Semi-empirical models

5.1.1.1. Oh model

Based on theoretical backscattering models, scatterometer measurements and airborne SAR data, Oh elaborated over the years a semi-empirical backscattering model for full-polarisation microwave over bare soil fields (Oh et al. 1992; Oh et al. 1994; Oh et al.

2002; Oh 2004). (Oh et al. 1992) used a truck-based scatterometer (LCX POLAR-SCAT (Tassoudji et al. 1989)) and recorded full-polarimetric signals at L-, C- and X-band with incidence angles ranging between 10° and 70°. This version of the model covers the conditions of surface roughness and soil moisture as: 0.1 < ks < 6.0, 2.6 < kl

< 19.7 and 9 vol. % < mv < 31 vol. %. Two polarised ratios are introduced: the co-polarised ratio p (= σ⁰_HH/σ⁰_VV) and the cross-polarised ratio q (= σ⁰_HV/σ⁰_VV). Through observations of radar backscatter data and scattering behaviour knowledge, the cross-polarised ratio q is empirically described as:

𝑞 = 0.23 Γ₀ 1− 𝑒^−𝑘𝑠

5.1 where Γ0 denotes the Fresnel reflectivity of the soil surface at nadir, s is the rms height of roughness in cm, k (=2π/λ) is the wave number of radar signal with wavelength of λ in cm, εr denotes the dielectric constant of the soil surface as:

Γ₀ = 1− 𝜖𝑟

1 + 𝜖𝑟 2

5.2 In addition, the co-polarised ratio p is found to be ≤ 1 for all angles and empirically de-termined as:

𝑝= 1− 2𝜃 𝜋

1/3Γ₀

𝑒^−𝑘𝑠

2

5.3

- 75 - where θ is the incidence angle in radian.

Both ratios p and q are sensitive to dielectric constant and rms height s, while the former effect is dominant. Furthermore, σ⁰HH, σ⁰VV and σ⁰HV are empirically expressed by Equa-tions 5.4–5.6:

𝜍_𝐻𝐻⁰ =𝑓 𝑝𝑐𝑜𝑠³𝜃 Γ_𝑣 𝜃 +Γ_𝑕 𝜃

5.4

𝜍_𝑉𝑉⁰ =𝑓𝑐𝑜𝑠³𝜃

𝑝 Γ_𝑣 𝜃 +Γ_𝑕 𝜃

5.5 𝜍_𝑕𝑣⁰ =𝑞𝜍_𝑣𝑣⁰

5.6 where

𝑓 = 0.7 1− 𝑒^{−0.65(𝑘𝑠)1.8}

5.7 By ignoring the imaginary part of dielectric constant ε''r, Γ0, ε'r and hence m_v and ks can be easily inverted from above equations based on Hallikainen et al. (1985).

Oh et al. (1994) extended the previous model by introducing phase difference statistics into surface parameter expressions and surface ACF. The soil moisture estimation is found to be improved by using polarimetric radar data from the same scatterometer as in Oh et al. (1992). Measured correlation function can be expressed as:

𝑝 𝜉 = 1− 𝜉²/(𝑎𝑙)² 𝑒^{−𝜉/𝑏𝑙}

5.8 where a and b are constants depending on the ACF and l is the correlation length. Based on the behaviour of theoretical models – the SPM and the KA (Ulaby et al. 1986) – σ⁰VV can be expressed as:

- 76 - 𝜍_𝑉𝑉⁰ = 13.5𝑒^{−1.4 𝑘𝑠 0.2} 1

𝑝Γ_h ks ²(𝑐𝑜𝑠𝜃)3.25−0.05𝑘𝑙𝑒^{−(2𝑘𝑠𝑐𝑜𝑠𝜃)0.6}𝑊_𝑘

5.9

where Wk denotes the roughness spectrum corresponding to Equation 5.8 as:

𝑊_𝑘 = (𝑘𝑙)²

1 + (2.6𝑘𝑙𝑠𝑖𝑛𝜃)² 1−0.71 1−3(2.6𝑘𝑙𝑠𝑖𝑛𝜃)² [1 + 2.6𝑘𝑙𝑠𝑖𝑛𝜃 ²]²

5.10 and Γh is expressed by:

Γ_h = cosθ − ϵ_r −sin²θ cosθ+ ϵ_r −sin²θ

2

5.11

The co-polarised ratio p and cross-polarised ratio q are adjusted as:

𝑝= 1−(2𝜃 𝜋)^0.314Γ0

2

5.12 𝑞= 0.25 Γ₀(0.1 +𝑠𝑖𝑛^0.9𝜃) 1− 𝑒^{−(1.4−1.6Γ0)𝑘𝑠}

5.13

Oh et al. (1993) empirically determined the degree of correlation α as:

𝛼= 1−0.2(𝑠𝑖𝑛𝜃)^{𝐴 𝑘𝑠,Γ0} (𝑐𝑜𝑠𝜃)^{𝐵(𝑘𝑠,Γ0)}

5.14

where

𝐴 𝑘𝑠,Γ₀ = (16.5Γ₀+ 5.6)e^{−41.6ksΓ02}

5.15

- 77 - and

𝐵 𝑘𝑠,Γ₀ = 8.1Γ₀kse^−1.8ks

5.16 Therefore, surface soil moisture m_v, rms height s and correlation length l can all be eas-ily inverted from the above equations with known polarimetric data.

Oh et al. (2002) incorporated airborne SAR data and improved the model for the degree of correlation α and the co-polarised phase difference δ as well as backscattering coeffi-cients. This version of the model is able to invert volumetric soil moisture instead of the complex dielectric constant. The ensemble-averaged differential Mueller matrix for mi-crowave backscattering model over bare field is introduced. It is found that the probabil-ity densprobabil-ity function (PDF) of the co-polarised phase angle (Øc = Øhh - Øvv) other than the cross-polarised phase angle (Øx = Øhv - Øvv = Øvh - Øvv according to the reciprocity rela-tion), is sensitive to incidence angle, the radar wavelength, the soil moisture and surface roughness, while the PDF of Øc is characterised by the degree of correlation α and mean value of the co-polarised phase difference δ as (Sarabandi 1992):

𝑓_Φ 𝜙_𝑐 = 1− 𝛼²

2𝜋(1− 𝑋²) 1 + 𝑋 1− 𝑋²

𝜋

2+𝑡𝑎𝑛⁻¹( 𝑋 1− 𝑋²)

5.17 where

𝑋 =𝛼cos⁡(𝜙_𝑐 − 𝜁)

5.18 Empirically, σ⁰HV, p and qand are proposed as:

𝜍_𝐻𝑉⁰ = 0.11𝑚_𝑣^0.7𝑐𝑜𝑠^2.2𝜃 1− 𝑒^{−0.32(𝑘𝑠)1.8}

5.19

- 78 - 𝑝= 1−( 𝜃

90°)^{0.35𝑚𝑣−0.65}𝑒^{−0.4(𝑘𝑠)1.4}

5.20

𝑞= 0.1(𝑠

𝑙+𝑠𝑖𝑛1.3𝜃)^1.2 1− 𝑒^{−0.9(𝑘𝑠)0.8}

5.21 Therefore, σ⁰VV and σ⁰HH are expressed as:

𝜍_𝑉𝑉⁰ = 𝜍_𝐻𝑉⁰

𝑞 = 0.11𝑚_𝑣^0.7𝑐𝑜𝑠^2.2𝜃 1− 𝑒^{−0.32(𝑘𝑠)1.8} 0.1(𝑠

𝑙+𝑠𝑖𝑛1.3𝜃)^1.2 1− 𝑒^{−0.9(𝑘𝑠)0.8}

5.22

𝜍_𝐻𝐻⁰ =𝑝𝜍_𝐻𝐻⁰ =𝑝 𝑞𝜍_𝐻𝑉⁰

= 1−( 𝜃

90°)^0.35𝑚𝑣

−0.65

𝑒^{−0.4(𝑘𝑠)1.4} 0.1(𝑠

𝑙+𝑠𝑖𝑛1.3𝜃)^1.2 1− 𝑒^{−0.9(𝑘𝑠)0.8} 0.11𝑚_𝑣^0.7𝑐𝑜𝑠^2.2𝜃 1

− 𝑒^{−0.32(𝑘𝑠)1.8}

5.23 The degree of correlation α and the co-polarised phase difference δ are described as:

𝛼= 1− 0.17 + 0.01𝑘𝑙+ 0.5𝑚_𝑣 (𝑠𝑖𝑛𝜃)^{1.1(𝑘𝑠)−0.4}

5.24

𝜁= (0.44 + 0.95𝑚_𝑣−1.0𝑠 𝑙)𝜃

5.25

while the differential Mueller matrix is approximated as:

- 79 - 𝑀⁰ =

𝑆^{𝑣𝑣0 2} 𝑆_𝑕𝑣^{0 2} 𝑆_𝑕𝑣^{0 2} 𝑆_𝑕𝑕^{0 2}

0 0 0 0 0 0

0 0

𝑅𝑒(𝑆_𝑣𝑣⁰ 𝑆_𝑕𝑕^0∗) + 𝑆_𝑕𝑣^{0 2} − 𝐼𝑚(𝑆_𝑣𝑣⁰ 𝑆_𝑕𝑕^0∗) 𝐼𝑚(𝑆_𝑣𝑣⁰ 𝑆_𝑕𝑕^0∗) 𝑅𝑒(𝑆_𝑣𝑣⁰ 𝑆_𝑕𝑕^0∗) − 𝑆_𝑕𝑣^{0 2}

5.26 and its elements can be described using α and δ as (Sarabandi et al. 1992; Ulaby et al.

1992):

𝑅𝑒(𝑆_𝑣𝑣⁰ 𝑆_𝑕𝑕^0∗) =𝛼𝑐𝑜𝑠𝜁 𝑀₁₁⁰ 𝑀₂₂⁰

5.27

𝐼𝑚(𝑆_𝑣𝑣⁰ 𝑆_𝑕𝑕^0∗) =−𝛼𝑐𝑜𝑠𝜁 𝑀₁₁⁰ 𝑀₂₂⁰

5.28

Therefore, all elements can be described from backscattering coefficients and the two phase difference parameters α and δ as:

𝑀₁₁⁰ = 1 4𝜋𝜍_𝑣𝑣⁰

5.29

𝑀₂₂⁰ = 1 4𝜋𝜍_𝑕𝑕⁰

5.30

𝑀₁₂⁰ = 𝑀₂₁⁰ = 1 4𝜋𝜍_𝑕𝑣⁰

5.31

𝑀₃₃⁰ = 1

4𝜋(𝛼𝑐𝑜𝑠𝜁 𝜍_𝑣𝑣⁰ 𝜍_𝑕𝑕⁰ +𝜍_𝑕𝑣⁰ )

5.32

- 80 - 𝑀₄₄⁰ = 1

4𝜋(𝛼𝑐𝑜𝑠𝜁 𝜍_𝑣𝑣⁰ 𝜍_𝑕𝑕⁰ − 𝜍_𝑕𝑣⁰ )

5.33

𝑀₄₃⁰ =−𝑀₃₄⁰ = 1

4𝜋𝛼𝑐𝑜𝑠𝜁 𝜍_𝑣𝑣⁰ 𝜍_𝑕𝑕⁰

5.34 A comparison between the measured and estimated differential Mueller matrix found good agreement (Oh et al. 1992). Note that Equation 5.21 is in contrast with Equation 5.13, where the cross-polarised ratio q is sensitive to the soil moisture.

Due to the insensitivity to s/l, Oh (2004) ignored the correlation length l in a new ex-pression of the cross-polarised ratio q that,

𝑞= 0.095(0.13 +𝑠𝑖𝑛1.5𝜃)^1.4(1− 𝑒^{−1.3(𝑘𝑠)0.9})

5.35

It is also claimed that the co-polarised ratio p is not suitable for retrieving soil moisture readings on very rough or dry conditions due to its insensitivity in those conditions.

5.1.1.2. Dubois model

Dubois et al. (1995) introduced a semi-empirical model to reproduce backscattering coefficients σ⁰_hh and σ⁰_vv for bare soil surfaces. The model is developed based on a large dataset of scatterometer data (LCX POLARSCAT and RASAM systems (Tassoudji et al. 1989; Wegmuller et al. 1994)) and assessed on both airborne and space-borne SAR data (i.e. AIRSAR and SIR-C). The large selection of different instruments increased the probability of the model‘s applicability to different soil surfaces and instruments.

The expressions are described as

𝜍_𝐻𝐻⁰ = 10^−2.75 𝑐𝑜𝑠^1.5𝜃

𝑠𝑖𝑛⁵𝜃 10^{0.028𝜀𝑡𝑎𝑛𝜃} 𝑘𝑠𝑠𝑖𝑛𝜃 ^1.4𝜆^0.7

5.36

- 81 - 𝜍_𝑉𝑉⁰ = 10^−2.35 𝑐𝑜𝑠³𝜃

𝑠𝑖𝑛³𝜃 10^{0.046𝜀𝑡𝑎𝑛𝜃}(𝑘𝑠𝑠𝑖𝑛𝜃)^1.1𝜆^0.7

5.37 where ε is the real part of the dielectric constant. The model is recommended in condi-tions where 1.5 GHz ≤ f ≤ 11 GHz, ks ≤ 2.5, θ ≥ 30° and mv ≤ 35 vol. %. This is due to several discrepancies being found between the Dubois model and theoretical models:

the model predicts that the co-polarised ratio p increases with roughness which is in contrast to the SPM prediction, where roughness is not taken, accounted for by p. In addition, the model predicts that σ⁰HH will be larger than σ⁰VV when ks*sinθ is large, which is also in contrast with the geometric-optical model (GO) and the IEM and SAR observations.

5.1.1.3 The semi-empirical model for ERS imagery

Loew et al. (2006a) presented a semi-empirical surface soil moisture retrieval scheme for ENVISAT ASAR Wide Swath Mode imagery, which is adapted from an existing algorithm originally designed for the ERS imagery (Rombach and Mauser 1997). Due to the restriction of the original algorithm to the incidence angle of ERS data (approxi-mately 23°), incidence angles from different image swathes are normalised to compen-sate for variability in imaging geometry to reference geometry by a simple statistical approach, which calibrates existing images to reference geometry without roughness information, rather taking into account the impact of land use (biomass) and soil texture.

For the statistical approach, backscattering coefficients based on a six-year database were derived according to different land uses and regressed with incidence angle vari-ability (Loew et al. 2006a). A 5° step is chosen between incidence angle range 15° and 45°, while backscattering coefficients of pixels with homogeneous land use type, are averaged, e.g. for bare soil and wheat. A linear relationship between σ⁰ and incidence angle θ is found as in the modified Equation 5.38. In addition, season effect, i.e. sum-mer and winter, is treated separately.

)

* /(

) 23

* ( / ⁰

0

23      

 _   

5.38

- 82 - where _⁰ is the backscattering coefficient of specific land use type bare soil in this study at the incidence angle θ. α and β denote model regression coefficients, which are set to -62.3 and 3341.7 respectively for bare soil for the summer period. Note that the backscattering coefficient unit is linear in Equation 5.38.

Furthermore, the shape of the relationship between the backscattering coefficient to di-electric constant of the soil is very similar for various land cover types (Loew et al.

2006a). Therefore, the real part of the dielectric constant εr is empirically related to land use specific backscattering coefficients at an incidence angle of 23° as:

) ( ) ( )

( ^{0 2 2}

0 dB c dB

b

ra   



5.39 where a, b and c are empirical coefficients for specific land use based on extensive sta-tistical analysis, which equals 34.20, 4.42 and 0.15 respectively for bare soil and σ⁰ as the normalised backscattering coefficient in dB. The coefficient of determination for this statistical model is high (R²=0.90) for bare soil fields in the published experiment.

Since the empirical database is based on different surface roughness conditions, the re-lationship between dielectric constant and backscattering coefficient is considered to represent the mean surface roughness of the bare soil surface over the two test sites in western and southern Germany (Loew et al. 2006a).

With additional in situ soil texture information, mv can be converted from the real part of the dielectric constant ε commonly through a polynomial expression (Hallikainen et al. 1985).

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Im Dokument Operational Retrieval of Surface Soil Moisture using Synthetic Aperture Radar Imagery in a Semi-arid Environment (Seite 97-106)