Another large source of error that needs to be corrected in the salinity retrieval algorithm is the reflected radiation from the galaxy [

20]. It can be large (5 Kelvin) and is difficult to deal with as it requires an accurate knowledge of the location and strengths of the galactic radiation, an analytic model for the wind induced roughness of the ocean surface and the antenna gain pattern.

#### 5.1. Geometric Optics Model

For a flat ocean the contribution of the reflected galactic radiation to the antenna temperature

${T}_{A,gal,ref}$ is given by integrating radiation from the galactic sources and reflected at the ocean surface over the antenna gain pattern. Location and strength of the galactic sources at L-band are taken from the galactic map [

20,

21], which was derived from radio-astronomy observations. In actuality, bistatic scattering from a rough ocean will result in galactic radiation entering the mainlobe of the antenna from many different directions. In effect, a rough ocean surface tends to add additional spatial smoothing to

${T}_{A,gal,ref}$. Modeling of this effect is based on the geometric optics (GO) approach, in which the rough ocean surface is modeled as a collection of tilted facets, with each facet acting as an in-dependent specular reflector. The formulation for this model is given by [

22] and the technical details of its application to calculating the TB of reflected galactic radiation at a rough ocean surface are spelled out in [

3]. A crucial input to the GO model is the distribution of the slopes of the tilted facets of the rough ocean surface, which depends on wind speed

$W$. The Aquarius algorithm uses the slope variance from [

11]. At L-band frequencies this value represents about a 50% reduction in the slope variance from the classic Cox and Munk experiment [

23], which measured the ocean sun glitter distribution.

The accuracy of the GO model can be estimated by looking at the zonal differences between the morning (descending) and the evening (ascending) swaths over the same ocean (

Figure 4) and comparing them with the size of the reflected galactic correction (not shown). Ideally, the differences between ascending and descending swaths should vanish when averaged over weekly to monthly time scales. From

Figure 4, we conclude that the GO model removes the reflected galactic radiation correctly to about 90%. The remaining 10% shows up as spurious signal in the Aquarius salinity retrievals.

#### 5.2. SMAP Fore—Aft Analysis

Observations from SMAP provide an opportunity to improve the correction for the reflected galactic signal. SMAP performs a full 360° scan in less than 5 s and thus observes each location in forward (fore) and backward (aft) direction within a couple minutes. The (relatively) strong reflected radiation emanating from the plane of the galaxy can appear in both the forward and the backward look but usually not at the same time. Radiation from directions other than the plane of the galaxy are generally quite small [

20]. If all other signals that depend on look direction (Faraday rotation, wind direction, solar and lunar radiation) have been accurately removed [

7], then taking the difference between fore and aft measured TA produces the reflected galactic radiation:

Here,

$\varphi $ denotes the azimuthal look angle. This equation can be used to derive an empirical galactic correction separate for the SMAP fore and aft looks. For example, looking for cases where the signal from the aft look is small (<2 K) and assuming that the model (theory) for the SMAP aft look reflected galactic radiation is correct if it is smaller than 2 K, then the empirical correction for the fore look can then be obtained from (4) as:

Likewise, assuming that the computed SMAP fore look galaxy model is correct if it is smaller than 2 K, then the empirical galaxy model for the aft look can then be obtained from (4) as:

When performing the analysis, observations were discarded for which the reflected solar radiation is not negligible. Reflected solar radiation differs between fore and aft looks and currently the correction for reflected solar radiation in the SMAP algorithm is not accurate enough to correct for the difference. It is possible to find observations for all times and orbit positions for which both the reflected solar radiation is negligible and either the TA galaxy of the fore or the aft look are less than 2 K. Therefore, it is possible to derive empirical galactic corrections with the SMAP sensor for both look directions using Equations (5) and (6). Separate derivations are performed for different wind speed regimes using 5 m/s intervals.

The largest part of the SMAP fore—aft results can be reproduced using a tilted facet calculation as explained in

Section 5.1 but adding 2 m/s to the wind speed when calculating the RMS slope variance. The effective increase in slope variance increases the surface roughness at L-band frequencies and this increase brings the slope variance from the value in

Section 5.1 closer to the Cox-Munk value [

23].

Based on the prescription to add 2 m/s to the wind speed when deriving the reflected galactic correction from the geometric optics calculation (

Section 5.1) a revised correction for the reflected galaxy for Aquarius can be derived, which takes the results from the SMAP fore—aft analysis into account [

3].

Figure 5 shows the significant improvement in the biases between ascending (evening) and descending (morning) swaths when using the SMAP fore—aft result compared with the original geometric optics calculation (

Figure 4). However,

Figure 5 also shows that even with this improvement in the galactic reflected model based on SMAP, some residual ascending—descending biases still remain, which would cause unacceptably large inaccuracies in the retrieved Aquarius salinity. Mitigating these residual ascending—descending biases is the goal of the empirical zonal symmetrization correction, which is the subject of the following section.

#### 5.3. Emprirical Zonal Symmetrization

There are several possible reasons for the remaining inaccuracies in the reflected galaxy correction:

The value of the variance of the slope distribution is not completely correct, even after effectively increasing the roughness by adding 2 m/s to the wind speed based on the SMAP fore—aft results.

Errors in the antenna gain patterns used to derive the tables of the GO model.

Other ocean roughness effects, which cause reflection of galactic radiation but cannot be modeled with an ensemble of tilted facets (e.g., Bragg scattering at short waves, breaking waves and/or foam, and net directional roughness features on a large scale).

The galactic tables themselves, which were derived from radio astronomy measurements [

20,

21]. For example, there is a small polarized component and Cassiopeia A is very strong and variable.

Such effects are very difficult or impossible to model. We have therefore decided to derive and use an empirical correction for the reflected galactic radiation, which is added to the GO calculation. The danger in doing this is that other geophysical issue (i.e., not associated with reflected radiation from the galaxy) could be masked. But, it was decided to accept this risk for V5.0.

This empirical correction is based on symmetrizing the ascending and the descending Aquarius swaths. The basic assumptions are:

There are no zonal ascending—descending biases in ocean salinity on weekly or larger time scales.

The residual zonal ascending—descending biases that are observed are all due to the inadequacies (either over or under correction) in the GO model calculation for the reflected galactic radiation.

The size of the residual ascending—descending biases is proportional to the strength of the reflected galactic radiation.

Assumption A is based on current understanding of the structure of the salinity field for which there no known physical processes that would cause such a difference. Assumption B results from analyses of the salinity fields and known limitation of the GO model. Assumption C is based on theory for scattering from rough surfaces and the assumption that the source of any difference is reflected galactic radiation and the fact that the source and surface are independent. It is expected to hold in some mean sense over the footprint.

A symmetrization of the ascending and descending Aquarius swaths can be done on the basis of a zonal average. According to Assumption C above the symmetrization weights will be determined by the strength of the reflected galactic radiation. We describe the symmetrization procedure for the 1st Stokes parameter, which is the sum of the brightness temperatures at the ocean surface and will be denoted by

${T}_{B}$. In the equations below,

$\langle \dots \rangle $ denotes the zonal average and the variable

$z$ denotes the orbital angle (

z-angle). If

$z$ lies in the ascending swath, then

$-z$ (or

${360}^{\xb0}-z$) lies in the descending swath and vice versa.

${T}_{B}\left(z\right)$ is first Stokes parameter as measured by Aquarius at the surface at

$z$.

${T}_{A,gal,ref}\left(z\right)$ is the value of the reflected galactic radiation received by Aquarius as computed based on the SMAP fore—aft results (

Section 5.2). The symmetrization term,

$\Delta \left(z\right)$, which is the basis of the empirical correction, is given as:

The probabilistic channel weights

$p$ and

$q$ add up to 1:

$p+q=1$. The symmetrized surface brightness temperature, called

${T}_{B}^{\prime}$, is given by:

It is not difficult to see that this symmetrization has the following features:

Assume that $z$ lies in the ascending swath and therefore $-z$ lies in the descending swath. If there is no reflected galactic radiation in the ascending swath, i.e., $\langle {T}_{A,gal.ref}\left(z\right)\rangle =0$, then $p=1$ and $q=0$. That means that the symmetrization term and thus the whole empirical correction $\Delta \left(z\right)$ vanishes, and therefore: ${T}_{B}^{\prime}\left(z\right)={T}_{B}\left(z\right)$.

If, on the other hand, there is no reflected galactic radiation in the descending swath, i.e., $\langle {T}_{A,gal.ref}\left(-z\right)\rangle =0$, then $p=0$ and $q=1$. That implies $\Delta \left(z\right)=\langle {T}_{B}\left(-z\right)\rangle -\langle {T}_{B}\left(z\right)\rangle $ and thus $\langle {T}_{B}^{\prime}\left(z\right)\rangle =\langle {T}_{B}\left(-z\right)\rangle $.

The zonal average of ${T}_{B}^{\prime}$ is symmetric: $\langle {T}_{B}^{\prime}\left(z\right)\rangle =\langle {T}_{B}^{\prime}\left(-z\right)\rangle $.

If the reflected galactic radiation is the same in ascending and descending swaths $\langle {T}_{A,gal.ref}\left(z\right)\rangle =\langle {T}_{A,gal.ref}\left(-z\right)\rangle $, then $p=q=\frac{1}{2}$ and thus the global average (sum of ascending and descending swaths) does not change after adding the symmetrization term: $\langle {T}_{B}{}^{\prime}\left(z\right)\rangle +\langle {T}_{B}^{\prime}\left(-z\right)\rangle =\langle {T}_{B}\left(z\right)\rangle +\langle {T}_{B}\left(-z\right)\rangle $.

If the zonal ${T}_{B}$ averages are already symmetric $\langle {T}_{B}\left(z\right)\rangle =\langle {T}_{B}\left(-z\right)\rangle $, then the symmetrization term and thus the whole empirical correction $\Delta \left(z\right)$ vanishes, and therefore: ${T}_{B}^{\prime}\left(z\right)={T}_{B}\left(z\right)$. That means that our method will not introduce any additional ascending—descending biases that were not already there.

Figure 6 shows the size and pattern of the empirically derived symmetrization

$\Delta $ in relation to the value of

${T}_{A,gal,ref}$ from the GO in time—

z-angle space. For the GO computation we have assumed an average wind speed of 7.5 m/s. Sizeable contributions for

$\Delta $ are observed in the vicinity of the galactic pattern that is obtained from the GO model. The magnitude of the peak values of

$\Delta $ is about 0.2 K compared to about 3 K in

${T}_{A,gal,ref}$ from the GO model.

An important feature of this symmetrization procedure is the fact that it is derived from Aquarius measurements only and does not rely on or need any auxiliary salinity reference fields such as ARGO or HYCOM.

It is assumed that the galactic radiation itself is unpolarized and polarization occurs only through the reflection at the ocean surface. Ignoring Faraday rotation of the galactic radiation in the empirical correction term, its 2nd (Q) and 3rd Stokes (U) components are:

where

${R}_{V,H}$ are the reflectivity for V and H polarization of an ideal (i.e., flat) surface.

Figure 7 shows the final ascending—descending biases after including the empirical zonal symmetrization. It is evident that all residual zonal biases have been effectively removed. That means, the empirical zonal symmetrization procedure is working as designed.