A simple method that was developed by has been widely used in the past to convert satellite microwave measurements to layer-averaged RH e.g.,. In this simple relation, the satellite Tb's are linearly related to the natural logarithm of layer-averaged humidity as follows:

ln⁡(RHch)=ach+bch×Tbch, where a and b are empirical coefficients that change with the earth incidence angle unless the satellite Tb's are corrected for the limb effect, and the ch stands for the channels. Since calculating the empirical coefficients as a function of earth incidence angle introduces a very large look-up table; similar to , we first applied a limb-correction technique to SAPHIR Tb's then used the same empirical coefficients for all the incidence angles. We calculated the limb-darkening (ΔTb) as the difference between Tb for each beam position and corresponding nadir Tb using data averaged over a long period of time:

Tbn=Tb(θ)-ΔTbΔTb=c×ln⁡(cos⁡θ), where Tbn is Tb at sub-nadir footprint, and Tb(θ) is Tb at any given θ . Since the SAPHIR data do not suffer from scan asymmetry, we preferably used the satellite data to develop the limb-correction technique. As shown in Fig. , the limb-darkening is stronger for the channels operating near the center of the water vapor absorption line than channels operating near the wings of the line. Figure  also shows the values for the coefficient c in Eq. () for different SAPHIR water vapor channels.

Microwave satellite data are less sensitive to clouds than IR data, however microwave measurements may also be affected by optically thick clouds. Since the empirical coefficients are only valid for clear-sky radiances, the data affected by clouds should be excluded from the analysis. We used the same thresholds proposed by to screen-out the clouds using the differences between Tb's of an upper channel (Tb2, channel 2 operating at 183 ± 1.10 GHz) and a lower channel (Tb5, channel 5 operating at 183 ± 6.8 GHz). The satellites Tb's are cloud free if Tb2–Tb5 < -15 K and Tb2 > 240 K. More details are provided in .

In addition to clouds, some of the satellite radiances may be affected by the surface emissivity which results in high (low) Tb values over land (ocean). Because of the inverse relation between the RH and TB, this corresponds to artificially low (high) RH values over land (ocean) for the measurements that are affected by the surface. We used a threshold for Tb's to exclude the measurements that are affected by the surface. Because the emissivity in microwave frequencies is low over ocean (0.5–0.7) and high over land (about 0.9), the Tb's affected by the surface are normally high over land and low over ocean. We used a subset of atmospheric radiation measurement program (ARM) radiosonde data and also radiative transfer calculations to determine the thresholds for Tb's that are affected by either land or ocean. We performed two sets of radiative transfer calculations using the same radiosonde profiles but different emissivity values for land and ocean. Figure  shows the Tb's histograms for different SAPHIR channels, when the difference between simulated Tb's for land and ocean is less than 0.01 K. Therefore, the histograms show the range of Tb's that are not affected by the surface. Based on these histograms we defined the following thresholds for excluding surface affected observations from analysis for channels 1–6, respectively: 230–270, 240–280, 250–290, 255–295, 265–295, and 270–300 K. The lowest threshold is applied when the observations are affected by the sea surface and the maximum applies when the observations are affected by the land surface. No filter is applied for topography, thus the results over mountainous terrains should be considered with caution since the satellite radiances are averaged over a spatially very inhomogeneous region.

Fourier series are traditionally used to model the diurnal cycle of meteorological variables such as temperature and humidity. Fourier series are periodic functions expressed in terms of sine and cosine functions as follows:

F(x)=a0+∑k=1Kakcos⁡(kx)+bksin⁡(kx), where x represents time of the day in radians (x∈[-π,π]) and can be calculated as x=t-t1t1π, where t1 is equal to 12, t is time of the day in hour, and a0, ak, and bk are the Fourier coefficients that can be calculated using the following relations:

a0=1∑wi∑wiyiak=2∑wi∑wiyicos⁡(kxi)bk=2∑wi∑wiyisin⁡(kxi), where yi are the measurements and wi indicates the weights given to each measurement. The weights are calculated as 1σ, where σ is the standard deviation of all the measurements within each individual bin. We assume that the data are equally spaced and the points define the middle of each interval, so that n data points can be used to divide the space [-π,π] into n equal intervals so that Δx=2πn. Some of the previous studies, e.g., and , have used least square techniques to determine the Fourier coefficients. However, as shown in Eq. (), the coefficients can be mathematically and directly calculated form the measurements. It is also required to determine number of terms that Fourier series should be expanded (i.e., K in Eq. ). However, there is no standard method to determine this number. We evaluated the mean absolute difference between the measurements and the values calculated using Fourier series to determine number of terms that Eq. () needed to be expanded (see Sect.  for more information). It was found that one term is sufficient over regions with small diurnal variation, but the series need to be expanded by two terms to properly cover the diurnal variation over regions with a larger diurnal amplitude.

We used a subset of the ARM radiosonde data to calculate the empirical coefficients (a and b) for Eq. (). Since the saturated vapor pressure can be calculated with respect to either liquid (temperatures above the freezing point of water) or ice phase (temperatures below the freezing point of water), the empirical coefficients can be defined the same way with respect to saturated vapor pressure over either liquid or ice. We use RHI to refer to RH with respect to ice and RHL for RH over liquid. It is expected that at least in the middle and upper troposphere (channels 1–4), the air temperature is generally below the freezing point thus we need to use the saturated vapor pressure over ice. Additionally, for the lower channels (channels 5 and 6) the saturated vapor pressure expressions for ice and liquid approach each other. Therefore, in most cases we only present the results for the ice phase (RHI) and the results for the liquid phase (RHL) are provided in the Supplement.

Figure shows an example of the relation between satellite Tb's and natural logarithm of layer averaged RHI for SAPHIR channel 2. Channel 2 of SAPHIR operates at 183 ± 1 GHz which is similar to a channel on many humidity sounders such as Advanced Technology Microwave Sounder (ATMS), AMSU-B, and MHS. Therefore, the results can be directly compared with the previous studies. For instance, the coefficients shown in Fig.  are consistent with , , and .

The empirical coefficients for all the channels are presented in Table  with respect to both liquid and ice. Coefficient a for Channel 1 over both ice and liquid is smaller than the same coefficient for other channels, but all other coefficients are very close for all the channels. Both a and b coefficients are greater over ice than over liquid.

SAPHIR observations are available for the tropical region expanding from about 35∘ S to 35∘ N. However, we limited the study to the region between 25∘ S to 25∘ N, because the frequency of the revisits is limited outside this region. We first binned the data based on local observation time into a grid of 1.0 × 1.0 degree. The data points within each grid-box were gridded into 24 bins based on local observation time. The local time was calculated using Coordinated Universal Time (UTC) and longitude which are both provided in SAPHIR data. Finally, we averaged the data within each grid-box of 1.0∘ × 1.0∘ × 1.0 h. Figure  shows average number of observations per hour after the data are filtered for clouds and surface effect. See Sect.  and Table  for details on the boxes shown on the maps. Since the satellite inclination is about 20∘, maximum number of observations occurs around 10–20∘ N and 10–20∘ S. As shown, on average, 100 to 300 observations are retained for each bin per hour. Over very high elevations the weighting functions for all the channels peak very close to the surface, therefore, the minimum number of observations occurs over mountains such as the Andes in South America.

Figure shows the mean daily RHL for different SAPHIR channels. As expected, RH significantly changes from upper troposphere (14–36 %) to lower troposphere (41–85 % for Channel 6). Note that in order to avoid the outliers especially over the Andes, in most cases, the upper limits printed on the color bars show the ninety-ninth percentile. The dry regions are observed over subsidence regions, e.g., South Pacific Ocean, South Atlantic Ocean, as well as Arabian Sea, and are consistent with previous studies e.g.,. Additionally, several moist regions are observed over convective regions, e.g., South America, Central Africa, and South Asia also known as Maritime Continent which is located within the Tropical Warm Pool. The RH generally decreases with distance from the convective regions which is consistent with previous studies. For instance, reported that the upper tropospheric humidity rapidly decreases with distance from the convective clouds. The pattern does not change from the upper to lower troposphere, but generally the decrease in moisture with distance from the convective region, especially over the Maritime Continent, is faster in the lower troposphere than in the upper troposphere. It is also evident that moist regions are connected so that water vapor can be transported across the Equatorial region.

Figure shows the layer-averaged tropospheric RHI. Because the saturated vapor pressure is lower over ice than over liquid, the RHI values are greater than the RHL values. So that RHI ranges between 18 and 52 % in upper troposphere (compared to 14–36 % for RHL), and 38–90 % in lower troposphere (compared to 41–85 % for RHL). Since we excluded the data that are affected by the surface, no difference between land and ocean is observed along the coastlines. The Andes show a large impact on the lower tropospheric RH but a small impact on upper tropospheric RH. The results for regions such as the Andes should be interpreted with cautious because of uncertainty in the measurements over mountainous regions. The fact that the results do not show a significant land–sea contrast shows that the filter for the surface effect properly removes the surface affected observations. The values reported in Fig.  are about 10–15 % higher than the values reported in . As stated before, this is because the moist regions are removed from the IR observations due to strict cloud screening. This is also consistent with who reported about 10 % dry bias in clear-sky IR observations in the upper troposphere.

Polar orbiting satellites orbit the earth twice a day, thus the daily average of relative humidity estimated from the measurements of these satellites may be biased depending on the crossing time. In order to evaluate the impact of crossing time on the estimated daily averages, we used the observations from only two overpasses being 12 h apart, similar to ascending and descending orbits of polar-orbiting satellites. We then computed the mean difference of the daily averages calculated using only two overpasses and the daily averages calculated using all the hourly data. Figure  shows the results for the measurements from 01:30 (13:30) local time and Fig.  shows the results for 09:30 (21:30) local time. In fact we collected all the measurements 30 min before and after the aforementioned local time. The midnight/afternoon orbit (01:30/13:30 local time) matches with several satellites including NOAA Joint Polar Satellite System and NASA A-Train, and the early morning/late evening orbit (09:30/21:30 local time) matches with the orbit for the MetOp satellites. As shown, especially for the lower tropospheric channels, the error is generally larger for 01:30/13:30 local time, because as shown later, in tropical region the peak of RH generally happens in early morning over many regions. In both cases, the error is generally less than 2 % RHI in upper troposphere and slightly increases in middle troposphere. However, in lower troposphere the difference between the two cases is considerable. The error due to eliminating diurnal variation is more than 4 % RHI especially over land for the measurements from 01:30/13:30 local time and overall less than 2 % RHI for the measurements from 09:30/21:30 local time. These results show that measurements from polar-orbiting satellites can be used to derive the mean tropospheric humidity in the tropical region with a good accuracy. However, polar-orbiting satellites may not provide a good picture of peak time and amplitude, because these parameters show a large spatial inhomogeneity and obviously depend on the satellite overpass time (see Sect.  for more details). The same figures are included in the Supplement for the error in RHL.

Figure shows the diurnal amplitude of layer-averaged RHI as the difference between maximums and minimums of RH derived from the Fourier series fit over the course of the day. We employ the diurnal amplitude and peak time derived from the Fourier series, because they are more stable and less noisy than the amplitude and peak time derived from the measurements, though both Fourier series and measurements yield very similar results. The diurnal amplitude derived from the measurements is included in the Supplement. The pattern and magnitude of the amplitude significantly change from upper to lower troposphere. In upper and middle troposphere, i.e., Channels 1–4, the diurnal amplitude is less than 15 % with the maximum occurring over the Andes, South Africa, Madagascar, and Australia as well as some scattered places over South America and Arabian Desert. The diurnal amplitude in upper troposphere is consistent with who reported up to 8 % change over tropical land regions in upper troposphere. However, in lower troposphere, the amplitude can be up to 29 % over land. In lower troposphere, the diurnal amplitude is generally less than 10 % over ocean, but over deserts and mountains the amplitude is greater than 15 % with maximum occurring over deserts of South and North Africa, Australia, Middle East, the Andes in South America, and the Sierra Madres mountains in Mexico. The diurnal amplitude for RHL (see the Supplement) is a few percent smaller than that for RHI but the pattern is very similar. The amplitude slightly changes from channel 1 to channel 5, but it is much larger for channel 6 than for channel 5 (29 % vs. 19 % over land). However, the pattern is very similar for channels 5 and 6. Generally, a stronger diurnal amplitude is found over land than over ocean for the lower channels. This can be explained by the fact that the diurnal variation of RH in the lower layers of the troposphere is enforced by the change in surface and boundary layer temperature which are larger over land than over ocean. Over ocean, the diurnal amplitude tends to be larger over convective regions than over subsidence regions which is consistent with previous studies e.g.,.

Since the location of ITCZ changes with time, the diurnal amplitude and peak time are expected to be seasonal-dependent. The ITCZ location is furthest away from the equator during the summer solstice (towards north) and winter solstice (towards south), therefore, in addition to examining the annual averages of the diurnal amplitude and peak time, we separately evaluated the diurnal amplitude and peak time for the months of December and January (Winter Solstice, Fig. ) as well as June and July (Summer Solstice, Fig. ). Comparing the diurnal amplitude of RH derived from the annual averages (Fig. ) with Figs.  and shows that the change in diurnal amplitude during Summer Solstice is larger than the change during the Winter Solstice. This is because due to the presence of more continents in the Northern Hemisphere, ITCZ moves further away from the equator during the Summer Solstice . The main difference between the diurnal amplitudes of Winter and Summer Solstices can be summarized as follows. The amplitudes in all layers are about 2 % higher in Winter Solstice than in the Summer Solstice. Shift in ITCZ forces the convective region over Central Africa to move to the southern latitudes during Winter Solstice. Therefore, North Africa is dominated by a dry subsidence region during Winter Solstice which causes a larger diurnal amplitude especially in the lower troposphere likely due to a larger diurnal cycle in air temperature. In contrast, in Summer Solstice, ITCZ moves to the northern latitudes so that most of the African continent is dominated by a convective region, except South Africa, which causes a smaller diurnal amplitude during Summer Solstice. In addition, the diurnal amplitude over North America is considerably lower during Summer Solstice than the Winter Solstice.

Figure shows the RHI diurnal peak time for different channels derived from the Fourier series fit. The peak time derived from the measurements is shown in the Supplement. Generally, over most regions, the peak time is delayed from upper to lower troposphere. For instance, over Africa the diurnal peak time changes from midnight in the upper troposphere to early morning in the lower troposphere. Overall, the peak time occurs around midnight over most continental regions which is consistent with previous studies e.g.,. Over oceans, the peak time normally happens in the morning over subsidence regions but mid-night over the convective regions. It should be noted that there is a large uncertainty in estimating the peak time when the diurnal amplitude is very small. Generally, the early morning peak time that has been reported before e.g.,, only occurs in some regions and it is not common for the entire tropical region. The early morning peak time is due to a peak in deep convective activities . In upper troposphere, the nighttime peak over land is generally consistent with , however reported a late-afternoon/evening peak time over the convective regions which is not consistent with the current study. This can be due to the fact that the peak time derived from IR data is affected by the cloud screening method. Channel 3 in (middle panel in Fig. 5) can be directly compared with channel 2 in this study. Over South America (Amazonian region) both studies show night-time peak time. However, over some other regions, e.g., Indian Ocean, the results are not consistent. We show a peak time before 00:00 LT over most part of Indian Ocean but reported a night-time peak (between 00:00 and 02:00 LT). We have filtered cloud contaminated data, therefore the results are not dominated by cloud diurnal regime in the tropical region. Since did not exclude the surface affected data, the results for the lower channels cannot be compared.

reported a large inhomogeneity in the diurnal variation of precipitation which is consistent with the inhomogeneity we have found for relative humidity. Using Tropical Rainfall Measuring Mission (TRMM) Precipitation Radar measurements, reported a large difference between diurnal variation of precipitation over coastal lands (early afternoon peak) and near shores (early morning) influenced by the land/sea breeze. These fine structures cannot be detected in MW observations due to coarse spatial resolution, but it highlights the fact that the results over inhomogeneous regions such as coastal regions may not be very representative. Although, the spatial resolution of the SAPHIR observations do not provide a sharp transition from ocean to land, a distinct peak time is observed along some of the shorelines. For instance, on the west coast of South America, the peak time over lands near shore occurs a few hours earlier than the peak time for both inland and free waters.

Figures and shows the peak-time for Winter and Summer Solstices respectively. The most distinguishable difference between the peak-times during Winter and Summer Solstices are as follows. Over Eastern Pacific, in the middle and lower troposphere, the peak mostly happens in the early morning during Summer Solstice but late evening and midnight during Winter Solstice. A nighttime peak is observed over Africa during Summer Solstice but the peak-time changes to early morning during Winter Solstice. Over the Indian Ocean, there is an early morning peak-time during Summer Solstice that changes to a late-evening and midnight peak during Winter Solstice.

Based on the mean and diurnal amplitude of RH, five regions are selected that will be used to further investigate some of the results. The coordinates (minimum and maximum of latitudes and longitudes) of these rectangular regions are shown in Table . These regions from west to east are located over South Pacific Ocean, Amazon, South Atlantic Ocean, North Africa, and the Maritime Continent. These regions are selected in a way to present a high diversity in diurnal amplitude as well as mean tropospheric RH. The boundaries for these regions are shown on all the maps.

The diurnal cycle of tropospheric RH is modeled using Fourier series and the series themselves can be demonstrated using the Fourier coefficients. As mentioned before, Fourier series can be theoretically expanded by infinite terms but in practice only a few terms are required for diurnal variation of meteorological variables. In this study, we expanded the Fourier series in only two terms, i.e., k=1, 2. As mentioned earlier, this number was determined by analyzing the difference between the measurements and the values estimated using Fourier series. For instance, Fig.  shows the mean absolute difference between the measurements and the Fourier series fit. Over most regions the difference is less than 1 % when the series are expanded in two terms.

Since the coefficient a0 is equal to the mean of the measurements, its distribution is already shown in Figs.  and . The coefficients ak and bk present information about the phase and amplitude of the signal and are included in the Supplement. If we rewrite the Fourier coefficients as a complex number zn=an+ibn then the amplitude and phase can be expressed using |z| and arg(z), respectively:

|z|=zz*=a2+b2arg(z)=x=tan⁡-1ba, where arg(z) is the phase difference ranging from -π to π and can be converted back to the time of the day using t=(12x/π)+12.

In the following, the selected regions will be used to further investigate the diurnal variation of tropospheric RH using Fourier series. Figure  shows the diurnal cycle of the layer-averaged RHI (absolute values) in selected regions for different SAPHIR channels. Figure  shows the RH values scaled between -100 and 100 to better distinguish the diurnal cycle for regions with a small diurnal amplitude. As shown, in most regions the diurnal amplitude is very small (less than a few percent), but the diurnal amplitude in middle and lower troposphere over North Africa is greater than 10 % (consistent with Fig. ). In upper troposphere (Channels 1 and 2), the diurnal variation is less than 5 % over all selected regions. As shown in Fig. , some regions such as SP and SA experience two peaks. Over SP, one of the peaks occurs in the early morning and the other one in the evening. The evening peak is stronger in upper troposphere, but in lower troposphere the morning peak becomes stronger than the evening peak. A similar pattern exists over SA, however the second peak occurs around early afternoon and becomes weaker from upper to lower troposphere. However in both cases the actual amplitude of the diurnal cycle is a very small percentage. The double peak is consistent with a double peak in precipitation reported in . Note that peaks in precipitation normally coincide with a minimum in RH. As reported in the RH trend, at least in upper troposphere, follows the trend in convective clouds except the minimum of RH that occurs when the precipitation starts to increase. As shown in Fig. , the peak time slightly shifts from upper to lower troposphere. The amplitude over North Africa also changes significantly from a few percent in the upper troposphere to more than 20 % in the lower troposphere which is due to change in the diurnal variation of air temperature. Figure  also includes an example for a grid box with a large difference between the measurements and the fit for the Fourier series. The grid box is located near Lago Salar de Arizaro (24.5∘ S, 67.5∘ W), a small salt flat of the Andes in Argentina. The surface area of the flat is only 1600 km2, therefore the grid box covers a mix of the salt flat and the surrounding terrains. As shown, the Fourier series are perfectly fitted to the data and the difference between the measurements and the Fourier series fit is due to noise in the data.

Figure shows the distribution of layer-averaged RH with respect to both liquid and ice over selected regions. In addition, the cumulative distribution functions are also provided in the Supplement. As shown the distributions of RH with respect to ice and water are similar in the lower troposphere (channels 5 and 6). This is because the relative humidity values calculated using Eq. () are nearly the same over ice and liquid for the expected range of brightness temperatures in lower troposphere. This is an indication that the transition from ice to liquid is performed smoothly. When necessary, we use RH over ice for channels 1–4, and over water for channels 5–6 to discuss the results. Overall, the distribution functions are very similar for the Amazon and the Maritime Continent regions. Some small differences exist between the two regions especially in the upper troposphere, where the Maritime Continent tends to be more moist than Amazon. The distribution functions are also similar for the South Pacific and South Atlantic, but the South Atlantic tends to be slightly more moist than the South Pacific in all layers. Therefore, we only explain distribution functions for South Atlantic, the Maritime Continent, and North Africa in more detail. The first and third quartiles of the distribution functions are used to explain the range of RH in each layer. The range of RH between first and second quartile includes 50 % of the data points. For channel 1, the first (third) quartiles are 15 (25), 40 (55), 20 (40), over the South Atlantic, the Maritime Continent, and North Africa. None of the regions show a normal distribution function. The distribution is left-skewed (the left tail is longer) for Amazon and the Maritime Continent, and right-skewed over the rest of the selected regions. The shape of the distribution remains almost similar for other channels. The quartiles are very similar between channels 1 and 2, but increase about 5 % per layer for channels 3–5. So that the quartiles for each lower channel are about 5 % higher than the same quantities for the channel peaking above it. From channels 5 to channel 6, the increase in quartiles is about 10 %. Note that in many cases the minimum and maximum are very close to first and third quartile meaning that 25 % of the data points lie within a small range of RH. A small percentage of the data shows supersaturation over liquid (up to 110 %) for Channel 6. This can be explained by the methodological error as well as the error in the satellite observations. We estimate that the error introduced by different sources (see Sect. ) can be up to 15 %. Our findings for SAPHIR Channels 1 and 2 are generally consistent with , especially for the results presented for AURA-MLS instrument. For instance both studies show that the RHI can reach up to 80 % over Africa. However, reported a maximum RHI of 80 to 100 % over the Maritime Continent but our results show a maximum of 60 to 80 % for SAPHIR Channel 1 and 80 to 100 % for SAPHIR Channel 2. Overall, it is expected that the cloud filter removes some supersaturated regions in our study, because those regions are normally associated with the cloud formation.

In this section, the error sources are discussed, though it is not possible to quantitatively estimate most of the errors. One obvious source of error is bias and noise in the satellite data. According to the bias in SAPHIR data should be less than 0.5 K which is roughly equal to 5 % in RH space. Although, the bias affects the RH values, it does not affect the results for diurnal amplitude and peak time. Since a large volume of the data is averaged, the noise in the satellite data should cancel out if the relation between RH and Tb was linear. However, due to the non-linear relation between the two, the random noise may not completely cancel out. Another source of error is from the Tb to RH transformation method which according to is estimated to be less than 10 %. Other sources of error include the surface and cloud effects. Although we have applied appropriate filters, it is still possible that some clouds are not filtered out and at least for the lower channels there are still cases that are affected by the surface. The limb-correction technique is based on satellite data averaged over the entire tropics. Therefore, it may introduce at least some noise in the correction for the individual measurements. However, it is expected that the limb-correction does not introduce a systematic bias. The random error introduced by the limb-correction technique may not completely cancel out due to a non-linear relation between RH and Tb. However, the overall impact of the random error on the results is expected to be negligible. Finally, diurnal variation of RH is highly influenced by the diurnal variation of air temperature, thus sources such as inversion in temperature lapse rate can contribute to the error because satellite data are averaged over a wide layer.