GW Variance Analysis

To analyze the AMSU-A radiances, we first remove the systematic scan-angle dependence by fitting the radiances T_b within the same scan to a polynomial of scan angle q, given by

                                                          (1)

where a_i are the fitting coefficients and  are the fitted radiances. Because AMSU-A radiances have shown an unexpected cross-track asymmetry about the nadir [Goldberg et al., 2001; Mo, 1999], the fitting in (1) is applied only to one half (15 FOVs) of the scan measurements, which are located on the same side of the track. The radiance residual  is defined as the difference between the measured and fitted radiances, namely, . As shown in Figure 2 for a model simulation, the residuals for an ideal atmosphere are very small (< ~0.01 K) if there are no air temperature fluctuations in the region. In other words, the polynomial (1) can adequately remove the scan-angle dependence in the cross-track radiances as well as atmospheric variations with horizontal scales > ~200 km (see below).

However, the averaged radiance residuals from fitting the real data with (1) are much larger (0.05-0.2 K) than expected and nearly symmetric about nadir. An example for channel 13 is shown in Figure 3, where these systematic FOV biases are comparable to the specified instrument accuracy and cannot be reduced statistically by averaging all the collocated measurements. These biases vary somewhat with frequency and satellite but little with latitude and time. Between equatorial (small atmospheric variability) and polar (large atmospheric variability) regions, the deduced biases may differ by < 0.02 K.

Figure 2. Modeled AMSU-A radiances (symbol) under a homogeneous atmosphere and the radiance residuals after the fitting with (1), both as a function of viewing angle from nadir. The line through the symbols depicts the fitted curve using the function (1). The radiance residuals from the fit are typically less than 0.01 K.

For GW variances, these radiance biases are significant and must be removed in the calculation. One way to determine the bias is to apply the fitting (1) to the measurements in the equatorial region (where large-scale atmospheric variability is small) and use the averaged residuals as the radiance biases. Because short-scale atmospheric variability is unlikely to be coherent on a global scale, the averaged residuals can sufficiently reduce random fluctuations from the atmosphere and yield the biases due to systematic instrument errors. Such empirically-determined biases are then subtracted from the radiances in each scan to yield “unbiased” radiance residuals, i.e.,  where  is the derived bias as a function of scan angle and frequency channel.

In the next step, a linear fit is applied to the unbiased radiance residuals  to further reduce large-scale atmospheric contributions. This fitting is similar the procedure used for the MLS data [Wu and Waters, 1996] to remove any linear components from large-scale atmospheric waves. In the AMSU-A case, the radiance residuals from each scan are divided into six groups of five, and each group are fitted to a linear function, namely,

                                                       (2)

where b0 and b1 are the fitting coefficients. The residuals  after the linear fit are used to compute variance , which is defined by

(3)

where  and  are the normalization factors (reduction in degrees of freedom) associated with the fittings in (1) and (2), respectively. In this radiance variance, power from large-scale waves is strongly damped, and the cutoff horizontal wavelengths are ~250 km in near-nadir cases and ~360 km in near-limb cases. For MLS limb-tracking data, the cutoff wavelength is somewhat shorter (~100 km) [Wu and Waters, 1997].

The radiance variance  estimated from a single scan can be very statistically uncertain, and this uncertainty must be reduced to observe weak GW variances. This can be achieved by averaging independent variance measurements over the same region, defined within a longitude-latitude grid box, during some period of time.  As shown in Wu and Waters [1997], uncertainties in  is proportional approximately to the true variance , as given by

                                                 (4)

where M is the number of independent measurements for . In this study, data from N15, N16, and N17 satellites are all used to improve the statistics, and are sufficient to produce reliable GW variance maps on a 0.5º×0.5º latitude-longitude grid. With the three satellites, each grid box typically has 24 samples from a month worth of data in the equatorial region and 36 in the polar region, and the averaged variance can bring down the noise by factors of 3-4 to < 0.15 K² for channel 13.

As suggested by Wu and Waters [1996], the radiance variance is composed primarily by two components: atmospheric variance  and instrument variance, namely

                                                       (6)

where ε represents other measurement error: this extra component is normally very small compared to the first two and only occasionally becomes important such as calibration malfunctioning. The channel-dependent variance , which is the random component from instrument noise, is stable throughout the entire mission and relatively easy to evaluate. Although it was measured before launch, more accurate estimates can be obtained from the radiance data during atmospheric measurements. A method for noise estimation from flight data has been described for MLS by Wu and Waters [1996], based on the average of radiance variance minima in monthly zonal means. The minimum variance of a monthly mean usually occurs near the equatorial region for MLS and AMSU-A stratospheric channels where resolved wave activity is weak and negligible. Therefore, the deduced radiance variance in this region is very close to the instrument noise. Table 1 lists the AMSU-A noise estimated with this method for N15, N16, and N17 satellites. The estimated values for AMSU-A noise/precision show little month-to-month variations and appear slightly smaller than those previously measured.

Table 1.  AMSU-A instrument noise for channels 9-14.

Channel number	Pressure of weighting function peak^a  (hPa)	Measured noise^b (K)	Precision estimated in this work (K)
			N15	N16	N17
9	~80	0.24	0.16	0.15	0.15
10	~50	0.25	0.20	0.20	0.20
11	~25	0.28	0.23^c	0.23	0.23
12	~10	0.40	0.33	0.35	0.35
13	~5	0.54	0.47	0.49	0.50
14	~2.5	0.91	0.79^c	0.81	0.82

Note:

a) Corresponding to the weighting function of the outermost viewing angles;

b) From Goldberg et al. [2001];

c) From the early period of N15 operation.

The atmospheric component , hereafter referred to as GW variance, is induced by wave-related temperature fluctuations. Because the microwave radiances are a direct measure of air temperature from the layer defined by the weighting function, the atmospheric component  reflects fluctuations of average air temperature in that altitude layer. Roughly speaking, the AMSU-A variances are contributed mostly from waves of horizontal wavelengths between 50-250 km for near-nadir cases (100-400 km for near-limb cases) and vertical wavelengths >10 km. The horizontal wavelengths are determined by the FOV size (at short scales) and the truncation length used in fitting (3) (at large scales). The sensitivity reduces sharply for waves with vertical wavelengths < ~10 km, as a result of vertical smearing by the temperature weighting functions.

Figure 3. Systematic radiance biases for AMSU-A channel 13 after the scan-angle dependence is removed. The results are averaged for the data in the tropical region (30ºS-30ºN) during January 2003. The AMSU-A data from N15, N16, and N17 satellites show only slightly different FOV-to-FOV variations. The cause(s) of these residuals is unclear but spillovers from the antenna sidelobes are capable of producing systematic biases with this magnitude [Mo, 1999].