization when s/l changes from 0.01 to 0.30, indicating brightness temperature and polarization coupling coefficient at V polarization is not sensitive to the variation of surface roughness. The variation of Q H (the Q p for H polarization) is 0.01 which is 10 times than the minimum distinguishable value of polarization coupling coefficient, and the variation of brightness temperature is K that is much larger than 0.1 K sensitivity, when frequency ch- anges from 6.95GHz to 10.65GHz. This demonstrates that the change of surface roughness parameter with frequency can be estimated by H polarization brightness temperature from pencil beam radiometer under the same random rough surface. 3.3 Random rough surface simulation The method for measuring roughness parameter at different frequencies is to carry out rough surface equidistant spatial sampling using pencil beam radiometer with different frequencies, thus the one-dimension spatial power spectrum density distribution of roughness can be obtained. The Fourier transform of power spectral density is the spatial auto-correlation function of one-dimension of rough surface height distribution. Combination of Equation (8) and the definition of surface correlation length, l and s can be computed by surface spatial auto-correlation function. Previous studies have shown that the spatial autocorrelation function of natural terrain height distribution is generally Gaussian-type and Exp exponential function (Taconet and Ciarletti, 007), and in this paper the Exp exponential function was selected as spatial autocorrelation function of rough surface. Assuming that l and s have been known, we also can obtain the mathematical expression of the spatial autocorrelation function. Based on the Fourier transform relationship between the spatial autocorrelation function and po- Fig. 1 Variation of brightness temperature (T BH and T BV ) difference between smooth and rough surfaces with s/l at horizontal polarization (a) and vertical polarization (b) The solid line is for 6.95 GHz and the dash line is for GHz Fig. Variation of Q p (Q H ) (a) and brightness temperature (T B ) (b) with s/l at H polarization

6 350 ZHENG Xingming, ZHAO Kai wer spectral density, the power spectral density function can be calculated. Define correlation length of 0 cm, the RMS heights of 1 cm and 0.5 cm, simulated one-dimension random rough surface length 10 m, sampling interval of 1 cm, Fig. 3 represents the randomly rough surface simulation result using the randomly surface generation method by power spectral density function and Gaussian random number series, proposed by Jin (008). The result shows that the simulated surface with RMS height of 0.5 cm is smoother than that of 1 cm, and has a smaller range of fluctuation. The numerical simulation result of randomly rough surface is consistent with the status of real surface. 4 Discussion For the dash line surface correlation length is 0 cm and RMS height is 1 cm; for the solid line surface correlation length is 0 cm and RMS height is 0.5 cm Fig. 3 Simulation result of one-dimension randomly rough surface The surface roughness causes the variation of surface brightness temperature received by satellite sensor, about 5 K for H polarization and 11 K for V polarization at 6.95 GHz frequency when the value of s/l change from 0.01 to This variation of brightness temperature not only introduces uncertainty and complexity into surface variables retrieving process, but also lowers the precision of surface variables estimated from passive microwave remote sensing data. The accurate measurement of surface roughness can decrease this uncertainty and improve the precision of parameter retrieval. For soil moisture retrieval from radiometer data, radiation emitted from soil surface is redistributed as for surface roughness effect. This roughness effect could be separated into two categories: one is geome- tric roughness effects in relation with spatial variations in the soil surface height, the other is dielectric roughness effects in relation with the variation of the dielectric constant at the soil surface and within the soil which can be caused by non-uniformities in the soil characteristics (Panciera et al., 009). The frequentlyused method for surface roughness measurement is pin meter 1, and the measured roughness result by pin meter is independent of frequency and surface heterogeneity and just the expression of "geometric" roughness. Thus the empirical formula is used to describe the relation between surface roughness parameter and frequency (Shi et al., 00). The "dielectric" roughness can not be detected by traditional pin meter method. The integrated surface roughness need to be explored for improving the precision of soil moisture retrieval. The brightness temperature of bare soil is the global response of soil characteristics including geometric roughness and dielectric roughness, and in relation with frequency. The spatial distribution of brightness temperature is the integrated distribution of surface roughness to a large extent. So using pencil beam antenna radiometer to sample one-dimension profile of soil surface, one-dimension profile brightness temperature distribution related with roughness and frequency can be obtained, that is to say, we obtained the power spectrum distribution of one-dimension rough soil profile. According to the Fourier relation between power spectrum density and autocorrelation function, the spatial autocorrelation function is computed from one-dimension brightness temperature distribution. Form Equation (8) and the definition of correlation length, the RMS height and correlation length of soil which is related to geometric and dielectric roughness and frequency are achieved. The simulation result of Q/P model illustrated that the accuracy of pencil beam antenna radiometer is enough for roughness measurement. The integrated surface roughness estimated from pencil beam antenna radiometer data is related to soil moisture because the brightness temperature of soil is also influenced by soil moisture. This is consistent with the research result that a simple linear correlation exists between roughness and soil moisture (Panciera et al., 009). Saleh et al. (007) made the same conclusion for grasslands. An improvement in soil mois- 1 Walker J P, Panciera R, 005. National Airborn Field Experiment 005

7 A Method for Surface Roughness Parameter Estimation in Passive Microwave Remote Sensing 351 ture retrieval accuracy was achieved after the calibration of the soil moisture dependence of roughness. To sum up, the roughness is related to "geometric" roughness, "dielectric" roughness, frequency and soil moisture, and pin meter method for roughness measurement can only detect geometric roughness, while the frequency and soil moisture dependence of roughness is modified by empirical formula (Shi et al., 005; Panciera et al., 009) and dielectric roughness is studied by refractive mixing model considering the heterogeneity of soil-air mixing media (Schwank et al., 010). For solving this complex problem about surface roughness, a method based on the spatial sampling by pencil beam antenna radiometer is proposed and the roughness retrieved by this method contains the effects of all the four factors presented in the frontier. This integrated surface roughness offers a more comprehensive explanation for scattered radiation by earth s surface, and cut down the complexity of soil moisture retrieval. Microwave thermal radiation received by radiometer with different frequencies originates from different thermal radiation depth, thus the roughness estimated from the radiometer sampling brightness temperature corresponds to the specific frequency, indicating the roughness dependence on frequency. However, in this study the surface roughness sample by radiometer occurs in a given soil moisture that affects the thermal radiation depth, so the dielectric roughness can not reflect the heterogeneity change with thermal radiation depth, and this need to be discussed in the future. 5 Conclusions The soil roughness affects soil moisture retrieval accuracy and the precise result of roughness measurement can decrease the uncertainty of soil moisture inversion. A new method for integrated surface roughness measurement is proposed based on pencil beam antenna radiometer data in this paper. The feasibility of this method is testified by numerical simulation. Unlike traditional method that can only measure geometric roughness, the integrated surface roughness estimated by the proposed spatial sampling method is a coupling roughness including geometric roughness, dielectric roughness and frequency dependence, and this integrated roughness can be directly used to model the relationship between the reflectivity of smooth surface and rough surface. This method solved the problem of dielectric roughness measurement to some extent, but can not distinguish geometric roughness and dielectric roughness from the measured integrated surface roughness. Instead of empirical regression, numerical calculation can be used to compute polarization coupling coefficient for H and V polarizations from beam pencil antenna radiometer data and ground-based experimental data. The follow-up work will be carried out including soil surface roughness field measurements in Northeast China and validation of the proposed roughness measuring method here. References Barré H M J, Duesmann B, Kerr Y H, 008. SMOS: The Mission and the System. IEEE Transactions on Geoscience and Remote Sensing, 46(3): Chen K S, Wu T D, Tsang L et al., 003. Emission of rough surfaces calculated by the integral equation method with comparison to three dimensional moment method simulation. IEEE Transactions on Geoscience and Remote Sensing, 41(1): Chen K S, Wu T D, Tsay M K et al., 000. A note on the multiple scattering in IEM models. IEEE Transactions on Geoscience and Remote Sensing, 38(1): Dobson M C, Ulaby F T, Hallikainen M T et al., Microwave dielectric behavior of wet soil Part II: Dielectric mixing models. IEEE Transactions on Geoscience and Remote Sensing, 3(1): Fung A K, Li Z, Chen K S, 199. Back scattering from a randomly rough dielectric surface. IEEE Transactions on Geoscience and Remote Sensing, 30(): Jin Yaqiu, 008. Theory and Method of Numerical Simulation of Composite Scattering from the Object and Randomly Rough Surface. Beijing: Science Press. (in Chinese) Kerr Y H, Waldteufel P, Wigneron J P et al., 001. Soil moisture retrieval from space: The soil moisture and ocean salinity (SMOS) mission. IEEE Transactions on Geoscience and Remote Sensing, 39(8): Lane J A, Saxton J A, 195. Dielectric dispersion in pure polar liquids at very high radar frequencies III. The effect of electrolytes in solution. In: Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 14: Liu Ning, Li Zongqian, 003. Bi-spectrum scattering model for dielectric randomly rough surface. Tsinghua Science and Technology, 8(5): Moreno R G, Dìaz Álvarez M C, Alonso A T et al., 008. Tillage and soil type effects on soil surface roughness at semiarid cli-

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