A Statistically Optimal Analysis of Systematic Differences between Aeolus

34 The European Space Agency Aeolus mission launched the first of its kind spaceborne Doppler 35 wind lidar in August 2018. To optimize assimilation of the Aeolus Level-2B (L2B) Horizontal 36 Line-of-Sight (HLOS) winds, systematic differences (referred as biases hereafter) between the 37 observations and numerical weather prediction (NWP) background winds should be removed. 38 Total least squares (TLS) regression is used to estimate speed-dependent biases between Aeolus 39 HLOS winds (L2B10) and the National Oceanic and Atmospheric Administration (NOAA) 40 Finite-Volume Cubed-Sphere Global Forecast System (FV3GFS) 6-h forecast winds. Unlike 41 ordinary least squares regression, TLS regression optimally accounts for random errors in both 42 predictors and predictands. Large well-defined, speed-dependent biases are found particularly in 43 the lower stratosphere and troposphere of the tropics and Southern Hemisphere. These large 44 biases should be corrected to increase the forecast impact of Aeolus data assimilated into global 45 NWP systems. 46


Key Points
23  There are speed-dependent systematic differences in the Aeolus M1-bias corrected 24 Level-2B HLOS winds compared to short-term (6-h) FV3GFS forecasts. 25  The total least squares (TLS) regression provides a statistically optimal analysis of the 26 differences. 27  A bias correction based on the TLS bias analysis proposed here is tested in a 28 companion paper to optimize Aeolus wind assimilation and thus the impact of Aeolus 29 winds on global NWP forecasts.

TLS Linear Regression 153
In this section, we review the TLS regression method [Ripley and Thompson, 1987]  Here we assume that and are independent and that the random error variance ratio 166 known. Also, we assume the true relationship between the 167 Aeolus and FV3GFS winds is described by a linear function: 168 where is an offset or constant bias and is a speed-dependent bias coefficient. 170 The TLS regression finds an optimal estimate of the , and by minimizing the 171 cost function 172 To determine the , set the derivative of J with respect to to zero, to obtain 175 Eq.
(3) thereby reduces the problem to a minimization in terms of and . A similar equation 177 holds even if the error variances vary with i, but then there is no closed form solution for and 178 , as there is in the current case, which is known as the Deming problem [Ripley and 179 Thompson, 1987]. When the coefficients and are obtained, the TLS estimate for the new or 180 within-sample observation is given by Eq. (3). Finally, the estimate of the bias for the kth 181 observation, either for a new or within-sample observation, is given by 182 We will refer to and ( 1 as the constant and speed-dependent bias coefficients, 184 respectively, hereafter. 185 Note that the error variance ratio is a crucial parameter in the TLS bias analysis. If 186 0 or 0, then the TLS solution is equivalent to the OLS regression of the O-B on the 187 Aeolus winds or on the FV3GFS winds, respectively. 188

Estimation of the random error variance ratio 189
The random error variance ratio ( ⁄ ) 2 used in the TLS bias analysis is estimated 190 from the O-B innovations from the BASE experiment using the Hollingsworth-Lonnberg (HL) 191 method [Hollingsworth and Lonnberg, 1986]. It is assumed that there is no correlation between 192 the random errors in Aeolus and FV3GFS winds and no horizontal correlation in the random 193 errors in Aeolus winds at 90 km distance and beyond. For more details, see Hollingsworth and 194 Lonnberg [1986] and Garrett et al. [2021]. 195 The random error variance ratio δ is estimated at the middle height of each vertical range 196 bin using the Aeolus samples for 1-7 September 2019, separately for Mie and Rayleigh winds.

Variation of Biases with Height 210
The variation of the TLS solution with height and orbital phase is described here. The 211 TLS samples are over all latitudes. The vertical distribution of the TLS constant and speed-212 dependent bias analysis coefficients for the innovation in terms of the background in Eq. (4) is 213 shown in Figure 7. The speed-dependent bias coefficient ( 1 varies substantially with height 214 and orbital phase. For Mie winds, the coefficient is quite large at most heights, ranging from 3 to ranges from 1 to 3% in ascending orbits and 1 to 5% in descending orbits, with maxima around 217 the 3.5 and 16 km.

Variation of Biases with Latitude 231
The variation of the TLS solution with latitude and orbital phase is described here. The 232 TLS samples are over all heights for 10-degree latitude bands. In general, the coefficients 233 obtained are large and vary considerably with latitude and orbital phase, with maxima found in 234 the tropics (Figure 10). For example, the speed-dependent bias coefficient ( 1 for Mie 235 winds in the tropics can be quite large, ranging from 0% to a maximum of 11%. The coefficient 236 ( 1 is smaller for Rayleigh winds, ranging from -1% to 5%, with maxima found in the 237 tropics and at northern high latitudes. The constant bias coefficient for Mie winds also varies 238 considerably with latitude and orbit, ranging from -1.0 m/s to +1.6 m/s. The coefficient is 239 smaller for Rayleigh winds.

Discussion 247
The results presented in this section indicate that the speed-dependent bias coefficient is 248 quite large, with ( 1 reaching up to ~10% and 5% for Mie and Rayleigh winds, 249 respectively, particularly in the lower stratosphere and lower troposphere of the tropics. This 250 suggests that there exist large speed-dependent biases in FV3GFS background winds and/or in 251 the Aeolus winds. Given that there exist large uncertainties in the FV3GFS (and ECMWF) 252 background winds in the tropics (see Figure 1), it is likely that the FV3GFS may be a significant 253 source of the large biases and this will require further investigation. In any case, these large 254 speed-dependent biases should be corrected to optimize Aeolus wind assimilation and the impact 255 The vertical distributions of the average biases as a function of Aeolus winds are shown 271 in Figure 12 for the descending orbits for three methods: The top panels are for OLS regression 272 using FV3GFS winds as a predictor, the middle panels, which repeat the bottom two panels of 273 The bias estimates of OLS regression using Aeolus winds only as a predictor (not shown) are 282 even larger (than the bottom panel). 283

284
In this section, a TLS bias correction for O-B is proposed to optimize Aeolus wind data 285 assimilation. For each assimilation cycle, the bias coefficients are computed by TLS regression 286 for the O-B in the week before the cycle (i.e., for the previous 28 cycles). One week provides a 287 large enough sample for the regression. As shown by Ripley and Thompson [1987], the TLS 288 solution only involves solving a quadratic equation with coefficients given by sample sums. 289 Therefore, an efficient approach is to calculate and save these sums for every cycle and 290 accumulate them over the 28 cycles. Because the findings in this study show substantial variation 291 of the bias coefficients with latitude, vertical layer, and orbital phase, the bias coefficients are 292 calculated from the winds in 19 discrete bins of latitude (centered every 10º between 90° S to 90° 293 N) for each vertical range/layer and for ascending and descending orbits separately.         Figure 4 but after the TLS bias correction is applied. Note that the remaining 488 bias in several bins are due to small sample size, and the TLS bias correction is not applied in 489 these bins in Aeolus wind assimilation.