Polarization Rotation Correction in Radiometry: An Error Analysis
Yueh proposed a method of using the third Stokes parameter TU to correct brightness temperatures such as Tv and Th for polarization rotation. This paper presents an extended error analysis of the estimation of Tv , Th, and TQ equiv Tv - Th by Yueh's method. In order to carry out the analysis, we first develop a forward model of polarization rotation that accounts for the random nature of thermal radiation, receiver noise, and (to first order) calibration. Analytic formulas are then derived for the bias, standard deviation (STD), and root-mean-square error (RMSE) of estimated TQ, Tv , and Th, as functions of scene and radiometer parameters. These formulas are validated through independent calculation via Monte Carlo simulation. Examination of the formulas reveals that: 1) natural TU from planetary surface radiation, of the magnitude expected on Earth at L-band, has a negligible effect on correction for polarization rotation; 2) RMSE is a function of rotation angle Omega, but the value of Omega that minimizes RMSE is not known prior to instrument fabrication; and 3) if residual calibration errors can be sufficiently reduced via postlaunch calibration, then Yueh's method reduces the error incurred by polarization rotation to negligibility.
3212 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 10, OCTOBER 2007 Polarization Rotation Correction in Radiometry: An Error Analysis Derek Hudson, Jeffrey R. Piepmeier, Member, IEEE, and David G. Long, Senior Member, IEEE Abstract Yueh proposed a method of using the third Stokes parameter TU to correct brightness temperatures such as Tv and Th for polarization rotation. This paper presents an extended error analysis of the estimation of Tv , Th, and TQ Tv Th by Yueh s method. In order to carry out the analysis, we first develop a forward model of polarization rotation that accounts for the random nature of thermal radiation, receiver noise, and (to first order) calibration. Analytic formulas are then derived for the bias, standard deviation (STD), and root-mean-square error (RMSE) of estimated TQ, Tv, and Th, as functions of scene and radiometer parameters. These formulas are validated through independent calculation via Monte Carlo simulation. Examination of the formulas reveals that: 1) natural TU from planetary surface radiation, of the magnitude expected on Earth at L-band, has a negligible effect on correction for polarization rotation; 2) RMSE is a function of rotation angle , but the value of that minimizes RMSE is not known prior to instrument fabrication; and 3) if residual calibration errors can be sufficiently reduced via postlaunch calibration, then Yueh s method reduces the error incurred by polarization rotation to negligibility. Index Terms Faraday effect, microwave polarimetry, polarization. I. INTRODUCTION THE EARTH S ionosphere and magnetic field cause Faraday rotation of the polarization of radiation emanating from the Earth s surface. This rotation mixes the vertical and horizontal polarization components of brightness temperatures Tv and Th, degrading the measurement of both. The oft-used second Stokes parameter TQ ( Tv Th) is doubly degraded. For L-band satellite measurements, the error in TQ due to uncorrected Faraday rotation can exceed 10 K, depending on solar activity, incidence angle, and the angle between the look direction and the Earth s magnetic field [2] (Faraday rotation is inversely proportional to the square of frequency; therefore, this source of polarization rotation is less important above L-band). Additional polarization rotation occurs if a sensor s antenna feed polarization basis is rotated with respect to the natural polarization basis of the Earth s surface. Such rotation may Manuscript received September 29, 2006; revised March 27, 2007. This work was supported by the NASA Goddard Space Flight Center. D. Hudson and D. G. Long are with the Department of Electrical and Computer Engineering, Brigham Young University, Provo, UT 84602 USA (e-mail: dlh8@et.byu.edu). J. R. Piepmeier is with the Microwave Instrument Technology Branch, NASA Goddard Space Flight Center, Greenbelt, MD 20771 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2007.898438 occur as an accidental misalignment [3] or may be deliberately permitted in order to simplify hardware [4]. Near-future L-band spaceborne radiometers, namely, SMOS [5] and Aquarius [6], are being designed to perform polarization rotation correction (PRC) in postprocessing. A basic method involves measuring the third Stokes parameter TU in addition to the usual Tv and Th. The method is introduced by Yueh in [1]. Previously developed forward models of polarization rotation [1], [3], [4], [7] are deterministic and neglect the role of receiver channel noise (although [8] includes noise in simulations). In the Appendix, we develop an extended model which takes into account the random nature of the radiation and also accounts for receiver noise. Simple and accurate expressions are derived for the means, variances, and covariances of the measurements in a three-channel (Tv, Th, and TU) radiometer. These are derived in the Appendix and summarized in Section II. In Section III, we review Yueh s correction technique. In Section IV, we derive the mean, variance, and root-meansquare error (RMSE) of the resulting estimate of TQ. Similar derivations for Tv and Th are presented in Section V. Insights from the resulting formulas are presented in Section VI, and conclusions are offered in Section VII. Throughout this paper, we illustrate with case studies of the Aquarius radiometer, whose deployment is expected in 2009. The Aquarius instrument will have three beams with respective incidence angles of 28.7 , 37.8 , and 45.6 [6]. For measurements of ocean emissions, these angles dictate a nominal TQ of about 20, 35, and 53 K, respectively [9]. The Aquarius instrument also has nominal integration time of 6 s. We also refer to the canceled NASA Hydros mission [10], whose nominal was 0.016 s (we adjust the Hydros incidence angle from 39.3 to 37.8 in our studies to match Aquarius). II. SUMMARY OF FORWARD MODEL As shown in the Appendix, the processes of receiving, detecting, and calibrating the first three Stokes parameters in a polarimetric radiometer can be summarized with the forward model TIa =TI + TRX,I + Tsys,I (1) TQa = +TQ cos 2 + TU sin 2 + TRX,Q + Tsys,Q (2) TUa = TQ sin 2 +TU cos 2 + TRX,U + Tsys,U . (3) 0196-2892/$25.00 2007 IEEE Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. HUDSON et al.: POLARIZATION ROTATION CORRECTION IN RADIOMETRY: AN ERROR ANALYSIS 3213 The quantities on the left-hand sides are our measurements of the first three Stokes parameters after rotation, detection, and calibration. On the right sides, the natural Stokes parameters of the scene TI , TQ, and TU are altered by polarization rotation [7] and perturbed by error sources, represented by quantities with a prefix. TRX,I , TRX,Q, and TRX,U are residual biases from the calibration process that is performed throughout data collection. For example, if the fourth calibration scheme described in [11] is used, then TRX,U corresponds to all but the first term on the right side of [11, eq. (42)]. That same calibration scheme also leads to TRX,v = TRX,h = TH TC TC TH TH TC (4) where TH and TC are the true temperatures of the hot and cold calibration sources, while TH and TC are the best available estimates of them (the calibration sources could be noise diodes or external targets, for example). Then, TRX,I and TRX,Q are defined as the sum and difference of TRX,v and TRX,h, respectively. Using (4) gives TRX,Q = 0. A more realistic description distinguishes between the calibration sources in the v and h channels, i.e., TRX,v = THv TCv TCv THv THv TCv TRX,h = THh TCh TCh THh THh TCh (5) so that TRX,Q is nonzero. This distinction also complicates the expression for TRX,U . The radiometer calibration process is such that TRX,I , TRX,Q, and TRX,U are slowly varying (e.g., over a period of many minutes or more) compared with the radiometer integration time . Even if estimates of the calibration parameters (e.g., TCv) are obtained as often as several times per , we assume that the predictable thermal environment of space allows us to average those estimates extensively to yield better estimates. Therefore, for estimating Tv, Th, and TQ measured in a single radiometer measurement cycle or even many cycles, we can consider TRX,I , TRX,Q, and TRX,U to be constants. It is also anticipated that this averaging (and other postlaunch calibration activities) will reduce TRX,I , TRX,Q, and TRX,U to such low magnitude that they are negligible compared to the other error sources. In the development of (1) (3), we have neglected the channel gains which are also estimated during data collection as part of the calibration process (see [11]). Although these gains and our uncertainties in them are relevant, we omit them in this paper, leaving their analysis for future work. The quantities Tsys,I , Tsys,Q, and Tsys,U in (1) (3) are zero-mean Gaussian random variables which correspond to the usual noise equivalent T (NE T) of radiometric measurements [12, p. 365]. They fluctuate significantly from one radiometer measurement cycle to the next. From their definition in (67) (69), we see that they have the same covariance matrix as Tsys,I , Tsys,Q, and Tsys,U (as well as TIa, TQa, and TUa), which is given in (66). In (66),N 2B , where B is the sensor bandwidth (about 20 MHz for Aquarius and Hydros). For future reference, we find the means of our calibrated measurements: TIa TIa = TI + TRX,I TQa TQa = +TQ cos 2 + TU sin 2 + TRX,Q TUa TUa = TQ sin 2 + TU cos 2 + TRX,U . (6) III. ROTATION CORRECTION TECHNIQUE In this section, we review PRC in the context of the forward model summarized in Section II. A. Estimation of TQ Yueh s model [1] does not include any of the terms in (1) (3). By noting that TU is much smaller than TQ in natural Earth scenes, he proposes to solve (2) and (3) for TQ by also neglecting the terms with TU. By assuming TU and all the quantities are zero, squaring both sides of (2) and (3), adding the two results, and then solving for TQ, we obtain Yueh s proposed estimate TQ = T2 Qa + T2 Ua (7) where we ignore the negative root since TQ is positive in geophysical circumstances. In reality, of course, TU and all the quantities are nonzero and constitute the error sources of the correction technique. Nevertheless, as demonstrated by the error analysis in Section IV, this equation provides a good estimate of TQ. B. Estimation of Tv and Th With TQ from (7) and TIa from (1), we can also find Tv and Th as ( TIa TQ)/2. An error analysis of Tv and Th is pursued in Section V. Yueh proposed an alternate method of estimating Tv and Th (although it is straightforward to show its equivalence to estimating Tv and Th via ( TIa TQ)/2). First, to estimate , we divide (3) by (2), then solve for . With the error sources (TU and all the quantities) assumed to be zero, this yields = 1 2 tan 1 TUa TQa (8) as an estimate of the angle of polarization rotation. Then, assuming the error sources are zero and solving (73) and (74) for Tv and Th, respectively, yields Tv = Tva + TQ sin2 (9) Th = Tha TQ sin2 (10) which are the corrected forms of [1, eqs. (15) and (16)], where TQ is given in (7). Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. 3214 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 10, OCTOBER 2007 IV. ANALYSIS OF TQ We now determine the probability density function (pdf), mean, and variance of the estimate TQ = T2 Qa + T2 Ua. We use the mean and variance to calculate RMSE. TQa and TUa are already well characterized. They are Gaussian (at least to a very good approximation) with means given in (6) and with variances and covariance given in (66). A. Rotation of Variables TQa and TUa are correlated, but we can rotate coordinates such that we have uncorrelated quantities. Define Z W = 1 T2 sys,Q + T2 sys,U Tsys,Q Tsys,U Tsys,U Tsys,Q TQa TUa . (11) This is useful because Z2 + W 2 = T2 Qa + T2 Ua = TQ. If we assume that TQa and TUa are jointly normal, then Z and W are also jointly normal, and it is straightforward to show that Z and W are uncorrelated and have the following means and variances: Z = Tsy s,QTQa + Tsys,UTUa T2 sys,Q + T2 sys,U Z (12) W = Tsy s,QTUa Tsys,UTQa T2 sys,Q + T2 sys,U W (13) Var( Z) = T2 sys,I + T2 sys,Q + T2 sys,U N 2Z (14) Var( W ) = T2 sys,I T2 sys,Q T2 sys,U N 2W . (15) The pdf of TQ is given by using Z, W, Z, and W in [13, eq. (2)], which is restated here in terms of the current problem: f TQ ( TQ) = TQ Z W e T2 Q +2Z2 4 2 Z e T2 Q +2W2 4 2 W j= Ij a T2Q I2j(d TQ) cos 2j (16) for TQ > 0 and 0 otherwise, where a 2Z 2W 4 2Z 2W d2 Z2 4Z + W2 4W tan W 2Z Z 2W and Ij is the modified Bessel function of the first kind and order j. B. Simple Result by Assuming 2Z 2W We attempted to find the first and second moments of Z2 + W 2 analytically but failed, even though the pdf is known. Fortunately, a small approximation leads to simple and accurate formulas, as shown now. T2 sys,Q + T2 sys,U has a worst case maximum value of about 4000 K2 for Aquarius; in more extreme cases, it might reach 10 000 K2 (this is for L-band radiometers with incidence angles less than about 50 ). But, this is small compared to T2 sys,I , which has a value of 660 000 K2 for typical Aquarius parameters (TRX,I = 620 K and TI = 190 K). Therefore, 2Z T2 sys,I/N and 2W T2 sys,I/N. If we define 2 T2 sys,I/N and use 2Z 2 and 2W 2, then TQ is the root of the sum of the squares of two independent Gaussians with the same variance and with nonzero, unequal means. Note that is the NE T for the total signal (first Stokes parameter). With this approximation, the pdf, mean, and variance of TQ can be described as functions of just and m, where m2 Z2 +W2 = T2 Qa + T2 Ua. In terms of the original parameters m2 =T2Q + T2U+ T2 RX,Q + T2 RX,U + 2 cos 2 (+TQ TRX,Q + TU TRX,U ) + 2 sin 2 ( TQ TRX,U + TU TRX,Q). (17) By either [14] or by [13, eq. (1)], the density of TQ is then f TQ ( TQ)= TQ 2 e T2 Q +m2 2 2 I0 TQm 2 , TQ>0 (0 otherwise). (18) The mean and variance of TQ are [15, p. 72] TQ = 2 e m2 2 2 1F1 3 2, 1; m2 2 2 (19) Var( TQ) =2 2 + m2 TQ 2 (20) where 1F1 is the confluent hypergeometric function. Equation (19) corresponds with the first line of (3.10 12) in [16]; the second line shows that we can rewrite TQ as TQ = 2 1F1 1 2, 1; m2 2 2 . (21) A difficulty with using either (19) or (21) is that for large , is small, and the argument of 1F1 has very large magnitude (e.g., 70 000 for the Aquarius = 28.7 case). Calculating the value of 1F1 to high precision presents a huge computational burden when its argument is so large. Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. HUDSON et al.: POLARIZATION ROTATION CORRECTION IN RADIOMETRY: AN ERROR ANALYSIS 3215 Fortunately, using (4B-9) in [16], (21) becomes TQ = 2 e m2 4 2 1 + m2 2 2 I0 m2 4 2 + m2 2 2 I1 m2 4 2 (22) which can be evaluated quickly. The final simplification comes by examining plots of Monte Carlo results (see Section IV-C2). These plots suggest that Var( TQ) 2. Hypothesizing that Var( TQ) 2 is correct and using this in (20) yields the simple formulas TQ 2 + m2 (23) Bias( TQ) 2 + m2 TQ (24) Var( TQ) 2 (our hypothesis) (25) RMSE of TQ = Var( TQ) + Bias2( TQ) 2 2 + m2 + T2Q 2TQ 2 + m2. (26) These equations are the key results of this paper. We note that the pdf of TQ given in (18) can be well approximated by a Gaussian pdf with the mean and variance of (23) and (25). Therefore, we are justified in ignoring higher moments hereafter and concerning ourselves with only the mean and variance (and the RMSE derived from them). C. Validation of (23) (26) In this section, we use numerical methods to validate the derivation of (23) (26). 1) Numerical Equivalence of (22) and (23): We can validate the final leap used to obtain (23) by showing that (23) matches (22). A mathematics software package finds the magnitude of the difference between (22) and (23) to be less than 20 nK in the Aquarius = 28.7 case and less than 60 nK in the other Aquarius cases (these are calculated with TU = TRX,Q = 0.5 K, TRX,U = 0 K, and typical Aquarius values of TQ, TI , TRX,I , and N for 180 180 ). 2) Validation by Monte Carlo Simulation of Electric-Field Model: The mean and variance of TQ can also be found by Monte Carlo simulation. This can be done using (39) and (48) directly, thus avoiding all the approximations used in deriving (23) (26) from (39) and (48). The precise procedure is to generate N samples of a, b, Ev, and Eh, which are all independent of one another except EvEh = TU/2. From these, N samples of x and y are formed according to (39) and then squared and averaged to produce a single sample each of Tsys,Q and Tsys,U as in (48). To simulate the calibration process, TRX,Q is subtracted off, while TRX,Q and TRX,U are added on, forming TQa and TUa as in (2) and (3). These are used in (7) to form a single sample of TQ. This entire procedure is repeated M times to form M independent samples of TQ. The empirical mean and variance of TQ can Fig. 1. (Top) Bias, (center) STD, and (bottom) RMSE of TQ, Tv, and Th as functions of , with TI , TQ, , and TRX,I chosen to be typical of the Aquarius = 28.7 beam over ocean. The values of the remaining parameters (TRX,Q, TRX,I , TRX,Q, and TRX,U ) were chosen arbitrarily within expected ranges. then be calculated from these samples. This method gives no formulas, but its results converge to the exact results as M increases. The Monte Carlo results match the analytic results from (24) (26) very well, for many values of each of the parameters. Figs. 1 4 show some of these results. The discrepancy can be attributed to the inherent imprecision in the Monte Carlo method. V. ANALYSIS OF Tv AND Th We now pursue an analysis of Tv and Th, which are defined as ( TIa TQ)/2. Using (1) and (23) Tv 1 2 TI + TRX,I + 2 + m2 (27) Th 1 2 TI + TRX,I 2 + m2 . (28) Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. 3216 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 10, OCTOBER 2007 Fig. 2. Same as Fig. 1 except for the Aquarius = 45.6 beam (different choices of TRX,Q, TRX,I , TRX,Q, and TRX,U were also used in order to demonstrate the variety of possible behavior in the error). A. Variance and RMSE of Tv and Th Using (66) and (25) Var( Tv) = 1 4 Var( TIa)+Var( TQ)+2Cov( TIa, TQ) 1 4 2T2 sys,I+T2 sys,Q+T2 sys,U N +2Cov( TIa, TQ) . (29) Similarly Var( Th) 1 4 2T2 sys,I+T2 sys,Q+T2 sys,U N 2Cov( TIa, TQ) . (30) Finding Cov( TIa, TQ) analytically appears to be intractable. But, from (29) and (30), we see that Cov( TIa, TQ) = Var( Tv) Var( Th). We can therefore find Cov( TIa, TQ) numerically by subtracting the Monte Carlo estimates of Var( Th) from the Monte Carlo estimates of Var( Tv). We studied such Fig. 3. Same as Fig. 1 except for the Hydros soil moisture sensing mission (different choices of TRX,Q, TRX,I , TRX,Q, and TRX,U were also used in order to demonstrate the variety of possible behavior in the error). numerical results and found patterns, then hypothesized the following formula for Cov( TIa, TQ) from those patterns: Cov( TIa, TQ) = 2Tsys,I N T2 sys,Q + T2 sys,U . (31) Using (31) in (29) and (30) Var( Tv) 2T2 sys,I+4Tsys,I T2 sys,Q+T2 sys,U+T2 sys,Q+T2 sys,U 4N (32) Var( Th) 2T2 sys,I 4Tsys,I T2 sys,Q+T2 sys,U+T2 sys,Q+T2 sys,U 4N . (33) These, together with (27) and (28), give Mean-square error of Tv 1 4 2+m2 TQ+ TRX,I 2 + 2T2 sys,I+4Tsys,I T2 sys,Q+T2 sys,U+T2 sys,Q+T2 sys,U 4N (34) Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. HUDSON et al.: POLARIZATION ROTATION CORRECTION IN RADIOMETRY: AN ERROR ANALYSIS 3217 Fig. 4. Same as Fig. 1 except with the Monte Carlo results generated using the Gaussian approximation, thus allowing much larger M (different choices of TRX,Q, TRX,I , TRX,Q, and TRX,U were also used in order to demonstrate the variety of possible behavior in the error). Mean-square error of Th 1 4 2+m2 TQ TRX,I 2 + 2T2 sys,I 4Tsys,I T2 sys,Q+T2 sys,U+T2 sys,Q+T2 sys,U 4N . (35) The RMSEs of the estimated Tv and Th are the positive square roots of these equations. They do not appear to simplify further, although the variances can be approximated as 2/2. B. Plots In Figs. 1 4, we illustrate the (top) bias, (middle) STD, and (bottom) RMSE for the estimated TQ, Tv, and Th, as functions of . The analytic results [given using (24) (28) and (32) (35)] are plotted as solid lines. The Monte Carlo results are plotted as symbols. Figs. 1 and 2 were computed using TI , TQ [9], TRX,I , and values that are typical of the innermost and outermost of the three Aquarius beams, respectively, while Fig. 3 uses values Fig. 5. Same as Fig. 4 except that we have chosen TRX,Q and TRX,U to be small compared to , which is the result anticipated from careful postlaunch calibration. As a consequence, the dependence on is weak in this figure. that are typical of the Hydros radiometer. The particular values of TU, TRX,Q, TRX,I , and TRX,Q were chosen arbitrarily within their expected ranges. All values are given at the top of each figure. The Monte Carlo results were generated as previously described (Section IV-C2). The discrepancies between the analytic and the Monte Carlo results decrease as M increases. But, it is difficult to increase M: generating a plot such as Fig. 1 currently requires days of computer time. Another option is to generate the Monte Carlo samples using the Gaussian approximation (see Section C in the Appendix). That is, rather than generating samples of the electric field, we generate samples of TIa, TQa, and TUa themselves as Gaussian random variables, with the means, variances, and covariances summarized in Section II. This method, although not quite as exact, is many orders of magnitude faster, allowing much larger M and more data points. Examples of the results obtained thereby are shown in Figs. 4 and 5. VI. INSIGHTS FROM THE EQUATIONS With the forward model we have developed, there are five sources of error to be considered in PRC: TU, TRX,I , Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. 3218 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 10, OCTOBER 2007 Fig. 6. RMSE (from the analytically derived formulas) of TQ, Tv, and Th as functions of TU, with TRX,I , TRX,Q, and TRX,U set to zero. TI , TQ, , and TRX,I are typical of (left) the Aquarius = 28.7 beam, (center) the Aquarius = 45.6 beam, and (right) the Hydros soil moisture sensing mission. TRX,Q, TRX,U , and NE T (manifested as ). In this section, we study some effects of these error sources, using the equations derived previously. We note that after PRC, can be viewed as merely a modulator of the error sources TRX,Q and TRX,U rather than as an error source in itself. A. Insignificance of TU Examining (17), we see that the first term is desired while the rest are sources of bias. However, TQ is more than an order of magnitude larger than the other components of (17); therefore, we can neglect terms without TQ, resulting in m2 T2Q + 2TQ( TRX,Q cos 2 TRX,U sin 2 ). (36) This eliminates TU from the equations, which suggests that natural TU is not a significant error source in PRC, at least at L-band. We can numerically examine the significance of TU as follows. We set the unknown error sources ( TRX,I , TRX,Q, and TRX,U ) to zero but retain since it is known. Then, we plot RMSE as a function of natural TU. The results are shown in Fig. 6 for the typical parameters of Aquarius beams. The RMSE at TU = 0 is due to NE T through . We can discern that the error caused by natural TU is negligible compared to NE T when |TU| < 1.5 K. Natural TU is reported to have a maximum magnitude of about 1.5 K over the oceans at intermediate and high wind speeds, 10.7 GHz, and an incidence angle of 50 [17]. Extensive measurements at L-band have not been made, but one group reports amplitudes of less than 1 K over wind-driven ocean [18]. These measurements, combined with Fig. 6, suggest that natural TU is not a significant error source for the Aquarius mission. This further suggests that the error allocation for other (wind) in the Aquarius error budget [6, p. 8] can be significantly reduced. At the right of Fig. 6, we plot the results for typical parameters of a soil moisture sensing mission such as the canceled Hydros mission. The short integration time of this conical scanning radiometer results in NE T being so large that the effects of natural TU are negligible for |TU| < 5 K. B. Optimal Value Examining (17) shows that near = 0 , the effect of TRX,Q is amplified compared to the effect of TRX,U , because of TQ being so much larger than TU. Similarly, the effect of TRX,U is amplified near = 45 . Consequently, if | TRX,Q| is significantly larger than | TRX,U |, then the RMSE of TQ is minimum near = 45 . Likewise, if | TRX,Q| is significantly smaller than | TRX,U |, then the RMSE of TQ is minimum near = 0 (see Figs. 1 4 for examples). If TRX,Q and TRX,U have the same magnitude and sign, then RMSE is minimum near = 22.5 . If they have the same magnitude but opposite sign, then RMSE is minimum near = 22.5 . But, in all these cases, if the magnitude of both is less than /2, then RMSE is approximately constant with respect to (and is for TQ), as shown in Fig. 5. Because TRX,Q and TRX,U are unknown (at least until instrument fabrication and initial calibration), there is no basis for claiming a priori that RMSE is better at any one value of than at any other value. This should correct the notion that it is best to sense the land or ocean at dawn because of low free electron content in the atmosphere (and, hence, small ), which is an assumption used in the design of the Hydros mission. We note that there may be other good reasons for sensing at dawn, such as the better known temperature profile of the atmosphere and planetary surface. C. Negligible Error Contribution of PRC If TRX,I , TRX,Q, and TRX,U are reduced to insignificance through postlaunch calibration, then the overall RMSE reduces to the NE T that exists regardless of polarization rotation. To see this analytically, let TRX,I = TRX,Q = TRX,U = 0 and also let TU = 0 since we know that its effect Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. HUDSON et al.: POLARIZATION ROTATION CORRECTION IN RADIOMETRY: AN ERROR ANALYSIS 3219 is not large. Then, m2 reduces to T2Q , and using a binomial expansion of (24) Bias( TQ) 2 + T2Q TQ 2 2TQ (37) RMSE of TQ 2 + 4 4T2Q . (38) The validity of these approximations is easily confirmed by numerical examples using Aquarius and Hydros parameters. Similar analysis shows that the RMSE of Tv and Th reduces to /2. Hence, error allocation for ionospheric effects can be greatly reduced [6]. VII. CONCLUSION We have extended the forward model of polarization rotation to include the random nature of radiation, radiometer channel noises, and (to first order) calibration. In particular, we have derived the means, variances, and covariances of the first three Stokes parameters TI , TQ, and TU (or their modified counterparts, Tv, Th, and TU) as measured by radiometers in which TU is measured as the correlation of Tv and Th. There are several known limitations to this forward model. First, it ignores the antenna sidelobe contributions to the apparent brightness temperature, which undergo a different amount of polarization rotation than the main beam contribution. Second, it ignores the mixing of the scene Stokes parameters by the antenna (i.e., the antenna cross-pol patterns). Third, it ignores the uncertainties in channel gains which remain after the calibration process. In this paper, we ignored these effects for tractability and because their effects are projected to be smaller than those effects which we have included. Using the forward model just described, we have analyzed the errors in using polarimetric measurements to correct for polarization rotation as proposed in [1]. We have derived closed-form equations for the bias, STD, and RMSE of the estimated TQ [see (24) (26)] and similar expressions for Tv and Th. These equations match the numerical results obtained by Monte Carlo simulation of our original electric-field model. These equations are the key results of this paper since they allow more accurate error analysis and error budgeting than has been possible previously. This analysis indicates several things about the five sources of error. First, the natural third Stokes parameter (of the magnitude expected at L-band for most natural Earth scenes) is an insignificant source of error compared to NE T (for 6 s over ocean). Second, the dependence of RMSE on rotation angle is determined by residual errors from the calibration process TRX,Q and TRX,U . Since these residuals are unknown (by definition), we cannot predict the dependence of RMSE on rotation angle (such as whether or not = 0 is the optimum angle). But, if the postlaunch calibration reduces TRX,Q and TRX,U to the level of NE T or less, then the dependence of RMSE on is weak. Third, if TRX,I , TRX,Q, and TRX,U are reduced to insignificance through postlaunch calibration, then the overall RMSE reduces to the NE T that exists regardless of polarization rotation. In other words, Yueh s method reduces the error incurred by the polarization rotation to negligibility. APPENDIX I In this Appendix, we derive (1) (3). These equations comprise the forward model of polarization rotation which is used in this paper. A. Electric-Field Model Our most basic foundation is a model of the electric fields x(t) y(t) = cos sin sin cos Ev(t) Eh(t) + a(t) b(t) . (39) Ev(t) and Eh(t) are the components of the total electric field emitted by the scene in the vertical and horizontal directions, respectively (hereafter, our notation suppresses the time dependence t of all quantities). Because the number of independent emitters in the scene is large in spaceborne radiometry, Ev and Eh are normally distributed, by the central limit theorem, with zero means [19, pp. 477 478]. We assume that they are real because we are concerned only with the first three Stokes parameter in this paper. Ev and Eh are rotated through an angle , modeling polarization rotation. We consider to be constant over the period of one radiometer measurement. Receiver noise is then added, which is represented by the electric field amplitudes a and b. Like Ev and Eh, we assume that a and b are normally distributed, zero mean, and normal random variables.We also assume that they are independent of one another and of Ev and Eh. They represent self-emission by the antenna and radiometer. This model neglects the sidelobe contributions (as they may undergo different amounts of rotation than the main beam radiation) and cross coupling of the polarization components that is caused by the antenna and radiometer nonidealities (cross-pol patterns). The quantities that are most commonly reported in radiometry are the first three modified Stokes parameters, as brightness temperatures Tv Th TU E2 v E2h 2 EvEh (40) to which we add, for this document TRX,v TRX,h a2 b2 . (41) In these and subsequent definitions, we ignore a proportionality constant which converts the product of two electric fields to a brightness temperature. This conversion also assumes a narrowband radiometer, so that the frequency spectrums of Ev(t), Eh(t), a(t), and b(t) are flat (see [12, p. 193]). Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. 3220 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 10, OCTOBER 2007 A quantity of high interest to users of radiometry data is the second Stokes parameter TQ Tv Th E2 v E2h , where denotes the expected value (ensemble average). In addition to the definition in (40), TU can be equivalently defined in a manner analogous to the definition of TQ. This definition is TU T+45 T 45, where T+45 is the brightness temperature of the component of the incident radiation that is linearly polarized at 45 with respect to the Ev and Eh axes. Our model assumes a radiometer architecture in which the signals at +45 and 45 linear polarization (in the radiometer polarization basis) are synthesized from x and y after enough amplification of x and y (by amplifiers) that the receiver noise added after this synthesis is negligible. Radiometers which create the signals at +45 and 45 earlier (such as from direct measurement of T+45 and T 45) require that additional noise terms should be added to the additional channels. This would add many terms to the final forward model and the error formulas. B. Description of Parameters In this document, we could express our results in terms of Tv, Th, TRX,v, and TRX,h. It is more concise, however, to use the related quantities TI Tv + Th, TQ Tv Th, TRX,I TRX,v + TRX,h, and TRX,Q TRX,v TRX,h. Note that TI , TQ, and TU comprise the first three Stokes parameters [7] as brightness temperatures. We also note that in the final expressions for bias and variance (and, hence, RMSE), TI and TRX,I always appear added together, never separately. Therefore, we reduce our parameter set by using Tsys,I TI + TRX,I . Besides TI , TQ, TU, TRX,I , and TRX,Q, other parameters are , N, TRX,I , TRX,Q, and TRX,U (N is defined early in Appendix I-C; TRX,I , TRX,Q, and TRX,U are defined in Appendix I-E). This collection of ten parameters can be used to completely describe the forward problem, and we, therefore, refer to them as the original parameters. Other quantities are defined for convenience but can be expressed in terms of these original ten. The symbols x and y represent the electric fields to be detected by the radiometer. By the construction of (39), they are also zero-mean normal random variables. We denote their expected squared values, as brightness temperatures, with Tsys,v Tsys,h Tsys,U x2 y2 2 xy . (42) Using (40), (41), and the facts that a and b are independent of all other quantities and are zero mean, we find Tsys,v = (Ev cos + Eh sin + a)2 =Tv cos2 +Th sin2 + TU 2 sin 2 + TRX,v. (43) By a similar process Tsys,h =Th cos2 +Tv sin2 TU 2 sin 2 + TRX,h (44) Tsys,U = TQ sin 2 + TU cos 2 . (45) C. Measured Temperatures, Tsys,v, Tsys,h and Tsys,U A conventional two-channel radiometer measures Tsys,v and Tsys,h by a time average Tsys,v 1 0 x2dt, Tsys,h 1 0 y2dt. (46) We use hats to denote measured or estimated quantities, which are random variables, as opposed to the unhatted quantities which represent the desired true quantities, such as the ensemble average of a random variable. A three-channel polarimetric radiometer also measures Tsys,U 2 0 xy dt. (47) As shown in [19, pp. 487 488], Tsys,v, Tsys,h, and Tsys,U can be rewritten as sums of independent samples Tsys,v = 1 N N i=1 x2i Tsys,h = 1 N N i=1 y2 i Tsys,U = 2 N N i=1 xiyi (48) where N = 2B , B is the sensor bandwidth, and is the integration time. We next proceed to find the distributions of Tsys,v, Tsys,h, and Tsys,U. For large N (for Aquarius, N 5e8), Tsys,v is so nearly Gaussian, by the central limit theorem, that we assume that it is Gaussian. Similar results apply for Tsys,h and Tsys,U . Therefore, they can be very well characterized by only their means, variances, and covariances, which we derive next. 1) Means of Tsys,v, Tsys,h, and Tsys,U : The ensemble average (expected value) of Tsys,v is Tsys,v = 1 N N i=1 x2i = 1 N N i=1 x2i = Tsys,v. (49) Similarly, Tsys,h = Tsys,h and Tsys,U = Tsys,U . 2) Var( Tsys,v) and Var( Tsys,h): Var( Tsys,v) = 1 N N i=1 x2i 1 N N j=1 x2j T2 sys,v = 1 N2 N i=1 N j=1 x2i x2j T2 sys,v (50) Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. HUDSON et al.: POLARIZATION ROTATION CORRECTION IN RADIOMETRY: AN ERROR ANALYSIS 3221 which we separate into terms for which i = j and for which i = j = 1 N2 N i=1 N j=1( =i) x2i x2j + 1 N2 N i=1 x4i T2 sys,v. (51) Using the independence of samples i and j and the known fourth moment of zero-mean normal random variables = 1 N2 N i=1 x2i N j=1( =i) x2j + 1 N2 N i=1 3 x2i 2 T2 sys,v. (52) Then, using (42) Var( Tsys,v) = 2 N T2 sys,v. (53) By a similar process Var( Tsys,h) = 2 N T2 sys,h. (54) 3) Var( Tsys,U ): By a process similar to (50) (53) Var( Tsys,U) = T2 sys,U N + 4 N x2y2 . (55) Consider x2y2 alone. Using the definitions of x and y in (39), it can be expanded to several dozen terms. The independence of a and b from Ev and Eh means that many terms can be factored as a , b2 , and so on. Then, using (40), (41), the fact that a and b are zero mean, and the known fourth moment of zero-mean normal random variables, many terms drop out or simplify, leaving x2y2 = 1 2 EvE3h E3 vEh sin 4 + 1 2TITRX,I + 3 4 E2 vE2h 3 8 T2 v + T2 h ! cos 4 1 2TRX,Q(TU sin 2 + TQ cos 2 ) + 3 8T2 v + 1 4 E2 vE2h + 3 8T2 h + TRX,vTRX,h. (56) Ev and Eh are marginally zero-mean Gaussians, with variances of Tv and Th and a covariance of TU/2. Assuming that they are jointly Gaussian, their joint pdf is completely specified.We can therefore determine E3 vEh , EvE3h , and E2 vE2h by direct integration E3 vEh = 1 4TvTh T2U E3 vEhe 2ThE2 v+2TUEhEv 2TvE2 h 4TvTh T2 U dEvdEh. (57) Using a table of integrals [20], the known second and fourth moments of zero-mean Gaussians, and much algebra, this reduces to E3 vEh = 3 2TUTv. (58) By similar processes, we find EvE3h = 3 2TUTh E2 vE2h =TvTh + 1 2T2U . (59) By using these results in (56) and then using (56) in (55), we obtain, after much algebraic manipulation Var( Tsys,U) = 1 N " T2 sys,I T2 sys,Q + T2 sys,U # (60) where Tsys,I Tsys,v + Tsys,h = TI + TRX,I and Tsys,Q Tsys,v Tsys,h. 4) Covariances of Tsys,v, Tsys,h, and Tsys,U : We wish to determine the covariances that exist between Tsys,v, Tsys,h, and Tsys,U . Similar to the derivation of (60), it can be shown that Cov( Tsys,v, Tsys,h) = T2 sys,U 2N (61) Cov( Tsys,v, Tsys,U) = 2Tsys,vTsys,U N (62) Cov( Tsys,h, Tsys,U) = 2Tsys,hTsys,U N . (63) D. Definition and Characterization of Tsys,I and Tsys,Q It is more convenient to work with the sum and difference of Tsys,v and Tsys,h than with these quantities themselves. Therefore, we define Tsys,I Tsys,v + Tsys,h, and Tsys,Q Tsys,v Tsys,h. Using the formulas given previously, it is straightforward to show that Tsys,I =TI + TRX,I = Tsys,I (64) Tsys,Q =TQ cos 2 + TU sin 2 + TRX,Q = Tsys,Q (65) and that the variances and covariances of Tsys,I , Tsys,Q, and Tsys,U can be summarized with the symmetric covariance matrix given in (66), shown at the top of the next page. E. Forward Model of Rotated and Calibrated Brightness Temperatures As discussed at the beginning of Appendix I-C, the measured temperatures are normal random variables with the means and variances that were just found. It is convenient to break them up into the sum of their means and zero-mean normal random variables Tsys,v Tsys,h Tsys,U Tsys,v + Tsys,v Tsys,h + Tsys,h Tsys,U + Tsys,U (67) Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. 3222 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 10, OCTOBER 2007 Var( Tsys,I) Cov( Tsys,I , Tsys,Q) Cov( Tsys,I , Tsys,U ) Var( Tsys,Q) Cov( Tsys,Q, Tsys,U ) Var( Tsys,U ) = 1 N T2 sys,I + T2 sys,Q + T2 sys,U 2Tsys,ITsys,Q 2Tsys,ITsys,U T2 sys,I + T2 sys,Q T2 sys,U 2Tsys,QTsys,U T2 sys,I T2 sys,Q + T2 sys,U (66) and similarly for the quantities defined for convenience Tsys,I Tsys,I + Tsys,I (68) Tsys,Q Tsys,Q + Tsys,Q (69) where Tsys,I Tsys,v + Tsys,h and Tsys,Q Tsys,v Tsys,h. Expanding these out in terms of the original parameters, we have Tsys,v =Tv TQ sin2 + TU 2 sin 2 + TRX,v + Tsys,v Tsys,h =Th + TQ sin2 TU 2 sin 2 + TRX,h + Tsys,h Tsys,U = TQ sin 2 + TU cos 2 + Tsys,U (70) Tsys,I =TI + TRX,I + Tsys,I (71) Tsys,Q =TQ cos 2 + TU sin 2 + TRX,Q + Tsys,Q. (72) Now, note that TRX,v and TRX,h (and, hence, also their sum and difference TRX,I and TRX,Q) are operationally estimated and subtracted off as part of the radiometer data calibration. Imperfection in this correction leaves residuals which we call TRX,v and TRX,h. It is convenient to also define TRX,I TRX,v + TRX,h and TRX,Q TRX,v TRX,h. With TRX,v, TRX,h, TRX,I and TRX,Q subtracted off and leaving only these residuals, we finally have a forward model for the outputs of the rotation, measurement, and calibration processes, which become the inputs to the PRC process of [1]. Using a notation similar to [1] for these inputs, where the subscript a can be interpreted as referring to temperatures after rotation, measurement, and calibration, Tva =Tv TQ sin2 + TU 2 sin 2 + TRX,v + Tsys,v (73) Tha =Th + TQ sin2 TU 2 sin 2 + TRX,h + Tsys,h (74) T Ua = TQ sin 2 + TU cos 2 + Tsys,U . (75) As explained in Section II, the measurement and calibration process also add a residual bias TRX,U to this last equation, as included in (3). Equations (73) and (74) are the generalizations of [1, eqs. (12) and (13)]. For convenience, we use the sum and difference of (73) and (74), as given in (1) and (2), respectively. REFERENCES [1] S. H. Yueh, Estimates of Faraday rotation with passive microwave polarimetry for microwave remote sensing of Earth surfaces, IEEE Trans. Geosci. Remote Sens., vol. 38, no. 5, pp. 2434 2438, Sep. 2000. [2] D. M. Le Vine and S. Abraham, The effect of the ionosphere on remote sensing of sea surface salinity from space: Absorption and emission at L-band, IEEE Trans. Geosci. Remote Sens., vol. 40, no. 4, pp. 771 782, Apr. 2002. [3] T. Meissner and F. J. Wentz, Polarization rotation and the third Stokes parameter: The effects of spacecraft attitude and Faraday rotation, IEEE Trans. Geosci. Remote Sens., vol. 44, no. 3, pp. 506 515, Mar. 2006. [4] A. J. Gasiewski and D. B. Kunkee, Calibration and applications of polarization-correlating radiometers, IEEE Trans. Microw. Theory Tech., vol. 41, no. 5, pp. 767 773, May 1993. [5] I. Corbella, F. Torres, A. Camps, A. 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New York: Wiley, 1964. [16] S. O. Rice, Mathematical analysis of random noise Conclusion, Bell Syst. Tech. J., vol. 24, no. 1, pp. 46 156, 1945. [17] S. H. Yueh, W. J. Wilson, S. J. Dinardo, and S. V. Hsiao, Polarimetric microwave wind radiometer model function and retrieval testing for WindSat, IEEE Trans. Geosci. Remote Sens., vol. 44, no. 3, pp. 584 596, Mar. 2006. [18] S. S. Sobjaerg, J. Rotboll, and N. Skou, Measurement of wind signatures on the sea surface using an L-band polarimetric radiometer, in Proc. IEEE Int. Geosci. and Remote Sens. Symp., 2002, vol. 3, pp. 1364 1366. Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply. HUDSON et al.: POLARIZATION ROTATION CORRECTION IN RADIOMETRY: AN ERROR ANALYSIS 3223 [19] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive, vol. 2. Norwood, MA: Artech House, 1986, ch. 7. [20] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed. New York: Academic, 1965. Sec. 3.462. Derek Hudson received the B.S. and M.S. degrees in electrical engineering from Brigham Young University (BYU), Provo, UT, in 2003. He is currently working toward the Ph.D. degree in electrical engineering at BYU. He has been with the Microwave Earth Remote Sensing research group at BYU since 1999. His research experience is in the application of inverse problem theory and in radar hardware design. Jeffrey R. Piepmeier (S 89 M 99) received the B.S. degree in engineering from LeTourneau University, Longview, TX, in 1993, the M.S. and Ph.D. degrees in electrical engineering from the Georgia Institute of Technology (Georgia Tech.), Atlanta, in 1994 and 1999, respectively. From 1993 to 1994, he was a Schakleford Fellow with the Georgia Tech Research Institute. Since 1999, he has been with the Microwave Instrument Technology Branch, NASA Goddard Space Flight Center, Greenbelt, MD. Currently, he is the Prelaunch Calibration Lead Engineer for the Aquarius/SAC-D mission. His technical interests include passive remote sensing and technology development for microwave remote sensing. Dr. Piepmeier was the third place winner in the 1998 IGARSS student Prize Paper competition. He was the recipient of an Excellence in Federal Career Gold Award (Rookie-of-theYear) in 2000 and was a 2002 NASA Earth Science New Investigator. In 2000, he served as the Conference Chair of the First Microwave Radiometer Calibration Workshop (MicroCal2000). He was the Cochairman/Chairman of the Instrumentation and Future Technologies Technical Committee of the IEEE Geoscience and Remote Sensing Society from 2000 to 2006. He is currently the Cochairman of the NAS Committee on Radio Frequencies. He is a member of Union Radio-Scientifique Internationale (Commission F) and the American Geophysical Union. David G. Long (S 80 SM 98) received the Ph.D. degree in electrical engineering from University of Southern California, Los Angeles, CA, in 1989. From 1983 to 1990, he worked for NASA s Jet Propulsion Laboratory (JPL), Pasadena, CA, where he developed the advanced radar remote sensing systems. While at JPL, he was the Project Engineer on the NASA Scatterometer (NSCAT) project which flew from 1996 to 1997. He also managed the SCANSCAT project, which is the precursor to SeaWinds which was launched in 1999 and 2002. He is currently a Professor in the Department of Electrical and Computer Engineering, Brigham Young University (BYU), Provo, UT, where he teaches upper division and graduate courses in communications, microwave remote sensing, radar, and signal processing and is the Director of the BYU Center for Remote Sensing. He is the Principal Investigator on several NASAsponsored research projects in remote sensing. He has numerous publications in signal processing and radar scatterometry. His research interests include microwave remote sensing, radar theory, space-based sensing, estimation theory, signal processing, and mesoscale atmospheric dynamics. He has over 275 publications. Dr. Long has received the NASA Certificate of Recognition several times. He is an Associate Editor for IEEE GEOSCIENCE AND REMOTE SENSING LETTERS. Authorized licensed use limited to: Brigham Young University. Downloaded on February 3, 2009 at 15:30 from IEEE Xplore. Restrictions apply.