Correcting diurnal cycle aliasing in satellite microwave humidity sounder measurements
Division of Space Technology, Department of Computer Science, Electrical and Space Engineering, Luleå University of Technology, Kiruna, Sweden
Corresponding author: A. Kottayil, Division of Space Technology, Department of Computer Science, Electrical and Space Engineering, Luleå University of Technology, Kiruna, Sweden. (email@example.com)
 Microwave humidity measurements from polar orbiting satellites are affected by diurnal sampling biases which are caused by changes in the local observation time of the satellites. The long-term data records available from these satellites thus have spurious trends, which must be corrected. Diurnal cycles of the microwave measurements have been constructed by combining data over the period 2001–2010 from five different satellite platforms (NOAA-15, -16, -17, -18, and MetOpA). This climatological diurnal cycle has been used to deduce and correct the diurnal sampling bias in Advanced Microwave Sounding Unit-B and microwave humidity sounder measurements. Diurnal amplitudes for channels which are sensitive to surface temperature variations show a sharp land-sea contrast with the amplitudes exceeding 10 K for land regions but less than 1 K for oceanic regions. The humidity channels sensitive to the upper and middle troposphere exhibit a seasonal variation with large diurnal amplitudes over convective land regions (often above 3 K) in comparison to oceanic regions. The diurnal peak times of these channels over land occur in the early mornings. The diurnal sampling bias correction has a greater impact over land regions when compared to oceanic regions due to the large diurnal amplitudes over land. The diurnal cycle of humidity generated as a part of this study could be used to evaluate diurnal cycles in climate models.
 Orbital drift of sun-synchronous satellites can introduce artificial trends in climate data records derived from their measurements. For example, Wentz and Schabel  have discussed how the orbital drift of National Oceanic and Atmospheric Administration (NOAA) satellites introduces an artificial cooling trend in the lower tropospheric temperature record derived from Microwave Sounding Unit (MSU) measurements. Orbital parameters of polar orbiting, sun-synchronous satellites are designed to measure the radiation emitted from any point on the Earth and its atmosphere at similar local times, thus sampling the same part of the diurnal cycle of the geophysical parameters measured. However, in reality, factors like atmospheric drag, the deviations from the Earth's spherical shape, the gravitational pull from celestial bodies, and solar activity causes the orbit height to decrease [e.g., Ignatov et al., 2004], unless it is manually maintained in a stable orbit (e.g., MetOpA). Such a drift in orbital height leads to changes in the orbiting period and hence a change in the local sampling time of the satellites leading to undesirable aliasing of the diurnal cycle into the time series of data from the polar orbiters.
 Different methods have been used to detect and correct for the orbital drift errors in satellite measurements and their derived products. For example, orbit drift corrections for the MSU temperature observations have been achieved using the diurnal cycle of simulated observations constructed from hourly outputs of a general circulation model [Mears et al., 2003]. Jackson and Soden  also corrected for the orbital drift error in High-Resolution Infrared Sounder (HIRS) observations using brightness temperatures simulated from the output of a climate model. They derived the diurnal sampling bias from the simulated brightness temperature time series and then removed these biases from the time series of observations. Waliser and Zhou  implemented a rotated empirical orthogonal function analysis (REOF) to detect the diurnal sampling bias in outgoing long wave radiation (OLR) and highly reflective cloud (HRC) datasets. They applied REOF to OLR and HRC anomalies and identified those eigenmodes which represent the equatorial crossing time changes. These eigenmodes were then used to remove the diurnal sampling bias in the OLR and HRC anomaly datasets. Lee et al.  corrected for the diurnal sampling bias in the HIRS OLR data product using the diurnal cycle of OLR which was created by aggregating inter-satellite bias corrected monthly mean OLR values from different NOAA satellites over a period of 25 years. Recently, Lindfors et al.  used monthly climatology of diurnal cycle of HIRS brightness temperatures generated by combining data for the period 2002–2007 from different NOAA satellites to derive the diurnal sampling bias in HIRS measurements.
 In this study, we focus on microwave humidity sounder measurements from NOAA and MetOpA satellites. Microwave data have the advantage of having an almost all-sky sampling as compared to only clear-sky sampling for infrared measurements [John et al., 2011]. Microwave observations of humidity from these satellites are available since 1998, and the data have been proven to have a significant impact for numerical weather prediction [e.g., Andersson et al., 2007]. Indeed, the microwave measurements of atmospheric humidity date back to 1993 with the launch of Special Sensor Microwave/Temperature and Humidity Profile (SSM/T2) as a part of the Defense Meteorological Satellite Program (DMSP). The microwave humidity data measurements obtained from the NOAA satellites can be composited for some climate applications [e.g., Xavier et al., 2010], but long-term homogeneous data are essential for monitoring the changes in tropospheric water vapor in a warming climate. However, data from the microwave humidity sounders are not homogeneous owing to factors like changes in spectral characteristics, sensor degradation, and drift in the satellite orbit. In this study, we focus on correcting the orbital drift error in satellite microwave humidity measurements.
 This study applies a methodology for the correction of diurnal sampling biases in microwave humidity measurements by first constructing climatological diurnal cycles of these measurements by combining data available from five different platforms for a period of over 10 years. The methodology used here is similar to those used in Lee et al.  and Lindfors et al. . This information is subsequently used for correcting the orbital drift errors in the humidity measurements.
 The data used in this study are from Advanced Microwave Sounding Unit-B (AMSU-B) and microwave humidity sounder (MHS) sensors onboard the NOAA and MetOpA satellites. AMSU-B is flown on NOAA satellite-15, 16, and 17, and MHS is on NOAA-18 and MetOpA satellites. They are both five-channel microwave radiometers dedicated to measuring atmospheric humidity, an important climate variable [e.g., Held and Soden, 2000], but not very well simulated by current climate models [e.g., John and Soden, 2007]. Out of the five channels, three (3, 4, and 5) are placed near a strong water absorption line at 183 GHz. The passbands of these channels are 183.31 ± 1.00, 183.31 ± 3.00, and 183.31 ± 7.00 GHz (183.31 + 7.00 GHz for MHS), respectively. Channels 1 and 2, at 89 and 150 GHz (157 GHz for MHS), respectively, look deeper through the atmosphere on to the Earth's surface. These channels are used for the retrieval of ice water path and rain rate [Weng et al., 2003]. Both AMSU-B and MHS sensors are cross-track scanning instruments with 90 Earth fields-of-view per scan line. Two of the five channels differ between MHS and AMSU-B; the differences are in channels 2 and 5, and the characteristics of these biases are described in [John et al., 2012]. These differences are taken into account during the inter-satellite bias correction procedure.
 We have used the data spanning the years 2001–2010. Only channels 1 and 2 of NOAA-15 and NOAA-16 have the full temporal coverage from 2001 to 2010 as shown in Table 1. This is due to the high noise in the measurements of some of the channels in these satellites after the selected time period. The near-nadir measurements from each satellite for each day, separately for ascending and descending orbits, were averaged onto 2.5° × 2.5° latitude-longitude grids. Only near-nadir measurements (three measurements on either side of nadir) are used in this study in order to avoid errors due to limb effect and scan asymmetry [Buehler et al., 2005; John et al., Monitoring scan asymmetry of microwave humidity sounding channels using simultaneous all angle collocations (SAACs), submitted to Journal of Geophysical Research, 2012a]. The gridded data contain information on the number of measurements (N), time of observation (t), and mean and standard deviation (σ) of brightness temperatures. The data from different satellites gridded in this manner was grouped on a monthly basis for 2001–2010. The grouping was done in such a way that a single grid point can have multiple data from different satellites for different local times (see Figure 1).
Table 1. The Time Period of the Data Taken from Different NOAA and MetOpA Satellites for Constructing the Diurnal Cycle of Different Channels
 A first-order correction of inter-satellite biases are done for these measurements using data taken during simultaneous nadir overpasses (SNOs; [Cao et al., 2004]) of these satellites. More details on application of SNOs to microwave humidity measurements are discussed in (John et al., Assessment of inter-calibration methods for satellite microwave humidity sounders, submitted to Journal of Geophysical Research, 2012b). All measurements were bias corrected to NOAA-18 measurements, which were taken as reference for all channels. MetOpA channels have very little bias, within ±0.2 K. Channel 1 has the least biases, always within ±0.5 K. For NOAA-15, -16, and -17, channel 2 biases are about 1.5 K due to the shift in frequency from 150 GHz for AMSU-B to 157 GHz for MHS. Note that these biases are state dependent and could be up to 3 K in some cases [John et al., 2012]. Channel 3 biases are within ±0.5 K. Channel 4 measurements from these satellites have significant time-varying biases ranging from −1.5 to 2 K. Channel 5 also has large biases ranging from −1.5 to 1 K. Note that for NOAA-15 and NOAA-16, channels 3–5 have larger errors than described here after 2006 [John et al., 2012]. Thus, we have not used data of those channels after 2006 in this study. An error analysis is done to account for the uncertainties in the estimated inter-satellite biases which are discussed in section 'Construction of Diurnal Cycle'.
3 Construction of Diurnal Cycle
 As can be seen from Figure 1, NOAA 16 has drifted by approximately 5 h over a period of 10 years. This drift can cause false trends in the time series [Jacobowitz et al., 2003]. On the other hand, this drift has the advantage of facilitating a better sampling of the diurnal behavior of the measurements as shown in Figure 2 and also can lead to short periods of global collocations [Holl et al., 2010] which are utilized to understand the radiance dependence of inter-satellite biases [John et al., 2012]. The figure shows measurements from all satellites (colored circles) for a grid box over the Sahara desert for all the Januaries in the study period. Each colored circle in the figure represents daily measurement average onto a 2.5° latitude-longitude grid point. Using these measurements aggregated for every month over several years, a diurnal cycle has been fitted (black curve) for each grid point using a second order Fourier series. The Fourier representation is written as follows [Lee et al., 2007; Lindfors et al., 2011]:
where Tb(t) is the brightness temperature, t is the observational local time in hours, a0 is the mean value of Tb, a1 is the amplitude of the 24 h oscillation, a2 is the amplitude of the 12 h oscillation, and t1 and t2 are the phases of the 24 and 12 h oscillations, respectively. Expanding equation ((1)) using simple trignometric identities yields
where b0 = a0, , , , and . Therefore, the coefficients a1, a2, t1, and t2 can be determined as
 The coefficients b0, b1, b2, b3, and b4 are determined using weighted linear least squares regression. The advantage of doing a regression for bi, instead of directly for ai and ti, is that the regression for bi is linear, whereas for ti it is nonlinear. The weights for regression are defined as follows. As mentioned earlier in section 'Satellite Data', at each grid point we have measurements from different satellites, and associated to each measurement, we have the information of its σ and N. Therefore, the estimated error in the measurement is . The least-square weighting for that measurement is given as the inverse of the square of estimated error in the measurement.
 While performing the weighted least squares regression, we avoid those measurements where the number (N) used for gridding the measurements is less than 10. Moreover, the regression is performed only for those grid points where the number of data points in each quarter of the 24 h local time period is greater than 10. This criterion restricts our analysis to the latitude belt within 70°S–70°N due to averaging daily data onto a 2.5° × 2.5° latitude-longitude grid. Unlike lower latitudes, at higher latitudes, a satellite can have more than one overpass separated few hours apart during both its ascending and descending orbits. Thus, taking the temporal average over a 2.5° grid will hinder the criteria of getting at least 10 data points within each quarter of the 24 h local time period. In addition, diurnal cycles are ill defined at very high latitudes because summer and winter have continuous daylight and darkness, respectively. Note that this is a limitation of our chosen data processing scheme, not a fundamental limitation of the method itself.
 A Monte Carlo error analysis was performed to determine the uncertainty of the derived fit parameters (amplitudes a1 and a2) from the Fourier series. Measurements in each grid point consist of data from the different satellites with each satellite having a different number of observations for their ascending and descending passes. Within a grid point, the data from each satellite for a particular pass containing “M” observations would then be replaced by a normally distributed random numbers having the same mean and standard deviation as that of the “M” observations. This process is repeated for all passes of all satellites. The fit parameters a1 and a2 were determined using equation ((1)). The process of randomizing the grid point data is repeated 300 times, thus generating 300 fit parameters (a1 and a2). The standard deviation of a1 and a2 thus generated is taken as the uncertainty of the amplitudes. If this uncertainty exceeds the original amplitude, then the fit cannot be accepted as real. Diurnal amplitudes of channel 1 and their associated signal-to-noise ratio (SNR; ratio of amplitude to standard deviation) are shown in Figure 3. Hereafter, we consider only those diurnal cycle fits with signal-to-noise ratio of both amplitudes greater than 1. The assumption of a Gaussian distribution for brightness temperatures for humidity sounding channels may not be a completely true representation [John et al., 2006]. Second-order Fourier series is adopted for diurnal cycle construction in order to have a better fit for the data under consideration. The actual diurnal cycle need not be sinusoidal in nature due to the real atmospheric variability present in the measurements. Therefore, addition of 12 h oscillations will aid in better representing the diurnal variation in the data rather than just using the 24 h oscillations. Our criterion demands that both amplitudes a1 and a2 should be significant, which implies that as long as a1 is significant there is a diurnal cycle due to solar insolation, and a2 could be greater or less than a1 depending on the real atmospheric variability at a particular grid point.
 Coastlines are excluded from our analysis, following the method adopted by Lindfors et al. . The land fraction within a 2.5° × 2.5° grid was calculated using a 1 km land-sea mask where the land fraction 0 is assigned for ocean, and values greater than 0.75 were assigned for land. The reason for avoiding coastlines is due to the differences in the surface emissivity between land and oceanic regions. For surface channels, land has comparatively larger surface emissivity (~0.95) in relation to ocean (~ 0.7), and therefore land appears radiometrically brighter than ocean. So combining data over land and ocean makes diurnal cycle look completely different from that generated using either of land or oceanic region alone. This affects surface channels 1 and 2 more than the humidity channels over the tropics.
 There are some differences in our approach of constructing a diurnal cycle compared to that of Lindfors et al. . Lindfors et al.  used individual measurements in a grid box from different satellites for the diurnal cycle, and therefore there is little scope for taking into account the outliers in the measurements. In our approach, since we are gridding the measurements over a 2.5° grid for each pass of the satellite for each day and doing a weighted least squares regression for determining diurnal cycle amplitudes, two things can be achieved. Taking a daily average will suppress most of the random variability in the measurements, and secondly, weighted least squares regression will take into account the outliers which mostly occur due to high scene inhomogeneity. If the scene is inhomogeneous, less weight is given for that data point during regression. The diurnal cycle amplitudes and phases determined using simple linear regression are not different from that determined by the approach followed by Lindfors et al. . But introduction of a weighted least squares regression changes the diurnal cycle amplitudes as compared to the simple linear regression approach. For example, for the surface channel 1, the differences in amplitude a1 between weighted and simple linear regression over tropical land is 4.7744 ± 2.8927 K, while for amplitude a2, this difference is found to be 1.3461 ± 1.1409 K. The amplitude differences in a1 and a2 in channel 1 for tropical ocean are found to be 0.8178 ±1.0720 and 0.2783 ± 0.8766 K, respectively. For channel 3, the amplitude differences in a1 and a2 for land is 0.3904 ± 0.6686 and 0.3698 ± 0.6510 K, respectively, and for ocean the respective differences in a1 and a2 are 0.3499 ± 0.5543 and 0.3392 ± 0.6239 K. A part of these differences in amplitudes arise due to the inclusion of all-sky data in the construction of diurnal cycle.
 The Monte Carlo error estimates for the amplitudes are also different from that of Lindfors et al. . Lindfors et al.  use the mean and pooled standard deviation of the measurements for generating random measurements at each grid point. They do not take into account the individual mean and standard deviation of the measurements which vary and are completely different from one another for different satellites and for their ascending and descending overpasses. However, we take into account these differences while generating the random measurements. The effect of choosing a slightly different approach for random measurements than that of Lindfors et al.  is reflected in the amplitude significance criterion. For example, in channel 3, the number of data points with amplitude a1 and a2 in the range of 0–2 K over the tropics and which satisfies an SNR greater than 1 with our approach is 12% and 20% less than that of Lindfors et al.  for a1 and a2, respectively.
 There are inter-annual variations in the diurnal cycle amplitudes for different microwave measurements. To test whether there are any significant differences in the diurnal cycle amplitudes due to inter-annual variations, we have compared the amplitudes generated from two consecutive years (2003 and 2004) of 6.7 µm channel brightness temperature data available from METEOSAT-7 for the month of July. The 6.7 µm channel is sensitive to upper tropospheric humidity variations. The average differences in the diurnal cycle amplitude a1 between two consecutive July months computed within 30°S–30°N latitudes separately for land and oceanic region is found to be 0.26 ± 0.24 and 0.21 ± 0.19 K, respectively. For amplitude a2, these differences are 0.12 ± 0.113 and 0.0971 ± 0.0967 K, respectively. These variations can be even larger between a normal year and an El Niño–Southern Oscillation (ENSO) year over the tropical Pacific Ocean [Chandra et al., 1998]. It is difficult to quantify these differences with our present approach since we rely on combining measurements over several years to get a good temporal resolution so as to construct the diurnal cycle. Moreover, combining data over a 10 year period to generate the climatology of diurnal cycle will provide an average behavior of the diurnal cycle amplitude taking into account all the natural variability occurring during those periods.
 We have also studied for any effect of inter-satellite bias on the diurnal cycle amplitudes. This was done by introducing a 1 K inter-satellite bias in each of the satellites. The bias was introduced to one satellite at a time, and the resulting diurnal amplitudes were compared against the original amplitudes. Such a comparison would show whether an inter-satellite bias in a particular satellite with respect to other satellites would significantly affect the diurnal cycle. Also it would highlight if any particular satellite has a greater bearing on the diurnal cycle construction. The sensitivity study was made on channel 3 of the different NOAA satellites, viz., NOAA-15, NOAA-16, NOAA-17, NOAA-18, and MetOpA. The results reveal that there is no significant difference in amplitude a1 with the introduction of a bias in each of the satellites. However, a2 seems to vary considerably with a maximum variation of up to 0.5 K, and this is true for all the satellites of interest. Inter-satellite biases therefore seem to have a greater impact on a2 but is insignificant for a1.
4 Results and Discussion
 In this section, we discuss our analysis of diurnal amplitudes of channels 1 to 5. The discussions begin with an overview of differences in the diurnal amplitudes and their peak times as observed in channels 1 to 5. This is followed by a presentation of the annual cycle modulation of the amplitudes. Finally, a depiction of the impact of diurnal correction is given by analyzing the time series of channel 1 brightness temperature on NOAA-16 and channel 3 brightness temperature time series on NOAA-17 satellites.
4.1 Diurnal Amplitudes
 Diurnal amplitudes a1 and a2 for the season December, January, February (DJF) for channels 1 to 5 are shown in Figure 4. The season DJF is selected here because the amplitude pattern is much more coherent than that obtained using any of the other months. In the figure, amplitudes are shown for those regions where the amplitudes a1 and a2 have been found to be significant from the Monte Carlo error analysis (signal-to-noise ratio >1). The first and second rows in the figure show the diurnal amplitudes a1 and a2 of channels 1 and 2, respectively, which are mostly sensitive to surface temperature variations. For these channels, the amplitude a1 is greater than a2 over land regions, whereas over ocean, these differences are not so distinct. The amplitude is larger over land than over ocean, which is consistent with those of Lindfors et al. . The dry land regions, as in the Australian desert and Sahara region, exhibit diurnal amplitude, which in a1 exceed 10 K and in a2 exceed 5 K. In addition, large amplitudes are also seen over convective regions such as the regions in South Africa and South America. The oceanic regions which lie in the descending branches of the Hadley cell circulation, such as Southeast Atlantic (20°W–20°E; 15°–30°S) and Southeast Pacific (120°W–70°W; 15°–30°S) show a1 and a2 to be over 2 K in channels 1 and 2. It is in these regions that the marine stratocumulus cloud is formed due to subsidence inversion [Wang et al., 2004]. Under the presence of such clouds with high liquid water content, the brightness temperatures of channels 1 and 2 tend to be warmer than under clear-sky conditions due to the low emissivity of oceanic surface [Sreerekha et al., 2008]. Therefore, the diurnal cycle seen over those regions is dominated by the diurnal cycle of low level clouds.
 The third, fourth, and fifth rows of Figure 4 show the diurnal amplitude of the humidity channels 3–5, which are sensitive to upper, middle, and lower tropospheric humidity, respectively. These channels are sensitive to relative humidity (RH) variations over broad pressure levels within different parts of the troposphere. For example, channel 3 is sensitive to RH changes within 200–500 hPa pressure level, whereas the sensitivity region for channel 4 lies within 300–800 hPa. Buehler and John  have shown that there exists a linear relationship between channel 3 brightness temperature and the channel's Jacobian weighted relative humidity which is roughly in the 200–500 hPa pressure level. A similar relationship exists between channels 4 and 5 brightness temperatures and relative humidity in the middle and lower troposphere, respectively.
 For channels 3 and 4, the diurnal amplitude is larger over land than over ocean. Higher values of diurnal amplitudes are seen over some of the regions over South America and South Africa, where convection is intense during the austral summer season (DJF). These observations are consistent with other studies which have shown the diurnal variations in the upper and middle tropospheric humidity using infrared measurements [Soden, 2000; Tian et al., 2004; Chung et al., 2007]. The diurnal amplitude a1 in channels 3 and 4 exceeds 3 K in convective regions over land. The oceanic regions show a diurnal amplitude of the order of 1 K in channels 3 and 4. The larger values of amplitude a1 in channel 4 as compared to channel 3 over the Himalayan region can be attributed to the differences in their sensitivity altitudes. Channel 4 has a peak sensitivity altitude matching the sensitivity altitude of the Himalayan summit, which is around 8 km, while channel 3 can partially detect the Himalayan surface with a peak sensitivity altitude at 250 hPa. The amplitude a1 gets larger whenever the sounding channels see the surface. This can be corroborated if one looks at the diurnal amplitude a1 of channel 5 (row 5, Figure 4), which detects the lower tropospheric humidity, and its diurnal amplitude is governed by changes in the surface temperature. In contrast to channel 1 diurnal amplitudes, the differences in amplitudes a1 and a2 for channels 3 and 4 are not so distinct over land regions free of convection. Over most of the oceanic regions, the diurnal amplitudes a2 of channels 3 and 4 are either comparable to or larger than that of a1. From Figure 4, it is evident that the diurnal amplitude variation of humidity in the upper and middle troposphere is comparable both in terms of magnitude and spatial features. The upper tropospheric humidity data available from microwave measurements thus must be corrected for diurnal sampling bias, and other issues like clear-sky bias and impact of ozone lines need also be taken into account for understanding their trend [Buehler et al., 2008; John and Buehler, 2004; Kottayil et al., 2012; John et al., 2011].
 The diurnal amplitude of channel 5, which is sensitive to lower tropospheric humidity variations, shows patterns slightly reminiscent of channels 1 and 2 but with different magnitudes. This is partly because in very dry regions (southwest of North America, Sahara, Gobi desert), channel 5 can detect the surface. In moister regions, it generally detects lower tropospheric humidity and therefore differs from surface channels.
4.2 Diurnal Peak Time
 Diurnal peak time determined for channels 1 to 5 for the season DJF is shown in Figure 5. Our definitions for the diurnal peak time are different for surface and humidity channels. This is because channels 1 and 2 are surface channels, and an increase/decrease in the surface temperature would have a similar effect on channel measurements under clear-sky conditions. However, for channels 3–5, the brightness temperature measurements are inversely related to relative humidity variations, i.e., higher humidity corresponds to a lower brightness temperature and vice versa. Therefore, for channels 1 and 2, diurnal peak time refers to the time corresponding to maximum brightness temperature inferred from diurnal fit. For channels 3–5, the peak time corresponds to the minimum brightness temperature inferred from diurnal fit or to the maximum humidity value.
 Diurnal peak times for channels 1 and 2 over land regions occur shortly after noon in local time (12.00 P.M.). This result is consistent considering that the maximum land temperature due to solar heating occurs in the early afternoon. The early morning peak time shown over most of the regions over Southeast Atlantic and Southeast Pacific coincides with the diurnal peak time of the marine stratocumulus clouds [Rozendaal et al., 1995; Wood et al., 2002]. Most of the land regions show a diurnal peak time in the early morning for channels 3 and 4, and this result is consistent with other studies which have shown an early morning peak time for upper and middle tropospheric humidity over land regions [Chung et al., 2007]. An early morning peak time for channel 3 is exhibited by 40% of oceanic regions within the tropics (30°N–30°S), while it is 20% for channel 4. Diurnal peak time over most of the regions for channel 5 occurs in the early morning. This is due to the surface RH values reaching their peak before sunrise and falling as the surface warms. Since this channel is sensitive to RH variations in the lower troposphere, the peak time will coincide with the peak time of surface RH value. In dry areas, the surface is coldest in the early morning, and this is well monitored by channel 5.
4.3 Seasonal Modulation of Diurnal Amplitudes
 Figure 6 shows the zonal average of the diurnal amplitudes a1 and a2 over the months for channels 1 to 5 over land. The first and second rows of the figure show diurnal amplitudes for channels 1 and 2, respectively. It is clear from the figure that for channels 1 and 2, amplitude a1 is much larger than a2. A seasonal shift in the diurnal amplitude a1 associated with the movement of Intertropical Convergence Zone (ITCZ) is seen. In the Southern Hemisphere, the diurnal amplitude a1 is maximum during austral summer season and reaches a minimum during austral winter. Similarly for the Northern Hemisphere, the amplitude a1 reaches its peak value during boreal summer and a minimum in boreal winter. This seasonal variation is not so apparent for diurnal amplitude a2. Diurnal amplitude variations with season for channels 3 and 4 are shown in the third and fourth rows of Figure 6. The spatial amplitude variations are almost similar for both channels 3 and 4. Here too, a seasonal shift in the amplitude is visible for a1. Higher a1 values for channels 3 and 4 are observed especially in convective regions lying close to the 30°S latitude band during the austral summer season. This holds also but to a lesser extent for the convective regions in northern equatorial latitudes during the boreal summer season. Such a seasonal variation is not very obvious in a2 for channels 3 and 4. The higher values of amplitude observed at higher latitudes cannot be considered as the real diurnal amplitudes of upper and lower tropospheric humidity due to dry atmospheric conditions prevailing in those regions. Dry atmospheres shift the sensitivity altitude of humidity channels to the surface. Unlike the observations from upper and middle tropospheric channels, no inference on seasonal variation can be made from channel 5 diurnal amplitudes.
 The zonal averages of diurnal amplitudes of oceanic regions are represented in Figure 7. The first and second rows of the figure show diurnal amplitudes of channels 1 and 2, respectively. Over the tropical oceans, the magnitudes of a1 and a2 are comparable for both the channels. The diurnal cycle amplitude of sea surface temperature is well below 1 K [Kennedy et al., 2007], and observations greater than 1 K shown over tropical oceanic regions are due to the presence of low-level clouds as explained earlier in section 'Diurnal Amplitudes'. For channels 1 and 2, a seasonal variation in the diurnal amplitudes is seen over the Southern Ocean and to a lesser extent over northern midlatitudes. The diurnal amplitudes greater than 1 K shown over the Southern Ocean cannot be entirely attributed to diurnal variations of sea surface temperature [Kennedy et al., 2007]; rather, changes in the sea surface emissivity could be a possible contributing factor. The third, fourth, and fifth rows in the figure show the diurnal amplitudes of channels 3, 4, and 5, respectively. Overall, for these humidity channels, there is little seasonal variation in the diurnal amplitude. Moreover, a2 is dominant over a1 for a major part of the oceanic region. The diurnal amplitudes of channels 3 and 4 often exceed 1 K over oceanic regions for all months.
4.4 Impact of Diurnal Correction
 In order to demonstrate the impact of diurnal correction, we have applied this correction to the channel 1 brightness temperature time series of NOAA-16 for its ascending and descending orbits separately for tropical (30°N–30°S) ocean and land regions. Brightness temperature time series for both ascending and descending orbits were corrected to 14.00 and 02.00 h in local times respectively, which were the equator crossing times when NOAA-16 was launched. The brightness temperature biases required for correcting the channel 1 brightness temperature were determined from diurnal fits obtained from the Fourier analysis as described in section 'Construction of Diurnal Cycle'. The tropical time series of channel 1 brightness temperature before and after diurnal correction are shown in Figure 8. The first column of the figure shows the tropical time series over land and ocean for the ascending orbit. The uncorrected ascending time series for channel 1 over land regions shows a strong cooling trend with a magnitude of −8.16 ± 0.1078 K per decade. Note that the equator crossing time of the ascending orbit of NOAA-16 was 14.00 local time on launch but had changed to 19.00 local time by 2010. A strong diurnal cooling in the uncorrected time series can be seen since 2006 where a notable drift in NOAA had begun to occur (see Figure 1). After diurnal correction, the trend over tropical land region is found to have a magnitude of only 0.32 ± 0.096 K per decade. Over tropical oceans, the differences in the linear trend shown for uncorrected and corrected time series is less distinct because the diurnal amplitudes over oceanic regions are small (less than 1.5 K). The trend in the uncorrected time series for oceanic regions is 0.6380 ± 0.0636 K per decade. This positive trend shown over oceanic regions in the uncorrected time series is due to the behavior of the diurnal cycle over this region. Due to the presence of low level clouds, the diurnal cycle over oceanic regions peaks in the early morning, and the diurnal warming in the evening is aliased onto the long-term records; this is captured clearly in the uncorrected time series since 2006. After diurnal correction, the trend over oceanic regions is only 0.19 ± 0.0625 K per decade, which is less than a third of the uncorrected trend.
 The second column of Figure 8 shows the tropical time series over land and ocean for the descending orbit. Over land, the uncorrected time series shows a cooling trend in the order of −0.26 ± 0.0831 K per decade, which is due to the orbital drift causing the equator crossing time of descending orbit to change from 2.00 h in local time at the time of launch to 7.00 h in local time by the end of 2010. Subsequent to orbital drift correction, the trend in the tropical time series of the ascending orbit is found to be 0.6454 ± 0.0768 K per decade. However over ocean, the trend in the uncorrected and corrected time series for the descending orbit does not show significant differences, which are −0.0294 ± 0.0632 and −0.0637 ± 0.0628 K per decade, respectively. The differences in the trends after diurnal correction shown for both ascending and descending orbits do not come as a surprise since the correction is for two different local times, at times of day when the temperatures are at their maximum and minimum, respectively. Earlier studies as in Vose et al.  have also shown that the trends in the maximum and minimum temperatures can be different. Moreover, the trends are also different for land and ocean due to the differences in diurnal cycle over these regions as pointed out earlier.
 We also investigate whether combining ascending and descending time series could possibly nullify the effect of diurnal cycle aliasing. For this purpose, we combined the time series of both ascending and descending orbits of NOAA-16 channel 1 brightness temperatures before and after diurnal correction, and the resulting time series for tropical land and ocean are shown in the third column of Figure 8. The observed trend in channel 1 brightness temperature for the combined uncorrected time series over land is found to be −4.2083 ± 0.0811 K per decade, which is one half of the trend observed in the uncorrected ascending time series. But combining corrected time series results in a trend in the order of 0.4882 ± 0.0773 K per decade, which is close to the corrected trend in ascending orbit over land. For tropical ocean, the combined uncorrected time series has a trend in the order of 0.3031 ±0.0567 K per decade, which is again a half of the trend observed in uncorrected ascending time series. Furthermore, combining corrected time series over ocean shows a trend of 0.0657 ± 0.0560 K per decade, which is within the uncertainty limit of the corrected trend in ascending time series. These results reveal that combining time series of ascending and descending orbits which are affected by diurnal sampling bias does not completely eliminate the spurious trend introduced in the time series. This is due to the nonsinusoidal nature of the diurnal cycle amplitude arising from a predominance of 12 h oscillations. On the other hand, combining time series of ascending and descending orbit eliminates to a large extent the 24 h component of the diurnal cycle. This aspect has been addressed in detail by John et al. (submitted manuscript, 2012b) in the context of inter-satellite biases.
 The impact of diurnal correction on humidity channel 3 is analyzed using the ascending time series of NOAA-17 satellite. NOAA-17 is chosen here because of the longevity of the humidity time series in comparison to the other satellite data used in this study. Orbital drift is not so significant for NOAA-17 in comparison to NOAA-16 as the former has drifted approximately by only 1 h during the period 2003–2010. The ascending times series of the humidity measurements were corrected to 22 h in local time corresponding to the initial equator crossing time of NOAA-17. The uncorrected and corrected trends in time series observed for channel 3 over tropical land are −0.1621 ± 0.1702 and −0.3728 ± 0.1695 K per decade, respectively, but over ocean these values are −0.6062 ± 0.0950 and 0.7259 ± 0.0951 K per decade, respectively. As expected, the trend differences between the corrected and uncorrected time series for channel 3 is not so significant due to a lesser orbital drift and smaller diurnal cycle amplitudes. This inference regarding the trends has been made from a short time series of humidity measurements, but a statistically significant trend can be borne out only when analyzing long time series of humidity measurements. A comprehensive analysis is needed to understand the trends in humidity with a comparatively longer humidity time series than considered in this study, which would be the focus of our future studies.
5 Summary and Conclusions
 A long-term monitoring of tropospheric humidity distribution over the globe is crucial for predicting the climate change in the wake of global warming. In this context, long-term microwave measurements of tropospheric humidity available from NOAA polar orbiting satellites play an important role in identifying the trend in tropospheric humidity. However, the measurements available from some of these NOAA satellites are affected by diurnal sampling bias resulting from satellite orbital drift. In this paper, we have applied a methodology for detecting and thereby correcting the diurnal sampling bias in NOAA satellite measurements. A diurnal cycle of satellite measurements was generated by combining data corrected for inter-satellite bias from different NOAA satellites and MetOpA satellite over a period of 10 years using Fourier analysis. A Monte Carlo error analysis has been performed to assess the significance of the diurnal amplitudes. Diurnal cycles were generated separately for all months for the five microwave channels.
 An analysis of diurnal cycle amplitudes for surface channels shows a clear land-sea contrast with the land regions showing very high amplitudes often exceeding 10 K over dry and convective regions. These channels also show a seasonal variation in the diurnal amplitude. The humidity channels sensitive to the upper and middle troposphere also show a seasonal variation in the diurnal amplitude, which can exceed 3 K in highly convective land regions and comparatively lower amplitudes over oceanic regions. For these channels, the magnitude of the diurnal amplitudes in 24 and 12 h oscillations are not so distinct. Over most of the oceanic regions within the tropics, the amplitude of the 12 h oscillation is either comparable to or greater than that of the 24 h oscillation. The channel sensitive to lower tropospheric humidity does not show a seasonal variation in the diurnal amplitude, and the diurnal amplitude variation in this channel is governed by changes in the surface temperatures and humidity. The diurnal cycle necessitates adequate sampling requirements for new satellite missions, and this should also be taken into account for future model-data comparison [Buehler et al., 2012, 2007; Eliasson, 2011]. The diurnal peak time for upper and middle tropospheric humidity over land regions has an early morning peak time, which is consistent with previous studies made using infrared measurements.
 The impact of diurnal correction has been evaluated by analyzing the channel 1 brightness temperature time series of NOAA-16 and channel 3 brightness temperature time series of NOAA-17 satellites. The results show that for channel 1 the diurnal sampling bias correction has a large impact on land regions due to high diurnal cycle amplitudes and a significantly lower impact on oceanic regions. Also for channel 1, combining the ascending and descending time series prior to diurnal sampling correction does not seem to completely eliminate the spurious trend in time series, which is most likely due to the predominance of the 12 h oscillations. The analysis of channel 3 brightness temperature time series reveals no significant difference in the trends in uncorrected and corrected time series. An exhaustive analysis of humidity time series is beyond the scope of our present study, but this would be taken up as a follow-up study in our future works. The main motive of this study was to correct the diurnal cycle aliasing in AMSU-B and MHS satellite measurements. This study is one of the major steps and a prerequisite towards generating a homogenized humidity dataset for climate studies. However, the diurnal cycle so derived can also be used for the evaluation of the diurnal cycle of humidity in global climate models.
 We thank our two anonymous reviewers and David Parker for their valuable comments. Ajil Kottayil was a visiting scientist at the UK Met Office where a part of this work has been done. V.O.J. was supported by the UK Joint DECC and DEFRA Integrated Climate Programme-GA01101, the UK JWCRP, and EUMETSAT CMSAF. A.K. and S.A.B. were supported by the Swedish Science Council and the Swedish Space Board. This work contributes to COST Action ES604–Water Vapor in the Climate System (WaVaCS) and to the EUMETSAT CMSAF activities. Thanks to Lisa Neclos, NOAA CLASS for AMSU-B and MHS Level-1b data and EUMETSAT NWP-SAF for the AAPP software to process the data.