### Bistatic scattering geometry

The scattering geometry of the bistatic radar system is shown in Fig. 1, where θ and \(\varphi\) are the zenith and azimuth angles of the incoming signal, and the subscripts *i* and *s* represent the incident and scattering components, respectively. Note that the SoOP-R forms a bistatic radar system, and the BackScatter Alignment (BSA) convention is the standard for radar polarimetry. In order to obtain the different polarization combinations using the wave synthesis technique, we transform the polarization coordinate systems accordingly.

### MIMICS model

The MIMICS model (Ulaby et al. 1988) is a very popular model used for backscattering systems, usually for monostatic radars. Unfortunately, this model cannot be directly used for SoOP-R, since the transmitters and receivers in SoOP-R follow a typical bistatic radar system (Fig. 1). Therefore, here we modify the backscattering model so that it can be used for the bistatic scattering systems. The method is based on adding the scattering geometry in the phase and extinction matrices implemented in the MIMICS model. The bistatic radar scattering model for the forest canopy (Bi-MIMICS model) uses an iterative algorithm to solve the radiation transfer equation (Ferrazzoli et al. 2011; Ulaby et al. 1988). The following equations are available in the original MIMICS handbook (Ulaby et al. 1988). However, the model is only provided in the backscattering mode. Here, we modify the model and obtain the bistatic radar form (Ferrazzoli et al. 2011; Ulaby et al. 1988), which is the typical form for SoOP-R. The development of the model is shown from Eq. 6 to Eq. 20. Note also that we have included both scattering angles zenith and azimuth.

The model simplifies the forest stands into 3 layers: the crown layer, the trunk layer, and the ground layer. As shown in Fig. 2, the crown layer is modeled in terms of the distribution of dielectric cylinders and disks, while the trunk layer is treated as cylinders of uniform diameter. To simplify the calculation, we assume that the incident azimuth is \({\varphi _i}\)_{i} = 0°. Then the angular relationship of backscattering is *θ*_{s} = *θ*_{i}, \({\varphi _s}\) = 180°; the angular relationship of mirroring scattering is *θ*_{s} = *θ*_{i}, \({\varphi _s}\) = 0°; and the angular relationship of forward scattering is *θ*_{s} = 180°—*θ*_{i}, \({\varphi _s}\)= 0°. After the incident energy is scattered by the particles, the intensity of the scattered energy *I*_{s} = (*θ*_{s}, \({\varphi _s}\)) and the intensity of the incident energy *I*_{i} = (*θ*_{i}, \({\varphi _i}\)) can be related through the modified Mueller matrix L_{m}:

$${I_s}({\theta _s},{\varphi _s}) = \frac{1}{{{r^2}}}{L_m}({\theta _s},{\varphi _s};{\theta _i},{\varphi _i};{\theta _k},{\varphi _k}){I_i}({\theta _i},{\varphi _i}).$$

(1)

In this equation, (*θ*_{k}, \({\varphi _k}\)) is the particle orientation, r is the distance between the incident energy and the particle, and the modified Mueller matrix *L*_{m} is defined by the electric field scattering matrix S as:

$${L_m} = {{\left[ {\begin{array}{*{20}{c}} {{{\left| {{S_{vv}}} \right|}^2}}{{{\left| {{S_{vh}}} \right|}^2}}{\Re \left( {S_{vh}^*{S_{vv}}} \right)}{\Im \left( {S_{vh}^*{S_{vv}}} \right)} \\ {{{\left| {{S_{hv}}} \right|}^2}}{{{\left| {{S_{hh}}} \right|}^2}}{\Re \left( {S_{hh}^*{S_{hv}}} \right)}{\Im \left( {S_{hv}^*{S_{hh}}} \right)} \\ {2\Re \left( {{S_{vv}}S_{hv}^*} \right)}{2\Re \left( {{S_{vh}}S_{hh}^*} \right)}{\Re \left( {{S_{vv}}S_{hh}^* + {S_{vh}}S_{hv}^*} \right)}{ - \Im \left( {{S_{vv}}S_{hh}^* - {S_{vh}}S_{hv}^*} \right)} \\ {2\Im \left( {{S_{vv}}S_{hv}^*} \right)}{2\Im \left( {{S_{vh}}S_{hh}^*} \right)}{\Im \left( {{S_{vv}}S_{hh}^* + {S_{vh}}S_{hv}^*} \right)}{\Re \left( {{S_{vv}}S_{hh}^* - {S_{vh}}S_{hv}^*} \right)} \end{array}} \right]} \mathord{\left/ {\vphantom {{\left[ {\begin{array}{*{20}{c}} {{{\left| {{S_{vv}}} \right|}^2}}{{{\left| {{S_{vh}}} \right|}^2}}{\Re \left( {S_{vh}^*{S_{vv}}} \right)}{\Im \left( {S_{vh}^*{S_{vv}}} \right)} \\ {{{\left| {{S_{hv}}} \right|}^2}}{{{\left| {{S_{hh}}} \right|}^2}}{\Re \left( {S_{hh}^*{S_{hv}}} \right)}{\Im \left( {S_{hv}^*{S_{hh}}} \right)} \\ {2\Re \left( {{S_{vv}}S_{hv}^*} \right)}{2\Re \left( {{S_{vh}}S_{hh}^*} \right)}{\Re \left( {{S_{vv}}S_{hh}^* + {S_{vh}}S_{hv}^*} \right)}{ - \Im \left( {{S_{vv}}S_{hh}^* - {S_{vh}}S_{hv}^*} \right)} \\ {2\Im \left( {{S_{vv}}S_{hv}^*} \right)}{2\Im \left( {{S_{vh}}S_{hh}^*} \right)}{\Im \left( {{S_{vv}}S_{hh}^* + {S_{vh}}S_{hv}^*} \right)}{\Re \left( {{S_{vv}}S_{hh}^* - {S_{vh}}S_{hv}^*} \right)} \end{array}} \right]} \eta }} \right. \kern-\nulldelimiterspace} \eta }$$

(2)

In this matrix, the subscripts v and h indicate the vertical and horizontal polarizations. The superscript * is thetions. The superscript * is the conjunction, and \(\Re \,and\,\Im\) are the real and imaginary parts of the complex value. Finally, \(\eta\) is the intrinsic impedance. The first-order bistatic transformation matrix that connects the incident intensity and the scattered intensity is as follows:

$${I_s}(\mu ,{\varphi _s}) = T(\mu ,{\varphi _s}){I_0}({\mu _0},{\varphi _0}),$$

(3)

where the transformation matrix T is represented by the phase matrix and the extinction matrix, \({\mu _0}\) and \({\varphi _0}\) are related to the incident angles. Both are calculated by the average modified Mueller matrix. The phase matrix is as follows:

$$P({\theta _s},{\varphi _s};{\theta _i},{\varphi _i}) = {N_k}\iiint f({s_k};{\theta _k},{\varphi _k}){L_m}({\theta _i},{\varphi _i};{\theta _s},{\varphi _s};{\theta _k},{\varphi _k})d{s_k}d{\theta _k}d{\varphi _k}.$$

(4)

In this equation, the size, orientation, and distribution function of the scatterer are \({s_k},({\theta _k},{\varphi _k})\), and \(f({s_k};{\theta _k},{\varphi _k})\), respectively. *N*_{k} is the density of the scatterer. The extinction matrix can be expressed as:

$$k = \left[ {\begin{array}{*{20}{c}} { - 2\Re ({M_{vv}})}0{ - \Re ({M_{vh}})}{ - \Im ({M_{vh}})} \\ 0{ - 2\Re ({M_{hh}})}{ - \Re ({M_{hv}})}{\Im ({M_{hv}})} \\ { - 2\Re ({M_{hv}})}{ - 2\Re ({M_{vh}})}{ - \Re ({M_{vv}} + {M_{hh}})}{\Im ({M_{vv}} - {M_{hh}})} \\ {2\Im ({M_{hv}})}{ - 2\Im ({M_{vh}})}{ - \Im ({M_{vv}} - {M_{hh}})}{ - \Re ({M_{vv}} + {M_{hh}})} \end{array}} \right],$$

(5)

$${M_{pq}} = \sum\nolimits_{k = 1}^K {\frac{{i2\pi {N_k}}}{{{k_0}}}{{\left\langle {{S_p}_{qk}({\theta _i},{\varphi _i};{\theta _s},{\varphi _s};{\theta _k},{\varphi _k})} \right\rangle }_k}} .$$

(6)

In this equation, \({\left\langle {{S_p}_{qk}({\theta _i},{\varphi _i};{\theta _s},{\varphi _s};{\theta _k},{\varphi _k})} \right\rangle _k}\) is the average scattering amplitude coefficient, *K* indicates the type of the scatterer, *pq* is the polarization, and k_{0} is the free space wavenumber.

### Scattering mechanisms

There are 8 scattering mechanisms in the MIMICS model (Ferrazzoli et al. 2011; Ulaby et al. 1988). These include the direct specular scattering term (SG); the random rough surface term (DG); the direct crown bistatic scattering term (DC); the ground reflection–crown scattering–ground reflection term (GCG); the crown scattering–ground reflection term (CG); the ground reflection–crown scattering term (GC); the ground reflection–trunk scattering term (GT); and the trunk scattering–ground reflection term (TG). The graphic description of these terms is shown in (Fig. 2; Ferrazzoli et al. 2011; Ulaby et al. 1988), and the combined term is as follows:

$$T(\mu ,{\varphi _s}) = {T_{SG}} + {T_{GCG}} + {T_{CG}} + {T_{GC}} + {T_{DC}} + {T_{TG}} + {T_{GT}} + {T_{RG}}.$$

(7)

In the SG term, the direct signals are propagated through the crown and trunk layers and then reflected by the ground layer in the specular direction. Then, the energy is propagated upwards through the trunk and the crown layers. The whole path can be formulated as follows:

$${T_S}_G(\mu ,{\phi _s}) = {e^{ - k_c^ + {d \mathord{\left/ {\vphantom {d \mu }} \right. \kern-\nulldelimiterspace} \mu }}}{e^{ - k_t^ + {H \mathord{\left/ {\vphantom {H \mu }} \right. \kern-\nulldelimiterspace} \mu }}}R(\mu ){e^{ - k_t^ - {H \mathord{\left/ {\vphantom {H \mu }} \right. \kern-\nulldelimiterspace} \mu }}}{e^{ - k_c^ - {d \mathord{\left/ {\vphantom {d \mu }} \right. \kern-\nulldelimiterspace} \mu }}}\delta (\mu - {\mu _i})\delta (\phi - {\phi _i}).$$

(8)

In the CGC term, the incident energy is firstly propagated through the crown and trunk layers, then scattered by the ground layer, and then through the crown layer, where the volumetric scattering occurs. Part of the energy is scattered by the ground layer and then the upward signal is propagated through the trunk and crown layers. This process can be formulated with the following equation:

$${T_{CGC}}(\mu ,{\phi _s}) = \frac{1}{\mu }{e^{ - k_c^ + {d \mathord{\left/ {\vphantom {d \mu }} \right. \kern-\nulldelimiterspace} \mu }}}{e^{ - k_t^ + {H \mathord{\left/ {\vphantom {H \mu }} \right. \kern-\nulldelimiterspace} \mu }}}R(\mu ){e^{ - k_t^ - {H \mathord{\left/ {\vphantom {H \mu }} \right. \kern-\nulldelimiterspace} \mu }}}{A_{CGC}}( - \mu ,\phi ,{\mu _0},{\phi _0}){e^{ - k_t^ + {H \mathord{\left/ {\vphantom {H {{\mu _0}}}} \right. \kern-\nulldelimiterspace} {{\mu _0}}}}}R({\mu _0}){e^{ - k_t^ - {H \mathord{\left/ {\vphantom {H {{\mu _0}}}} \right. \kern-\nulldelimiterspace} {{\mu _0}}}}}{e^{ - k_c^ - {d \mathord{\left/ {\vphantom {d {{\mu _0}}}} \right. \kern-\nulldelimiterspace} {{\mu _0}}}}}.$$

(9)

In the CG term, the incident energy is first propagated by the crown and trunk layers, scattered by the ground, and then the signals are scattered by the crown layer. It can be formulated as follows:

$${T_{CG}}(\mu ,{\phi _s}) = \frac{1}{\mu }{e^{ - k_c^ + {d \mathord{\left/ {\vphantom {d \mu }} \right. \kern-\nulldelimiterspace} \mu }}}{e^{ - k_t^ + {H \mathord{\left/ {\vphantom {H \mu }} \right. \kern-\nulldelimiterspace} \mu }}}R(\mu ){e^{ - k_t^ - {H \mathord{\left/ {\vphantom {H \mu }} \right. \kern-\nulldelimiterspace} \mu }}}{A_{CG}}( - \mu ,\phi ,{\mu _0},{\phi _0}).$$

(10)

In the GC term, the energy is first scattered in the crown layer and then propagated by the trunk layer. Then the energy is scattered by the ground layer and then propagated by the trunk and crown layers. This process can be expressed as follows:

$${T_{GC}}(\mu ,{\phi _s}) = \frac{1}{\mu }{A_{GC}}(\mu ,\phi ,{\mu _0},{\phi _0}){e^{ - k_t^ + {H \mathord{\left/ {\vphantom {H {{\mu _0}}}} \right. \kern-\nulldelimiterspace} {{\mu _0}}}}}R({\mu _0}){e^{ - k_t^ - {H \mathord{\left/ {\vphantom {H {{\mu _0}}}} \right. \kern-\nulldelimiterspace} {{\mu _0}}}}}{e^{ - k_c^ - {d \mathord{\left/ {\vphantom {d {{\mu _0}}}} \right. \kern-\nulldelimiterspace} {{\mu _0}}}}}.$$

(11)

In the DC term, the energy does not propagate in the boundary layer, but it is only scattered by the crown layer:

$${T_{DC}}(\mu ,{\phi _s}) = \frac{1}{\mu }{A_{DC}}(\mu ,\phi , - {\mu _0},{\phi _0}).$$

(12)

In the TG term, the energy first propagates through the crown and trunk layers and it is scattered at the ground. Then a bistatic scattering occurs at the trunk layer, finally, the energy is propagated upwards through the trunk and crown layers:

$${T_{TG}}(\mu ,{\phi _s}) = \frac{1}{\mu }{e^{ - k_c^ + {d \mathord{\left/ {\vphantom {d \mu }} \right. \kern-\nulldelimiterspace} \mu }}}{e^{ - k_t^ + {H \mathord{\left/ {\vphantom {H \mu }} \right. \kern-\nulldelimiterspace} \mu }}}R(\mu ){A_{TG}}( - \mu ,\phi , - {\mu _0},{\phi _0}){e^{ - k_c^ - {d \mathord{\left/ {\vphantom {d {{\mu _0}}}} \right. \kern-\nulldelimiterspace} {{\mu _0}}}}}\delta (\mu - {\mu _0}).$$

(13)

The GT term refers to the same path as the TG term, but in an inverse direction:

$${T_{GT}}(\mu ,{\phi _s}) = \frac{1}{\mu }{e^{ - k_c^ + {d \mathord{\left/ {\vphantom {d \mu }} \right. \kern-\nulldelimiterspace} \mu }}}{A_{TG}}( - \mu ,\phi , - {\mu _0},{\phi _0})R({\mu _0}){e^{ - k_t^ - {H \mathord{\left/ {\vphantom {H {{\mu _0}}}} \right. \kern-\nulldelimiterspace} {{\mu _0}}}}}{e^{ - k_c^ - {{{d \mathord{\left/ {\vphantom {d \mu }} \right. \kern-\nulldelimiterspace} \mu }}_0}}}\delta (\mu - {\mu _0}).$$

(14)

Finally, the RG term represents the energy first propagated through the crown and trunk layers, at the ground layer bistatic scattering occurs, and the energy finally is propagated upwards through the trunk and the crown layers:

$${T_{RG}}(\mu ,{\phi _s}) = {e^{ - k_c^ + {d \mathord{\left/ {\vphantom {d \mu }} \right. \kern-\nulldelimiterspace} \mu }}}{e^{ - k_t^ + {H \mathord{\left/ {\vphantom {H \mu }} \right. \kern-\nulldelimiterspace} \mu }}}G(\mu ,\phi , - {\mu _0},{\phi _0}){e^{ - k_t^ - {{{H \mathord{\left/ {\vphantom {H \mu }} \right. \kern-\nulldelimiterspace} \mu }}_0}}}{e^{ - k_c^ - {{{d \mathord{\left/ {\vphantom {d \mu }} \right. \kern-\nulldelimiterspace} \mu }}_0}}}.$$

(15)

For the above terms, *k* is the extinction matrix, ± indicate upward/downward directions, *c* and *t* indicate the crown and trunk layers, respectively, *R* is the reflectivity matrix of the specular surface, *G* is the rough surface scattering matrix, and *A* represents the scattering that occurs in the crown and trunk layers, which are calculated by the phase and the extinction matrices. For more details of the scattering models, please see the corresponding reference (Ulaby et al. 1988).

### Wave synthesis technique

In microwave SoOp-R, the reflected signal on the ground surface is relatively weaker than the direct signal, and the receivers need specific features. These include accounting for the different polarizations, so that the reception of the reflected signal is strong enough for measurements. Unfortunately, the existing microwave scattering models are only developed for linear polarization, because these are usually used for the traditional radiometry and monostatic radars. However, the new emerging SoOP-R remote sensing technique uses the circular polarization, since it needs to overcome the ionospheric effects. Therefore, by employing the wave synthesis technique, here we present the required modifications to the existing scattering models, so that we obtain a model capable of estimating the bistatic scattering at various polarization combinations. The scattering characteristics for different polarizations are obtained according to the method of wave synthesis (Liang and Pierce 2005) as follows:

$${\sigma _r}_t({\psi _r},{\chi _r},{\psi _t},{\chi _t}) = 4\pi \tilde Y_m^r{M_m}Y_m^t$$

(16)

In this equation, Mm is the modified Mueller matrix, and Ym is the modified stokes vector, where the upper subscript r and t indicate the polarization of transmitted and received signals. These depend on the variables of orientation angle ψ and ellipticity angle χ, respectively. After changing the orientation angles and ellipticity angle, we can get the bistatic scattering properties at various polarizations.

### Model validation and future prospects

The MIMICS model in its backscattering form is available to the scientific community and it has been validated with in situ measurements (Ulaby et al. 1988). However, since the bistatic form of the MIMICS model has not been developed yet, here we upgrade the MIMICS model to the bistatic form and validate the results with the references (Ferrazzoli et al. 2011; Ulaby et al. 1988). In SoOP-R the transmitted signals are in right hand circular polarization to overcome the ionospheric effects. Since the existing forms of the scattering models are in linear polarization (Ferrazzoli et al. 2011; Ulaby et al. 1988), here we employ the wave synthesis technique to obtain the various polarization combinations. We modify the Stokes vectors to the linear polarization form and find out that the results are similar to that of the linear form of the model. Future works are addressed to validate the model with in situ measurements.