Faraday rotation and L band oceanographic measurements



[1] Spaceborne radiometric measurements of the L band brightness temperature over the oceans make it possible to estimate sea surface salinity. However, Faraday rotation in the ionosphere disturbs the signals and must be corrected. Two different ways of assessing the disturbance directly from the radiometric measurements, and hence enabling a correction for the effect, are being discussed. Also, a method, which aims at circumventing the problem by using the first Stokes parameter in the salinity retrieval, is being discussed.

1. Introduction

[2] Significant progress in weather forecasting, climate monitoring, and extreme event forecasting depends on a good estimation of both soil moisture and sea surface salinity. It has for some time been known that a spaceborne L band (1.4 GHz) radiometer system is able to provide the necessary data from which soil moisture and sea surface salinity can be estimated. Such systems have traditionally required large and heavy antenna structures for appropriate spatial resolution. For this reason, there has not been and currently is no capability for directly and globally estimating these key variables from space.

[3] However, concepts and technology have developed over time, and today spaceborne systems are viable in the form of thinned aperture radiometers or even as more traditional real aperture systems. In thinned aperture systems, good ground resolution is achieved by interferometric processing of the outputs of relatively few and light antenna elements distributed within the physical aperture. Weight and bulk of a large real aperture are traded for many small receivers and substantial processing, which comes out favorably with present technology. Well-known systems in this category are NASA's proposed HYDROSTAR and the European Space Agency's Soil Moisture and Ocean Salinity (SMOS), which has been selected as an Earth Explorer Opportunity mission scheduled for launch in 2007. See, for example, http://www.esa.int/export/esaLP/smos.html. Also, significant developments have taken place concerning lightweight real aperture antennas, which are not the least driven by the communications industry. Hence recent NASA proposals like the Hydros (a conically scanning system) and the Aquarius (a three-beam push broom system) are of interest here.

[4] As already noted, an L band radiometer can supply data from which soil moisture and sea surface salinity can be estimated. The most stringent requirements to the radiometer are imposed by the salinity observations, and the present article is dealing with this subject. The L band brightness temperature (TB) emitted by the ocean depends not only on salinity (S) but also on sea surface temperature (SST) and a range of sea state variables including wind speed (WS).

[5] The following considerations are not aimed at any particular mission with its specific (range of) incidence angle(s) and polarization(s) but rather at illustrating some effects and problems in general as well as possible solutions. Hence, for simplicity, focus will be on vertical polarization (V pol) and a 50° incidence angle (a value very often used for spaceborne radiometers) unless otherwise indicated.

[6] The brightness temperature sensitivity to salinity is at best (open, warm ocean) approaching ΔTBS = 1 K practical salinity unit−1 (psu). Hence to estimate salinity with an accuracy of 0.1 psu, radiometric measurements better than 0.1 K are required. This includes the influence of other perturbing effects.

2. The Faraday Rotation

[7] Usually, within radiometry, ionospheric effects are ignored, but in the present case the frequency is so low and the measurement requirements so stringent that an investigation is necessary. The plane of polarization of microwave signals propagating from Earth through the ionosphere to a satellite is rotated by an angle θ. The amount of rotation depends on the direction and location of the ray path with respect to the Earth's geomagnetic field and on the state of the ionosphere. To get a feeling for the magnitude of the polarization rotation angle θ, a mean daytime value can be estimated from θ = 17/F2 (F in GHz) taken from Hollinger and Lo [1984]. Hence the average daytime rotation is found to be θ = 8.7° at 1.4 GHz; furthermore, it illustrates why the effect generally can be ignored at higher frequencies.

[8] A more in-depth consideration of the Faraday rotation is found in the work of Svedhem [1986], from which Figure 1 is taken. Figure 1 shows the worst-case average rotation on the 1200 UTC March equinox. The ground incidence angle is 50°. The average over 1 month of the daytime maximum rotation is estimated to be 28°, whereas monthly maximum averages at 0600 LT are around 5°. In addition, day-to-day variations can reach values within +100% to −50% of the averaged values because of the unpredictable nature of the ionosphere. Proposed missions generally have a 0600 LT orbit (and indeed SMOS has that), so from these considerations it is clear that the radiometer system has to cope with a Faraday rotation of at least 5° and possibly up to 10°.

Figure 1.

One-way Faraday rotation in degrees at 1.4 Ghz for March equinox, 1200 UTC. Ground incidence angle is 50°, and azimuth is worst case.

[9] The polarization rotation will result in a slight mixing of the true vertical (TBV) and true horizontal (TBH) brightness temperatures. The radiometer will measure

equation image

Typical values are TBV = 132 K and TBH = 66 K.

[10] Assuming θ = 10°, we find that equation image and that equation image The 2 K error in TBV translates into an error in retrieved salinity of ∼2–4 psu (depending on sea temperature), which is totally unacceptable.

3. Correction Based on Polarization Ratio

[11] Equation (1) can be solved with respect to TBV, for example. After some reductions, the following expression is found:

equation image

Hence if both the local vertical equation image and local horizontal equation image brightness temperatures are measured, and θ is known, the true vertical brightness temperature can be found.

[12] The true and the local polarization ratios are defined as

equation image

Inserting the expressions for the local brightness temperatures into the expression for the local polarization ratio leads after some reductions to the following expression for the angle of rotation:

equation image

Hence if the true polarization ratio is known, the Faraday rotation can be found from measurements of the local polarization ratio.

[13] Figures 2–4 have been generated from a model for the brightness temperature emitted from the sea which is based on the work of Klein and Swift [1977], including the Hollinger wind responses (0.2 K per m s−1 at V pol and 0.3 K per m s−1 at horizontal polarization (H pol)), and summarized by Sasaki et al. [1987]. Figure 2 shows the vertical brightness temperature sensitivity to salinity and sea surface temperature. From Figure 3 it is seen that the true polarization (R) only exhibits a marginal dependence on salinity and temperature: R varies between 2.00 and 2.09 for SST ranging from 0° to 30° and salinity ranging from 0 to 38 psu.

Figure 2.

Vertical brightness temperature (TV) as a function of sea surface temperature (SST) with salinity as parameter. Wind speed is zero.

Figure 3.

Polarization ratio (R) as a function of SST with salinity as parameter. Wind speed is zero.

Figure 4.

Polarization ratio (R) as a function of wind speed (WS) with sea surface temperature as parameter. Salinity is 34 psu.

[14] Typically, the variations are much smaller for realistic conditions where approximate temperature and salinity values for a given area are known (from climatology and from other measurements). For example, for SST = 21 ± 1°C and for S = 35 ± 1 psu, R varies between 2.061 and 2.068, corresponding to a 0.3% variation.

[15] Figure 4 shows a stronger dependence on wind speed with R variations from 2.06 down to 1.94 for wind speeds between 0 and 20 m s−1 (20°C and 34 psu), i.e., a 6% variation. Hence, if the wind speed is known, the true polarization ratio (R) can be estimated with good fidelity. By measuring the local ratio (R′), θ can then be found, and finally having this the true vertical brightness temperature is found using the formulas shown in this section. Similarly, the true horizontal brightness temperature may be found.

3.1. Simple Example With No Measurement Errors

[16] Consider the following example: S = 34 psu, SST = 20°C, and WS = 10 m s−1. From the ocean model the following is then found: TBV = 132.65 K, TBH = 66.40 K, and R = 1.998. Assume a 10° Faraday rotation and the measured (local) brightness temperature values are found to be TBV =130.65 K and TBH = 68.40 K.

[17] If these values were used without correcting for the Faraday rotation, the retrieval would be incorrect. The H pol value points toward lower salinity while the V pol value points toward higher salinity (bearing in mind that it is steadily assumed that the wind speed is known from other sources).

[18] The measured local polarization ratio is R′ = 1.910. The true polarization ratio R is known as the wind speed is known, so the Faraday rotation is found from equation (4) to be θ = 10.02°, and from equation (2) we then find that TBV = 132.65 K, meaning that the correct value of the 1.4 GHz vertical brightness temperature is recovered and good quality geophysical parameters may be found.

[19] The example is only a quick illustration of the retrieval procedure excluding radiometric and other errors. Some of these errors will be included in sections 3.2–3.4.

3.2. Example With Radiometric Errors

[20] A sensitivity analysis was carried out to assess the impact of radiometric errors of ±0.1 K. This was performed by modeling vertical and horizontal brightness temperatures, true polarization ratio, and resulting local vertical and horizontal brightness temperatures for the following ocean conditions: 34 psu salinity, 20°C temperature, 10 m s−1 wind speed, and 10° and 3° Faraday rotation. The local vertical and horizontal brightness temperature values are then perturbed by ±0.1 K to simulate the effect of radiometric errors. Using these more or less faulty values, the associated local polarization ratio, the retrieved Faraday angle, and, finally, the retrieved vertical brightness temperature are calculated. The result is that the Faraday angle and the brightness temperature are found quite well despite the radiometric errors: The average error in the retrieved vertical brightness temperature is 0.07 K (worst case: 0.13 K). The observed errors in the brightness temperature are largely due to propagation of the measurement errors and not to incorrect Faraday correction: Once a certain brightness temperature is measured wrongly because of instrument errors, the true value cannot be found.

3.3. Example With Error in Salinity and/or Sea Surface Temperature Assessment

[21] As already discussed, a realistic scenario might be that the sea temperature and salinity are known to a certain level of accuracy a priori, e.g., 21 ± 1°C and 35 ± 1 psu. The nominal polarization ratio R is then 2.065 with a variation between 2.061 and 2.068, as already stated. If, in the retrieval process as described above, the nominal value for R is being used but the salinity and/or the temperature are off by, e.g., one unit, an error will be introduced, as the true polarization ratio may be off by 0.002 on average. This results in an error on the retrieved vertical brightness temperature of 0.04 K.

3.4. Example With Error in Wind Speed Assessment

[22] Finally, the impact of faulty wind speed auxiliary data is analyzed assuming there are neither instrument errors nor errors in the S and SST assessment. The true wind speed is once again 10 m s−1, resulting in the vertical and horizontal brightness temperatures, hence polarization ratio, as given above. Using this information, the correct Faraday angle, hence vertical brightness temperature, is retrieved (as in the first simple example). However, assuming wind speed readings of 8 m s−1 instead of 10 m s−1, Figure 4 gives a wrong “true” polarization ratio, which in turn results in a wrong Faraday angle, hence an incorrect retrieved brightness temperature that is off by 0.22 K. This illustrates the importance of good wind speed data.

4. Correction Based on Polarimetric Measurements

[23] Until now, it has been assumed that the radiometer system in question measures only vertically and horizontally polarized signals. If, however, the radiometer is enhanced to be a fully polarimetric system, other options exist to correct for Faraday rotation.

[24] The (brightness) Stokes vector is

equation image

where λ is the wavelength, k is Boltzmanns constant, and z is the impedance of the medium in which the wave propagates. The first Stokes parameter I is the sum of the vertical and horizontal brightness temperatures (that is, it represents the total power in the field), while the second Stokes parameter Q is the difference of the same quantities. It is seen that I and Q represent the information also found in the simpler V- and H-polarized radiometers. The third Stokes parameter can be interpreted as either the real part of the correlation between the vertical and horizontal electrical fields or the difference between orthogonally sensed brightness temperatures skewed 45° with respect to the normal vertical and horizontal orientation. Finally, the fourth Stokes parameter is interpreted as either the imaginary part of correlation between the vertical and horizontal fields or as the difference between left-hand and right-hand circularly polarized brightness temperatures. Subjected to a Faraday rotation θ, the Stokes parameters, as sensed by the polarimetric radiometer system, can be expressed in terms of the true parameters

equation image
equation image
equation image
equation image

[25] It is seen that the first Stokes parameter is unaffected by Faraday rotation, which is not surprising bearing in mind that it represents the total power in the field and power is not affected by a rotation. Likewise, the fourth parameter, being based on circular polarized quantities, is, of course, unaffected. The interesting Stokes parameter in the context of Faraday rotation correction is the third, U. The locally sensed U′ first contains the natural third Stokes parameter and for small Faraday rotations with a weight of almost 1 owing to the cosine factor. However, U′ also contains a contribution from the difference between the natural vertical and horizontal brightness temperatures, strongly dependent of the Faraday rotation angle owing to the sinus factor. Taking again the typical numbers from section 2, TBV = 132 K and TBH = 66 K, meaning that Q = 66 K and a contribution for θ = 10° as large as 23 K is found. The natural third Stokes parameter value for the ocean may not yet be fully established, but it is generally agreed that it is small: below 1 K and possibly only a tenth of that. This means that in the present context it may be ignored, and having measured Q′ and U′ the Faraday angle can be found with good fidelity from equation (6). This concept was first been published and discussed by Yueh [2000]. Having found the Faraday rotation angle the true vertical brightness temperature is found using equation (2).

5. Avoiding the Problem by Using the First Stokes Parameter?

[26] It was suggested in section 3 that by measuring H and V polarization and assuming knowledge of the wind speed, the Faraday rotation can be estimated and corrected for by investigating polarization ratios. Likewise, it was suggested in section 4 that if the fully polarimetric signal (all 4 Stokes parameters) is measured, and azimuthal symmetry of the scene is assumed (i.e., very little natural third and fourth Stokes signals), the Faraday rotation can be assessed and hence corrected for.

[27] However, the fully polarimetric mode is not always the baseline mode for proposed systems, as it is technically more demanding. It may be proposed that one might circumvent the whole Faraday problem by retrieving the necessary geophysical parameters from the first Stokes parameter alone, as it is not affected by the rotation. The following discusses this retrieval option by checking the sensitivity of the first Stokes parameter to relevant geophysical parameters: salinity, temperature, and wind speed. Figure 5 shows how the first Stokes parameter depends on sea temperature in the no wind condition.

Figure 5.

First Stokes parameter (I) as a function of SST with salinity as parameter. Wind speed is zero.

[28] For open, warm ocean, 20°C and in the 30–38 psu range, a first Stokes parameter sensitivity to salinity of ΔIS = 1.10 K psu−1 is observed. This must be compared with the sensitivity of the vertical brightness temperature to salinity which is found to be (Figure 2) ΔTBVS = 0.69 K psu−1. The sensitivity of the first Stokes parameter must be compared with that of the vertical brightness temperature, bearing in mind that the radiometric ΔT associated with the first Stokes parameter is inherently a square root of 2 worse than that of the vertical brightness temperature. The ratio of sensitivities is seen to be 1.10/0.69 = 1.59, that is, larger than a square root of 2, and the “signal-to-noise ratio” is actually slightly improved by using the first Stokes parameter. This conclusion also holds for cooler ocean temperatures.

[29] Also, the sensitivity to sea surface temperature can be assessed. As usual, the sensitivity is very small for salinities around 30 psu. For 38 psu and the 20°–30°C range is found ΔI/ΔSST = 0.29 K/°C for the first Stokes parameter, while the sensitivity for the vertical brightness temperature is ΔTBV/ΔSST = 0.16 K/°C. The ratio ΔITBV is slightly larger for the temperature sensitivity than for the salinity, i.e., an unwanted feature, but the difference is small, and the knowledge of sea surface temperature from other sources so good that this is not considered a problem.

[30] Finally, the sensitivity to wind speed must be assessed. Recall the Hollinger data, which predict 0.2 K per m s−1 for V pol and 0.3 K per m s−1 for H pol at a 50° incidence angle. This means that the sensitivity of the first Stokes parameter to wind speed is assessed to be ΔIWS = 0.5 K per m s−1, which is a factor of 2.5 larger than that of the normal vertical brightness temperature. The value of ΔIS was typically a factor of 1.6 larger than ΔTBVS, so the wind speed influence on the salinity retrieval is a factor of 2.5/1.6 = 1.6 larger than before, which may be a problem.

6. Other Incidence Angles and Wind Speed Dependencies

[31] All considerations until now have been carried out with a 50° incidence angle. However, this value is not necessarily the optimum choice for a future L band ocean mission, be it a real aperture or a synthetic aperture system. Therefore it is of interest to assess to what degree the results discussed above are also valid for other angles. In the following example a 30° incidence angle is assumed.

[32] Again, the Klein and Swift model was used, and the results are summarized as follows: The basic vertical brightness temperature sensitivity to salinity is slightly reduced (by ∼10%). The sensitivity to sea surface temperature is practically the same as it was at 50° incidence. The polarization ratio is smaller than before, and its relative variation with temperature and salinity is much smaller than before, typically halved. This means that the requirements on accuracy for sea temperature and salinity estimates become less demanding when assessing the true polarization ratio. The Hollinger wind data indicate wind responses of 0.25 K per m s−1 at both polarizations (30° incidence angle). This means that to first order the polarization ratio does not change with wind speed, greatly simplifying the task of estimating the true polarization ratio.

[33] An example was carried out with radiometric errors as in section 3, and the average error in the retrieved brightness temperature is basically as before. In summary, first results show that the correction method based on the polarization ratio works better at steeper incidence.

[34] The first Stokes parameter sensitivity to salinity is practically as before. The sensitivity to sea surface temperature is the same as it was at 50° incidence. The wind speed dependence for the vertical brightness is 0.25 K per m s−1, and for the first Stokes it is 0.5 K per m s−1, i.e., a factor of 2 larger. At 50° incidence this factor was 2.5, so wind speed correction is easier. In summary, the use of the first Stokes parameter works as good or better at steeper incidence.

[35] Finally, it must be noted that the brightness temperature dependence on wind speed has played a rather significant role in the discussions. The old Hollinger data have been quoted and used: 0.25 K m−1 s−1 for both vertical and horizontal polarization up to some 30° incidence, whereupon the curves diverge, resulting in 0.2 K per m s−1 at V pol and 0.3 K per m s−1 at H pol at 50° incidence. Recent measurements in the Wind and Salinity Experiment (WISE) campaign [Camps et al., 2002] largely seem to confirm the Hollinger data. They indicate the same sensitivity (again, 0.25 K per m s−1) for both polarizations at steep incidence but slightly more divergent curves as the incidence angle increases. Once the final conclusions are reached concerning the WISE data, a reiteration of the calculations in this article must be carried out, but only details are expected to change.

7. Conclusions

[36] It is well established that important sea surface parameters like salinity can be remotely sensed by a spaceborne L band radiometer. Vertical polarization is generally preferred, as it is more sensitive to salinity than horizontal polarization. However, Faraday rotation will mix the horizontally and vertically polarized signals emitted from the sea surface as they are received by the radiometer. In case the Faraday rotation is known and the radiometer measures the local horizontal and vertical signals, the mixing can be untangled and the true vertical brightness temperature recovered for subsequent salinity estimates. The price is that a dual polarized radiometer is required despite the fact that only the true vertically polarized brightness temperature is needed. The Faraday rotation angle might be available from other sources, but that option is not discussed here. The issue here is to circumvent the problem without additional external information.

[37] First, it is demonstrated that by using the local polarization ratio (vertical divided by horizontal brightness signal) and estimating the true polarization ratio, the Faraday angle can be found. The true polarization ratio can be assessed if the wind speed is known from some auxiliary information source, data which is needed anyway for the basic salinity retrieval process.

[38] Second, the Faraday angle can be assessed by enhancing the radiometer to full polarimetric operation. Assuming a small natural third Stokes parameter value for ocean surfaces, the measured value will enable recovery of the Faraday angle. The additional price to pay is the complexity of polarimetric operation of the radiometer.

[39] Third, the Faraday problem can be circumvented by directly retrieving salinity from the first Stokes parameter which is invariant to rotation. The price is that this parameter is slightly more dependent on wind speed than was the case for the vertically polarized signal, which makes corrections more difficult. However, there is not a straightforward trade-off between a retrieval independent of Faraday rotation problems and with some sensitivity to wind and a retrieval that has to deal with uncertainties in relation to Faraday and having a slightly smaller sensitivity to wind speed. A more in-depth investigation of salinity retrievals based on the first Stokes parameter is required.

[40] During the design phase of a radiometer system, it must be decided whether to go for the polarimetric option, as this has severe hardware implications. Note that an early choice between the first and the third options is not needed. In the data retrieval phase after launch the optimum option can be selected as demanded by the situation.


[41] During several SMOS Science Advisory Group meetings, Faraday issues have been discussed and thus inspired and supported this work.