Notes on the Implementation

Reference Model

The reference model Duan2010 is based on Duan et al. [2010]. To understand what it does and how it is structured, the study should be read first. The key quantities that we want out of the model are the absorption coefficient and index of refraction as a function of height, since those can be converted to the range delay and attenuation, which are needed for the SAR processing.

Structure

From an implementation perspective, the code is structured as follows:

This functionality is provided in the instantiation of a Duan2010 model. All input datasets are part of references and are loaded at the package import time.

Once the two key quantities (refraction and absorption) are computed as a function of altitude, we can compute the quantities actually needed for the SAR processing:

This functionality is provided as part of the get_range_attenuation_angles() and geometry_from_central_angle() functions.

For more information about the usage of the code, please see the Quick Start Notebook and the API description itself. The remainder of this document is focused on deviations and improvements from the Duan et al. [2010] study.

Remark on polarization notation

To prevent confusion when comparing XVAMP to the paper when it comes to the computation of the polarization, the following is a quick recap of how the polarization terms are computed for each species and then combined.

We recall eq. (4):

\[P_\text{mix} = \sum_i \Phi_i^* P_i \left( T, \frac{\rho_{r,mix}}{\nu_i^*} \right)\]

as well as the two main equations used to derive polarizations, the one from Harvey and Lemmon [2005], eq. (8), using the virial expansion, reproduced partially in eq. (8) in Duan et al. [2010]:

\[\frac{P}{\rho} = A_\epsilon + A_\mu / T + B_\epsilon \rho + C \rho^D\]

and the one from Pitzer [1983], eq. (14):

\[P_\nu = \rho \frac{4\pi N_0}{3} \left( \alpha_T + \frac{\mu^2 g}{3kT} \right)\]

where we have substituted \(\rho = d/M\).

Now, inserting the virial expansion equation and the definitions for \(\Phi_i^*\) and \(\rho_{r,mix}\) into the mixing equation and then simplifying yields:

\[P_\text{mix} = \sum_i x_i \rho_{mix} \left( A_{\epsilon,i} + A_{\mu,i} / T + B_{\epsilon,i} \rho_i + C \rho_i^D \right)\]

where \(x_i\) is the molar fraction and \(\rho_{mix}\) is the molar density of the mixture. This can therefore be further simplified by using the molar density of the species, \(\rho_i\):

\[P_\text{mix} = \sum_i \rho_i \left( A_{\epsilon,i} + A_{\mu,i} / T + B_{\epsilon,i} \rho_i + C_i \rho_i^{D_i} \right) = \sum_i \rho_i P'_i\]

here, we have introduced the shorthand notation \(P'_i\) for later use. Note how the terms inside the sum are equivalent to the Pitzer [1983] model if we set

\[A_{\epsilon,i} = \frac{4\pi N_0}{3} \alpha_T\]

\[A_{\mu,i} = \frac{4\pi N_0 g}{9k} \mu^2\]

\[B_{\epsilon,i} = C_i = 0\]

which in turn also implies \(P_\nu = P\). (This equivalence is the reason the Duan2010 option use_virial_approximation has no effect, it just changes the notation of the parameters.)

For the species where we have detailed parameters according to the Harvey and Lemmon [2005] polarization equation, we should make use of them, which means the overall model is nonlinear in molar density \(\rho\). This is one of the reasons why internally, XVAMP always directly computes the product of \(\rho_i P'_i (\rho_i)\), instead of first calculating \(P'_i (\rho_i)\), and then later scaling it by \(\rho_i\). The other reason is the benefit that the \(\rho_i P'_i (\rho_i)\) formulation (compared to the \(\Phi_i^* P_i \left( T, \frac{\rho_{r,mix}}{\nu_i^*} \right)\) notation) does not require the knowledge of any characteristic volumes.

On the coding side, both types of describing the polarization parameters are implemented:

• Harvey and Lemmon [2005] formulation: eq8() to evaluate, and A_epsilon_from_eq8() to estimate \(A_\epsilon\) from a reference polarization.

• Pitzer [1983] formulation: eq14() to evaluate, and alpha_T_from_eq14() to estimate \(\alpha_T\) from a reference polarization.

The polarization parameters are evaluted according to the format they’re in inside evaluate_polarization_parameters(), and simply summed together (since their relative importance to the mixture polarization has already been taken into account) in sum_polarizations().

Modifications

In the following, the Duan et al. [2010] study is simply referred to as the “paper”, and Jessie Duan’s Matlab implementation as the “reference code”.

“Bottom” of the simulation

The reference code is only defined for altitudes of 0 and above, i.e., at the mean planetary radius of 6051.8 km or more. If delay or attenuation quantities for areas below the mean radius are desired, the final permittivity profile is extrapolated. XVAMP instead extrapolates the temperature, pressure, density, and mixing ratio profiles to negative altitudes instead.

Note

This should have a negligible impact on the final delay or attenuation values.

Cloud polarization and absorption

The paper describes a way to incorporate the effect of the cloud layer in sections 2.1.5 and 2.2.5. The reference code, however, follows a different approach: the one assuming shell-like droplets described by Cimino [1982]. The two approches differ by one and two orders of magnitude in the polarization and absorption profiles they yield, respectively, with the paper version being the higher one. For typical satellite heights, the difference amounts to a sub-millimeter change in the delay terms, but an approximately 1.5 dB two-way attenuation increase for the paper version.

The reason the code uses the Cimino [1982] model instead is because it was derived explicitly for the Venus conditions. As such, the Cimino model explains the observed discrepancy between cloud “mass contents derived from the absorption coefficient data” and “those measured by […] the Sounder probe” (section 9). Therefore, the Cimino model is preferred, and also implemented in XVAMP. However, the paper version is available as an option (use_clouds_from="duan").

Important

This has a significant impact on the final attenuation values, and a negligible one on the final delay values.

Kramers-Krönig rule for SO2 and OCS relative permittivity

The Kramers-Krönig relationship is used in the paper to derive the real part of the relative permittivity from the imaginary part (which, in turn, is derived from the absorption). In this step, the value of the real part of the relative permittivity at infinite frequency needs to be known or assumed. Both the paper and the reference code contain such values for both SO2 and OCS; however, their source is not known anymore. For backwards compatibility, XVAMP also uses these values but offers the option to choose unity instead, which is the theoretical value at infinite frequencies for all media.

Note

Since these are minor changes in the resulting real part of the relative permittivity of SO2 and OCS, and these in turn are only small contributors to the total absorption and delay profiles, this change yields only a centimeter-level delay difference and negligible attenuation difference.

Polarization and absorption of OCS

There are some discrepancies in the way OCS is added into the atmospheric model. The paper uses a Lorentzian line shape to compute its absorption and attenuation, while the reference code uses a Ben-Reuven line expression for the absorption (using line shape parameters of SO2 for lack of better data) but does not compute the attenuation. XVAMP, for backwards compatibility, defaults to the reference code. However, XVAMP also offers to use a Ben-Reuven line shape parameters derived for OCS (albeit very approximately).

Note

Since OCS is such a minor constituent, the different options have a sub-millimeter effect on the delay and a milli-decibel effect on the attenuation.

Polarization of H2SO4

When computing the the polarization of the gaseous H2SO4, the reference code relied on some unnecessary assumptions to derive the polarizability \(A_\epsilon\). The present code addresses this by instead following the polarization computation approach suggested by the study which provides the necessary experimental data, Kolodner and Steffes [1998], Section 3.2. This approach is implemented in kolodner_steffes_1998 (to derive the mass density and the real part of the relative permittivity of H2SO4) and then completed in a general sense in compute_polarization_parameters() (to compute the polarization parameters that can then be evaluated at given molar densities and temperatures of the atmospheric column).

Note

This has no impact on the attenuation (since absorption is computed from a different model; Section 7 in Kolodner and Steffes [1998]), but changes the polarization profile by about 20%. Since H2SO4 is only a minor contributor to the total polarization, this should have a negligible impact on the final delay values.

Limitations

There are many assumptions made in the model, most of which are described in the paper. Some others, however, come from the implementation. The following is a non-exhaustive list of those.

• There is currently no check being made that all species molar fractions add up to one (or equivalently, that the sum of all species mass densities equals the total mass density). Furthermore, the cloud mass density is considered “outside” the other species mass densities, i.e., it is not derived from H2O or H2SO4 abundances in the profile, nor does it change when H2O or H2SO4 changes.