Over the past decades, different surface backscattering models have been developed for a better understanding of the relationship between soil surface parameters (surface roughness and soil moisture) and radar signals. The great majority of models rely on simulations using as their input a constant value of soil moisture, assuming uniform moisture down to the penetration depth of the radio wave. This hypothesis is not realistic, particularly at low frequencies for which the penetration depth is only a few centimeters. In this regard, on clay loam soil, Bruckler et al. [1], found that the radar penetration depth in the C-band decreases from about 5 cm for a soil moisture content of 10%, to 1 cm for a soil moisture content of 30%. When the radar wavelength increases (L-band), the penetration depth increases but never exceeds a few centimeters in the upper soil layer. The use of surface backscattering models has often revealed various limitations in the backscattering simulations [2]. On the other hand, a linear relationship between radar signal and soil moisture is generally observed, but with different slopes from one study to another [3-6]. Various studies have attempted to analyze the effect of a non-uniform soil moisture profile in active and passive microwave modelling [7-11]. In general, they achieve only a slight improvement in the retrieval of the backscattering level. Fung et al., [8], has shown the contribution of multi-layer permittivity to improve backscattering simulations and to reduce discrepancy between radar measurements and radar signal over two different test fields. Walker et al. [9], have proposed also a variable dielectric profile showing an improvement in the simulation of data collected in the European Microwave Signature Laboratory (EMSL) experiments. Lu et al. [11], have solved Ricatti equation to introduce moisture profile in BSM (Bi-Spectrum model) backscattering model, showing the higher contribution of soil surface moisture in simulations, comparing to deep soil moisture. All these studies focus particularly in the analysis of moisture profile contribution in backscattering simulation, in field scale, without a study of the inhomogeneous moisture profile effect in correlation between radar signal and moisture measurements, particularly in larger spatial scale.

The aim of this paper is to discuss this scientific question and also to highlight the particular importance of the soil moisture profile measurement protocol in reducing the substantial differences in results given by various experimental studies, designed to analyse the correlation between moisture and radar signals. In this study, we propose to analyse the moisture profile effect, using ASAR SAR images, experimental soil moisture, and a modified version of the Integral Equation Model (IEM) with three layer permittivities.

The paper is organised as follows. In section 2, the methodology proposed to simulate radar signals with a three-layer soil moisture profile is presented. Section 3 deals with experimental measurements of soil moisture and satellite data acquired over two sites in France (Villamblain and Avignon). Section 4 provides details of results obtained using experimental data and modelling simulations. Finally, Section 5 gives our conclusions.

Assuming a permittivity profile ε(z) specifically representing a continuous variation of soil moisture as a function of depth z, the magnitude and phase of the reflection coefficient at any given depth inside the soil are computed according to the Reflection-Coefficient Equation known as the Ricatti Equation, written as Ulaby et al. [12]: (1) d R d z + j β R − β ′ 2 β ( A − R ) = 0where β = k₀ncosθ, and A is the amplitude of the incident electric field.

n = [ε(z)] ^1/2 is the refractive index, θ is the incidence angle and k₀ is the wave number.

Resolving this equation in the case of the natural soil medium is far from straightforward. Therefore, in this study, the problem is simplified by considering a soil medium having only three discontinuous permittivity layers and using the IEM model for backscattering simulations [8], [12].

First, the principles of the IEM model will broadly be recalled. The backscattering coefficient is expressed as follows: (1) σ p p 0 = k 2 2 exp [ − 2 k z 2 s 2 ] ∑ n = 1 ∞ s 2 n | I p p n | 2 W ( n ) ( − 2 k x , 0 ) n ! , p = h , v

Where k_z= k cosθ_i, k_x= k sin θ_i, θ_i is the incidence angle, k is the wave number, and s is the rms surface height. I p p n is a function of the incidence angle, the soil's dielectric constant, and the Fresnel reflection coefficient. W⁽ⁿ⁾ (−2k_x,0) is the Fourier transform of the nth power of the surface correlation function.

This version of the IEM model introduces the transitional function proposed by [13], in which the Fresnel coefficient is not estimated at the incidence angle θ_i, but at an angle ranging between θ_i and normal incidence, depending on the surface roughness parameters.

The IEM input parameters are the dielectric constant, deduced from surface volumetric moisture content and soil texture [14], and the correlation function of surface heights or its corresponding spectrum.

Figure 1 illustrates the sensitivity of backscattering coefficient IEM simulations as a function of soil moisture, for different roughness values (s=0.4 cm, s=0.6 cm, s=0.8 cm, s=1cm) in horizontal and vertical polarisations. First, we observe approximately an identical effect of soil moisture increasing on IEM simulations for different roughness. Second, the backscattering coefficients illustrate the saturation of radar signal for high soil moisture. This saturation is generally less important for real data which shows a linear increasing even for high soil moisture around 30% [4].

We propose to estimate an effective permittivity computed from the Fresnel coefficients at the interface between the different layers [8]. This effective permittivity will be used as an input to the IEM model.

The coefficients R₂, R₁ and R₀ are the Fresnel coefficients at the boundaries of the three layers (with respective depths d₃, d₂ and d₁). They are written as:

The Fresnel reflection coefficients, R₀, between the air (region 0) and layer 1 (with permittivity ε₁) are defined as: (2) R 0 h = cos θ i − ɛ 1 − sin 2 θ i cos θ i + ɛ 1 − sin 2 θ i (3) R 0 v = ɛ 1 cos θ i − ɛ 1 − sin 2 θ i ɛ 1 cos θ i + ɛ 1 − sin 2 θ i

The subscripts h and v denote horizontal or vertical polarisation (p), respectively.

The Fresnel coefficient R₁ between layer 1 and layer 2 (with permittivity ε₂) is: (4) R 1 h = ɛ 1 − sin 2 θ i − ɛ 2 − sin 2 θ 1 ɛ 1 − sin 2 θ i + ɛ 2 − sin 2 θ 1and (5) R 1 v = ɛ 2 ɛ 1 − sin 2 θ i − ɛ 1 ɛ 2 − sin 2 θ 1 ɛ 2 ɛ 1 − sin 2 θ i + ɛ 1 ɛ 2 − sin 2 θ 1with the angle of transmission θ₁ defined by Snell Law as: (6) sin 2 θ 1 = 1 ɛ 1 sin 2 θ iand finally the Fresnel coefficient at the boundary of layers 2 and 3 (with permittivity ε₃) is: (7) R 2 h = ɛ 2 − sin 2 θ 1 − ɛ 3 − sin 2 θ 2 ɛ 2 − sin 2 θ 1 + ɛ 3 − sin 2 θ 2 (8) R 2 v = ɛ 3 ɛ 2 − sin 2 θ 1 − ɛ 2 ɛ 3 − sin 2 θ 2 ɛ 3 ɛ 2 − sin 2 θ 1 + ɛ 2 ɛ 3 − sin 2 θ 2with the angle of transmission θ₂ defined by Snell Law as: (9) sin 2 θ 2 = ɛ 1 ɛ 2 sin 2 θ 1

The reflection coefficient for reflectivity of a layer of depth d₂ over a half space is written as: (10) R 12 ( θ i ) = R 1 ( θ i ) + e j 2 ψ 1 R 2 ( θ i ) 1 + e j 2 ψ 1 R 1 ( θ i ) R 2 ( θ i )where ψ₁ is an attenuation factor that depends on the thickness d₂ of the second layer, the electromagnetic wavelength, λ₀, and effective permittivity ε₂. (11) ψ 1 = 2 π d 2 λ 0 ɛ 2 − sin 2 θ 1

The expression (8) can be extended for the first layer of depth d₁ above it as: (12) R ( θ i ) = R 0 ( θ i ) + e j 2 ψ 0 R 12 ( θ i ) 1 + e j 2 ψ 0 R 0 ( θ i ) R 12 ( θ i ) (13) where ψ 0 = 2 π d 1 λ 0 ɛ 1 − sin 2 θ i

In practice, there are no physical boundaries between layers, and the soil medium is characterized by a continuous dielectric profile. In order to avoid an oscillatory behaviour caused by the assumption of discrete layers, an incoherent treatment of the reflections in equations (8) and (10) was chosen in our computations [8], [12]. The terms containing the phases were neglected. As a result, the final reflection coefficient, for example, can be written as: (14) R ( θ i ) = R 0 ( θ i ) + real ( e j 2 ψ 0 ) R 12 ( θ i ) 1 + real ( e j 2 ψ 0 ) R 0 ( θ i ) R 12 ( θ i )

The corresponding effective permittivity could be computed from this Fresnel coefficient: (15) ɛ eff = ( 1 + R ( 0 ) 1 − R ( 0 ) ) 2

This study relies on two databases: -

The first one of these was obtained at the Avignon site in the south of France, in 1995. It comprises ground measurements only, without satellite acquisitions.

The objective of this paper was to analyse the effect of the soil moisture profile on the estimation of radar signals over bare soils. Because it was not possible to take into account the full range of soil moisture variability as a function of soil depth, a modification of the IEM model is proposed, which involves three layers each having a different soil moisture value.

The relationship between the radar signal and soil moisture reveals a strong dependence on the depth of the considered measurements. Analysis based on a modified IEM model illustrates the effect on radar simulations of taking the soil moisture profile into consideration, in IEM model validity domain. Although it does not exceed 0.5dB in the experimental measurements described here, this effect could partially explain the differences in the parameters of the linear relationship between the radar signal and soil moisture. Simulations also show how the contribution of each of the three layers changes as a function of soil moisture range.

In conclusion, the main result of this study is that the protocol used when measuring soil moisture is of utmost importance. Different measurements performed without an accurate knowledge of depth lead to an inaccurate estimation of the correlation between soil moisture and the radar signal.