a. Swath geometry

Both MSU and AMSU are cross-track scanning radiometers that measure the upwelling brightness temperature at a number of different view angles as they scan the earth perpendicular to the satellite subtrack. MSU views the earth at 11 different viewing angles separated by 9.47°, yielding a range of view angles from 0.0° for the nadir view to 47.35° for the two views furthest from nadir (Kidwell 1998). On the earth’s surface, this corresponds to earth incidence angles ranging from 0.0° to approximately 56.19°. MSU has a half-power beamwidth of 7.5°, corresponding to a nadir spot size on the earth of 110 km, expanding to 178 km × 322 km for the near-limb view resulting from the increased distance from the satellite and oblique incidence angle. The AMSU instruments have significantly higher spatial resolution, viewing the earth at 30 different viewing angles separated by 3.33°, with view angles ranging from 1.67° to 48.33° (Goodrum et al. 2000). These view angles correspond to earth incidence angles ranging from 1.88° to 57.22°. The half-power beamwidth of the AMSU instrument is 3.3°, yielding a nadir spot size of 48 km × 48 km, increasing to 80 km × 150 km for the near-limb views. Because the MSU lower-tropospheric dataset is constructed using a weighted difference of measurements made at different view angles, these differences in viewing geometry between the MSU and AMSU instruments complicate the merging procedure.

b. Temperature weighting functions

AMSU5 is a double-sideband receiver that is sensitive to two sidebands centered at 53.71 and 53.48 GHz, each with a bandwidth of 0.17 GHz. MSU channel 2 (MSU2) is a single-sideband receiver centered at 53.74 GHz with a bandwidth of 0.20 GHz. Because (on average), ASMU5 uses a lower frequency than MSU2, the AMSU5 weighting function peaks nearer to the surface than the corresponding MSU weighting function for a given incidence angle. In Fig. 1, we plot the measurement bands for the two instruments, as well as the oxygen-absorption coefficient (Rosenkranz 1993) for the atmosphere at two representative pressures.

The brightness temperature T_b of the microwave radiation incident on the satellite is given by

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where W_s is the surface weight, T(z) is the temperature at height z, W(z) is the temperature weighting function, and the integral extends from the surface to the top of the atmosphere (TOA). The surface weight and the temperature weighting functions are dependent on the atmospheric absorption coefficient κ(z) as a function of height z, the surface emissivity e_s, and the local zenith angle θ of the radiation path through the atmosphere. We consider oxygen and water vapor in these calculations. Under the assumption of specular reflection from the surface, W_s and W(z) are given by

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where τ(z₁, z₂) is the zenith optical depth for a layer that extends in height from z₁ to z₂ (Ulaby et al. 1981). The temperature weighting function contains contributions from radiation emitted by each layer, which initially travels both in the upward and downward directions. The downward-travelling radiation propagates to the surface and reflects upward. Increasing the zenith angle causes the surface weight to be reduced and the temperature weighting function to move higher in the atmosphere. In Fig. 2, we plot temperature weighting functions for each view of MSU channel 2 and AMSU channel 5 for simulated land and ocean views using the U.S. Standard Atmosphere, 1976 and an assumed relative humidity profile. These calculations were made using a radiative transfer model based on Rosenkranz (1993, 1998) and a model of the ocean surface developed by Wentz and Meissner (2000). For a given incidence angle, the AMSU channel 5 weighting function peaks several hundred meters closer to the surface, and the contribution of the surface is increased substantially relative to the corresponding MSU channel 2 weighting function. This leads to slightly higher brightness temperatures for AMSU for a given incidence angle, especially over land areas, where the surface contribution is larger because of larger emissivity relative to the ocean.

The MSU channel 2 lower-tropospheric (2LT) dataset is based on a weighted difference of MSU views,

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This combination of views nearly cancels the stratospheric influence and moves the peak of the temperature weighting function lower in the troposphere (Spencer and Christy 1992). In Fig. 3, we show the 2LT temperature weighting functions for land and ocean surfaces on the same vertical scale as Fig. 2.

Our task now is to find a combination of AMSU measurements that provide the same brightness temperatures as would be found using the combination of MSU measurements presented in Eq. (3). We use a regression method to obtain weights a_fov for the AMSU views. The set of equations to be solved are given by

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where T_AMSU(fov) and T_MSU(fov) are the AMSU and MSU brightness temperatures for each field of view (FOV), respectively. The MSU weights c_fov are those given in Eq. (3). Additional equations of the form k(a_fov − a_fov+1) = 0 were added to the set of equations before their solution was determined. The effect of these equations is to reduce the effects of noise by introducing a nonparametric smoothness constraint on a as a function of FOV. We solved this equation simultaneously by using monthly zonal averages from NOAA-14 (MSU) and NOAA-15 (AMSU) as input data. The zonal averages were calculated over 5° zonal bands, and they were calculated separately for land and ocean scenes and for each FOV. Land areas with surface height averaged over the 2.5° × 2.5° cell that exceed a threshold altitude of 1500 m were excluded from the averages to reduce contamination from surface emission. The land and ocean averages were used to form separate equations to deduce a set of values of a_fov that produce a good brightness temperature match for both land and ocean scenes independently. Each equation was weighted according to the area of the earth that it represented; that is, for a given zonal band, the land equation was weighted by the land area in that band and the ocean equation was weighted by the ocean area in that band. The values of a_fov were constrained to be equal to the corresponding weight on the opposite side of the swath, and the weights for the central 14 views were set to zero so that the derived AMSU product would cover roughly the same part of the swath as the MSU product. (We also performed the calculation with only the central 12 views excluded, which would result in an improved match to the MSU measurement swath; the resulting weight for the innermost included view was so small that we decided to exclude it.) In Table 1, we show the regressed values for the weights of a_fov; in Fig. 4, we plot the MSU and AMSU weights as a function of incidence angle. The resulting temperature weighting functions are plotted in Fig. 3 for both ocean and land surfaces, along with the original MSU-derived weighting functions.

The validity of this procedure was evaluated in two ways. First, we studied the residual error between the weighted, zonally, and monthly averaged MSU and AMSU combinations. The standard deviation of the difference between these two combinations is about 0.12K, about 3 times the error we expect because of differences in temporal sampling between the two instruments. Thus, this difference suggests that, although it is impossible to exactly match these weighting functions with a single set of AMSU weights, a reasonably good match can be obtained. Second, Fig. 3 indicates that there is also a good match between the two weighting functions derived using these weights and the U.S. Standard Atmosphere, 1976, showing that our combination of views physically matches the temperature weighting of the original MSU 2LT product. Although the differences in weighting functions between the two instruments makes a globally valid, exact solution based solely on view weighting impossible, the weighting procedure developed here minimizes the magnitude of the location-dependent differences between MSU and AMSU measurements. These location-dependent differences will be removed empirically in a later step.

c. Construction of gridded monthly averages

All of the adjustment and merging steps discussed in the following sections are performed using monthly averages gridded on a 2.5° × 2.5° longitude and latitude grid. These averages are constructed using the following method: for each half scan (field of view 1–8 or 23–30), a “lower-tropospheric temperature (TLT) measurement” is calculated by applying the weights in Table 1 to the measured temperature for each view and summing. This TLT measurement is then assigned to each 2.5° grid cell that contains the center of a measurement footprint. Then, for each grid cell, all of the TLT measurements for each month are averaged together to form an estimate of the average TLT temperature over the month. Unfortunately, the accuracy of these monthly gridded averages is limited by the differencing procedure. For both MSU and AMSU, the TLT retrieval is a weighted average of near-limb temperatures subtracted from a weighted average of temperatures measured closer to the nadir view. Because the measurements from the various views are not made at the same location on the earth, each TLT measurement is a combination of the desired vertical extrapolation and an unwanted spatial derivative of temperature along the scan direction. In the midlatitudes and tropics, the effect of these spatial derivatives on the monthly gridded average is significantly reduced by cancellation of spatial derivatives in different directions, because the monthly average is a combination of right- and left-side views and ascending and descending satellite tracks. Near the poles, cancellation does not occur for the measurements made because the satellite velocity vector is close to being east–west. In this case, the north–south part of the derivative adds instead of canceling. The residual spatial derivative, combined with a large north–south gradient in the temperature, leads to significant errors in the retrieved TLT measurements. These can be reduced if the half scan located equatorward of the satellite is excluded from the average for polar latitudes. For a given latitude, the scanning direction for half scans that come from the equatorward half of their respective scans is closer to being north–south and thus contains a larger gradient-related error. We exclude this half scan completely for latitudes above 60°, and reduce its contribution with a linear latitude-dependent weighting factor for latitudes between 50° and 60°.

d. Instruments studied

In this work, we have investigated the use of the data from the nine MSU instruments (ending the use of the NOAA-14 data after December 2003) and the AMSU instrument on NOAA-15. We have determined that AMSU channel 5 on the NOAA-16 platform suffers from significant unexplained drifts (Mears and Wentz 2009). The premature malfunction of the AMSU instrument on the NOAA-17 platform yields a time series that is too short to contribute significantly to a long-term dataset. Because two instruments (MSU on NOAA-14 and AMSU on NOAA-15) continued to operate after the NOAA-17 failure, its use would bring little new long-term information to the data product. We have not yet included data from the NOAA-18, Meteorological Operation-A (MetOP-A), or Aqua satellites in our analysis.

a. Premerge adjustments

1) Incidence angle adjustments

Although the viewing angle of each observation ideally depends only on the view number, the earth incidence angle also depends on the height of the satellite above the earth’s surface and the local curvature of the earth. Both these factors change during a single orbit of the satellite due to both the oblate shape of the earth and the elliptical nature of the satellite’s orbit. In addition, the average height of the satellite tends to decay over time because of atmospheric drag on the satellite. These changes in incidence angle substantially change the pathlength of the near-limb views through the atmosphere, and thus can affect the weighting function and the measured brightness temperature. This is particularly important for TLT because weighted differences are used to retrieve the lower-tropospheric temperature. To minimize this effect, each observation is adjusted to correspond to a nominal incidence angle corresponding to its field of view. This procedure implicitly removes the effects of the decay in orbital height (Wentz and Schabel 1998). A first-order adjustment is made using simulated brightness temperatures calculated from an atmospheric profile climatology based on the NCEP reanalysis (Kalnay et al. 1996; Mears et al. 2003). This part of the adjustment is performed by the middle two terms in Eq. (5). For MSU channel 2, we found that we had to include an additional term that was well-modeled as a time-dependent instrument roll (Mears et al. 2003) to remove a persistent bias between measurements on opposite sides of the swath. For AMSU channel 5, we found that after performing the model-based adjustment, an additional empirical correction T₀(fov) for each field of view (not well described by an instrument roll) was needed to remove residual cross-track biases (Mears and Wentz 2009).

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We define the empirical correction T₀(fov) to be independent of location on the earth and time of year. Because this adjustment is independent of time, it does not directly affect long-term changes in the resulting dataset. It does serve to slightly reduce the variability in intersatellite differences, and thus may improve the accuracy of our merging procedure.

b. Determination of target factors

Differences between globally averaged measurements made at the same time by different MSU and AMSU instruments can have a component that is strongly correlated with the temperature of the warm calibration target of one or both satellites, suggesting that there may be a calibration error that is dependent on calibration target temperature. This effect was first noticed by Christy and coworkers (Christy et al. 2000). Possible causes for these calibration errors include residual nonlinearity in the radiometer response that was not adequately measured during ground calibration or an error in the specification of the effective brightness temperature of the calibration sources. Both these error sources imply an additional error that is dependent on the scene temperature (Mears and Wentz 2009). Our empirical error model that includes both target temperature– and scene temperature–dependent errors is given by

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where T₀ is the true brightness temperature, A_i is the temperature offset for the ith instrument, α_i is a small multiplicative “target factor” describing the correlation between the measurement error and the temperature of the hot calibration target, and T_TARGET,i is the target temperature anomaly for the ith satellite. The “scene factor” β_i describes the correlation between the measurement error and the scene temperature T_SCENE; ɛ_i is an error term that contains additional uncorrelated zero-mean errors resulting from instrumental noise and sampling effects. In our previous work, we found that the target factors α_i are necessary to accurately match the overlapping observations for the MSU instruments, which confirms the earlier results of Christy et al. (2000), though we sometimes find different numerical values for α_i.

We determine the target factors by using an analysis of globally averaged (50°S–50°N) TLT observations adjusted for satellite height and measurement time effects. As was the case for the simple nonextrapolated MSU–AMSU datasets, we choose to perform this calculation separately for the MSU and AMSU sets of satellites (Mears and Wentz 2009) because of small seasonal differences between the MSU and AMSU data.

For MSU, we closely follow the method described in Mears and Wentz (2009). For each month when two or more satellites are observing simultaneously, we form an equation by taking the difference between versions of Eq. (6) for each satellite pair,

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thus eliminating the true brightness temperature. For now, we ignore the scene temperature term, which will be discussed in section 4d. For MSU TLT, this results in a system of equations with 182 equations in 17 unknowns (8 offsets and 9 target factors; one offset, A_NOAA-10, is arbitrarily set to zero to prevent a singular system of equations). The system is then solved using singular value decomposition to find the target factors. The offsets values are discarded. The final values for the offsets will be determined as a function of latitude in the next step. The values of the target factors determined by this calculation are tabulated in Table 2.

The deduced value for the NOAA-9 target factor (0.0691) is significantly larger than the corresponding value we found for MSU TMT (0.0362). The TMT target factors were calculated using a simple average of near-nadir views, which is in contrast to the nadir-limb differencing used to construct the TLT temperatures. The differencing procedure amplifies any noise present so that we expect the noise in the TLT global averages to be larger than the TMT averages (Mears and Wentz 2005). Under the assumption of uncorrelated noise in each FOV, the noise in TLT averages is given by the quadrature sum of the noise in each FOV weighted by the absolute value of the weight for each FOV; for MSU, it is 55/4 ≅ 5.567 times larger than the noise in MSU TMT. Furthermore, investigation of the covariance matrix for the regression reveals that the NOAA-9 target factor is poorly determined because of the short overlap period with other satellites and the relatively small seasonal-scale fluctuations in the NOAA-9 target temperature. Taken together, these facts lead us to suspect that the fitted value for the NOAA-9 target factors is too large because of overfitting of noise. Examination of the regression results suggests that there are solutions that are nearly as good as the best fit but with smaller values of the NOAA-9 target factor. To reduce the values of the target factors, we add nine ad hoc equations to the regression of the form

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In our overdetermined system, these equations have the effect of “pulling” the target factors toward zero, with the amount of the pull determined by the constant C. If a given target factor is well determined by the other equations in the regression, the effect of these additional equations will be small. On the other hand, if a target factor is poorly determined (and thus subject to overfitting) its absolute value will be reduced significantly. When C is set to 1.0, the absolute values of the target factors for TIROS-N and NOAA-9 are reduced substantially, with little effect on the target factors for the other instruments. The values for the target factors with C set to 1.0 are shown in the third column of Table 2. We choose to use these “magnitude reduced” target factors to adjust the TLT data because the NOAA-9 target factor (0.0486) is closer to the value found for TMT. We note that this is a subjective decision that has the effect of increasing the final global trend by about 10%. For comparison purposes, we also show the TLT target factors we used in our earlier work (Mears and Wentz 2005) and used by Christy et al. (2000). Our current NOAA-9 target factor is larger than its earlier value but still much smaller than the value found by Christy et al. (2000). This difference is an important contributor to the differences between our respective datasets and an important source of “structural” uncertainty, the uncertainty that arises because of the choice of methodology (Thorne et al. 2005a).

As noted above, we have found that channel 5 for NOAA-16 is contaminated by an unexplained drift in measured temperature. This complicates the determination of the target factors for NOAA-15. Following Mears and Wentz (2009), we use detrended versions of the measured temperature and the target temperatures to find the value of the NOAA-15 TLT target factor to be 0.004, a value so small that it is unimportant to the final results.

c. Latitude-dependent offsets

Since developing the MSU-only version of our dataset (Mears and Wentz 2005), we have determined that zonally averaged intersatellite differences can be substantial when the constant terms (A_i) are held constant as a function of latitude. This led us to extend our empirical model so that the values of A_i are allowed to smoothly vary with latitude, which is similar to the approach used by Christy et al. (2000). To determine the latitude dependence of the offsets, we simultaneously solve a system of equations given by

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This equation is a version of Eq. (6) generalized so that each equation describes the difference between measurements made by the ith and jth satellites for the kth zonal band, where the values of A_i,k are allowed to vary with latitude. We performed the calculation for 2.5° wide bands and formed an equation for each latitude band and month where two or more satellites were observing simultaneously. The target factors α_i were set to the values found in the previous step. Singular value decomposition was used to determine the solution to the system of equations. The use of singular value decomposition finds the minimal variance solution for the offsets so that we change the measured temperatures by the least possible amount. To reduce the effects of noise on the retrieved offset values, we smooth the offsets in the north–south direction using a “boxcar” smooth with a width of 12.5°. Figure 5 shows the smoothed offsets as a function of latitude for each of the nine MSU satellites used in this study. The differences between the MSU and AMSU measurements are too complex to be described by latitude-dependent offsets and are addressed separately in section 3e.

d. Scene temperature–dependent errors

When we apply the target factors and offset determined in the previous steps to the data and evaluate the intersatellite differences, we find significant seasonal-scale fluctuations near the poles, where the seasonal cycle is large, but not near the equator, where the seasonal cycle is small. This suggests that part of the remaining differences is caused by a scene temperature–related calibration error. To attempt to correct for this error, we again take the difference between versions of Eq. (6) for each month that two or more satellites are observing simultaneously. Substituting the values already determined for A_i,j and the α_i into

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and keeping the T_SCENE dependence from Eq. (3), we obtain a system of equations given by

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for each zonal band. We can replace T_SCENE,i,k and T_SCENE,j,k with T_SCENE,k because the scene temperature is a property of the earth and thus independent of the satellite index. Here, T_SCENE is closely approximated by the measured antenna temperatures. We use an average value for T_SCENE found by averaging the results from all satellites over 1979–98 to form a TLT climatology that depends on latitude and month. These values are then used in the system of equations described by Eq. (11) to deduce the values for β. Because β_i only appears in the equations as the difference between the values of β for different satellites, its average value is arbitrary. We use singular value decomposition to choose the minimum-variance solution for β_i. We report the values of β for each satellite in Table 2.

To display the effects of these various intercalibration steps, we show in Fig. 6 the color-coded time–latitude plots of the intersatellite differences at different stages in the intercalibration process for an example pair of satellites, NOAA-11 and NOAA-12. Figure 6 shows that the reduction in intersatellite differences for these two satellites is quite small for the scene temperature step. We retain this step in this case for the sake of consistency with other satellite pairs and our other MSU-derived datasets. Because the scene temperature factors are fairly small, they have only a small effect in the tropics and lower midlatitudes, where the seasonal cycle is much smaller than near the poles. In the bottom panel of Fig. 6, we show time series of near-global averages (50°S–50°N) of the monthly differences between NOAA-11 and NOAA-12 for both the raw antenna temperatures and after all adjustments were applied. A comparison of this plot to the corresponding plots for the near-nadir views (Mears and Wentz 2009, their Fig. 6) shows that additional noise is present in the fully adjusted differences for TLT, as we expect because of the noise amplification that occurs during the calculation of the weighted FOV differences.

e. Location- and month-dependent offsets between MSU TLT and AMSU TLT

The AMSU FOV weights determined in section 2b match the results for MSU TLT and AMSU TLT as best as possible for average atmospheric and surface conditions. However, the match is not perfect and small differences remain. These differences are caused by a combination of 1) small differences in the temperature weighting function coupled with local values of the vertical structure of the atmosphere and of the emissivity and temperature of the surface and 2) different amounts of spatial smoothing and spatial derivative inadvertently accomplished by the FOV weighting for the two satellite types. To minimize the effect of these errors on the MSU–AMSU merging procedure, we need to find the average difference between MSU and AMSU TLT data and use this average to adjust the AMSU data so that they correspond to the MSU data before merging the two datasets together. We will perform this adjustment as a function of earth location and time of year to remove the most important part of these errors.

In Fig. 7, we plot the mean difference [NOAA-14 (MSU) minus NOAA-15 (AMSU)] in TLT brightness temperature for the month of June. Data from 1999 to 2003 are included in the average. The most obvious differences occur near the edges of continents—these are due to the different spatial smoothing and spatial derivative caused by using the weighting average of different FOVs. Similar maps are computed for all other months, and a two-harmonic seasonal fit is calculated at each point. These differences are used to adjust the AMSU data so that they match the MSU data. In practice, we use maps of the parameters of harmonic fits (i.e., constant offsets plus the amplitudes and phases of the seasonal cycle) to the seasonal cycle at each point to generate an adjustment for the AMSU data that is dependent on location and month.

f. Merging data from different satellites

After all the adjustments are applied to the MSU data, we evaluate the intersatellite differences for any remaining problems. As was the case in the simple, nonextrapolated case, we found that several satellites had averages for several months that appeared to be anomalously high or low (Mears and Wentz 2009). These typically occurred near the beginning or end of each satellite’s lifetime and were often associated with months where several days of data were missing (causing sampling errors) or with times when data quality was noted to have deteriorated by the satellite operations team at NOAA (Goodrum et al. 2000; Kidwell 1998). In other cases, no cause could be identified. These spurious months, which are listed in Table 3, were excluded from further processing. The data from different MSU satellites were then combined by using simple averaging when data from more than one satellite were present. For AMSU, only one satellite is currently used; therefore, no merging is necessary to produce an AMSU-only dataset. Data from the two satellite types are then combined by again using simple averaging when data from both satellite types are present. A record of which satellites are used for each month is kept and propagated through subsequent steps to become part of the final data product.

a. Comparison with the UAH TLT dataset

The UAH MSU–AMSU TLT dataset is made using the same raw measurements as our dataset, so any differences between the two datasets are the result of differences in methods used to construct the long-term climate data record. In Fig. 8, we plot the global and tropical anomaly time series for each dataset. On short time scales, the two datasets are very similar, as might be expected because they both originate from the same satellite measurements. Over longer periods, some small differences become apparent, with the RSS dataset showing more warming for both the global and tropical averages. The cause of these differences is likely to be a combination of several factors, including different quality-control algorithms, different values for the target factors (see Table 2), and the different methods used to perform the diurnal adjustment. In particular, the different target factor for NOAA-9 causes the differences in the 1985–87 period, whereas differences in the diurnal adjustment and the NOAA-11 target factor combine to produce the changes between 1991 and 1996. For both global and tropical trends, the differences between the two datasets are smaller than the estimated 2σ error in our earlier MSU-only TLT trends (Mears and Wentz 2005). The trend error estimates in Mears and Wentz (2005) were obtained by estimating the uncertainty in the merging parameters and the diurnal adjustment. Although these earlier estimates do not yet include the effects of including AMSU data in our procedure, they can serve as a first guess of the magnitude of the error. The trend differences between the RSS and UAH results are due to differences in merging procedure and primarily occur at time scales longer than one year. For this reason, the uncertainty in trend difference that can be inferred from the standard deviation of the difference time series is much smaller than the observed difference in trends.

In Fig. 9, we plot the 1979–2007 TLT trends as a function of latitude for both datasets. Each dataset shows the same general pattern, with little warming (or even cooling) south of 50°S and the largest warming in the Northern Hemisphere polar regions. The RSS data show significantly more warming in the tropics than the UAH dataset. Because the effects of the target factors are nearly constant with location, this difference is likely to be due to differences in the diurnal adjustments and the effect of these differences on the latitude-dependent offsets.

In Fig. 10, we show color-coded maps of 1979–2007 trends for both datasets and their difference. The overall patterns are very similar, though the UAH trends are noticeably smoother in the east–west directions because of their spatial smoothing procedure (Christy et al. 2000, 2003). The difference map suggests that much of the differences between these datasets are a function of latitude, again pointing to differences in the diurnal adjustment as the primary cause.

b. Comparison with homogenized radiosonde datasets

It is well established that temperature measurements made by radiosondes contain numerous inhomogeneities resulting from changes in instrumentation and observing practices over time. Before radiosonde measurements can be used to describe long-term changes, these inhomogeneities need to be characterized and removed to the largest possible extent. A number of groups have produced homogenized radiosonde datasets. For comparison with our satellite measurements of TLT, we chose four of the most recent homogenized datasets. These datasets are either available as gridded TLT measurements or contain enough information so that it is possible for us to construct a gridded TLT dataset. These datasets were constructed using automated methods to find and estimate the size of “breakpoints” in each radiosonde’s time series, which are then used to create adjusted versions of the radiosonde data with the effects of the detected breakpoints removed. We list each radiosonde dataset below.