a. Radiative transfer model

The radiative transfer model used, as well as the retrieval algorithm, is based on the models of Wilheit (1990) and Al-Khalaf (1995). The radiative transfer equation is approximated by dividing the atmosphere into layers of 200-m thickness up to a height of 20 km. Each layer is characterized by its temperature, pressure, and relative humidity at the center of the layer. Temperature and relative humidity are assumed to be linear with height between layers. A log-linear interpolation for pressure is used for the radiative transfer computations. The absorption coefficient of each layer can be determined by the concentration of absorbers in the layer. The principal absorbing constituents important at the SSM/I and SSM/T-2 frequencies are water vapor and hydrometeors; oxygen is of marginal importance. The retrieval algorithm allows for the presence of cloud, but the presence of large amounts of rain cannot be modeled accurately given our assumptions. In practice, the presence of rain usually causes a large estimated uncertainty in the retrieval.

The radiative transfer model is based on the method of Chang and Wilheit (1978). Surface reflectivity for a smooth ocean surface is computed via the Fresnel relations using dielectric constant data from Lane and Saxton (1952). Absorption for liquid cloud drops is handled using the Rayleigh approximation.

b. Algorithm overview

The humidity profile is represented by an array of relative humidities {R_j} at a set of up to 10 levels. Relative humidity is assumed to be linear with height between these levels and constant below the lowest level. The RH at 20 km is fixed at zero with a linear interpolation to the highest retrieved level. Note that since the temperature profile is held constant throughout the retrieval, specifying the relative humidity is equivalent to specifying any other moisture parameter, such as mixing ratio.

A flowchart for the retrieval algorithm is shown in Fig. 2. For an SSM/T-2 observation over ocean, a known temperature profile is taken from ECMWF analyses, spatially interpolated. A first-guess humidity profile is obtained using the method of section 4d. The T_B’s corresponding to this atmosphere are calculated for each of the channels. These are compared to the observed T_B’s by calculating a normalized brightness temperature error, C_j = Σ^M_i=1 [(T^ret_i − T^obs_i)/a_i]², where M is the number of channels used; T^ret_i and T^obs_i are the retrieved and observed brightness temperatures, respectively, for the ith channel; and a_i is the noise equivalent delta temperature (NEΔT) for the ith channel. If C_j < M, the measurements agree with the observations, on average, to within the noise level of the instrument.

A new moisture profile is calculated, solving for a set of changes to the relative humidities δR_j, which minimizes E in

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where N is the number of retrieved levels, b_j is the standard deviation of relative humidity for the jth level, R_j is the retrieved relative humidity for the jth level, and R_j is the mean relative humidity for the jth level. Note that the temperature differences are weighted by the NEΔTs and the RH differences by the standard deviation of layer RH, so each term is normalized in terms of the standard deviation of the appropriate parameter.

If the new humidity profile is supersaturated at any level, clouds are added to the model atmosphere as described in section 4e. Relative humidity means and standard deviations at each level were derived by Al-Khalaf (1995) from a global dataset of radiosondes.

At this point, the formal uncertainty of each component of δR (and R) can be determined by taking the root sum of squares of the appropriate terms in the equation that minimizes E. If the formal uncertainty at a level is not significantly less than the uncertainty of the relative humidity constraint (b_j), the layer is not important to the T_B’s and the statistics are dominant in selecting the retrieved RH. Otherwise, the physics is dominant (Wilheit 1990).

Next, the process is repeated beginning with the computation of normalized T_B error C_j. Note that while the new humidity profiles are calculated using both the observations and the statistical constraints, the value of C_j depends strictly on the agreement between retrieved and observed brightness temperatures. The statistical constraints are used to guide the choice of the next guess, but in determining the best profile of all the guesses so far, we use the physical error model without any statistics.

The process is continued, computing C_j for each iteration until C_j < 0.1 or for 25 iterations. For each set of observations, C is the best (minimum) C_j. If C < 1, the associated profile is taken as the retrieved profile. Otherwise, the algorithm is unable to bring the profile into agreement with the T_b’s and the retrieval is discarded. The yield of an algorithm is defined as the percentage of retrievals for which C < 1, or the percentage of retrievals that are considered to be “good.”

c. Retrieval heights

Because the level where a given SF peak occurs can vary from one observation to another depending on the actual water vapor profile, it is difficult to know a priori for which levels the brightness temperatures provide meaningful humidity information. This problem has been treated by making the retrieval levels as described in section 4c variable.

Wilheit and Al-Khalaf (1994) found that for each SSM/T-2 channel near 183 GHz, there is a nearly constant overburden (the integrated water vapor amount above a given level) above the height at which the atmospheric temperature equals the observed brightness temperature. This was the basis for a simplified algorithm, giving, for each of those channels, a height and a corresponding humidity.

The lowest retrieval level is set to 2 km below the lowest level from this single-height, single-channel algorithm, with the restriction that it is between 1 and 2 km. The upper bound of 2 km is used to ensure that there are retrieval levels low in the atmosphere, since the SF for TIWV (discussed in section 5b) or any semitransparent channel (e.g., 150 GHz) will peak at the surface with a scale length of about 2 km. The lower bound of 1 km is used because the algorithm cannot resolve features with a resolution below about 1 km. The highest retrieval level is set to 2 km above the highest level from the single-height, single-channel algorithm, with the restriction that it does not exceed 11 km. A constant spacing of 1 km between levels is used, adjusting the height of the highest retrieval level upward as much as 1 km to achieve this spacing. This method results in from 5 to 10 retrieval levels.

d. The first-guess profile

A climatological mean RH profile for ocean background (Al-Khalaf 1995) is used as a basis for the first-guess profile by initializing the values of R at each of the nine levels to

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where h is in km. The information from the single-height, single-channel algorithm of Wilheit and Al-Khalaf (1994) is then added for each of the three channels in turn by adding a Gaussian-shaped distribution to the RH profile, as follows:

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where R^fg-old_j and R^fg-new_j are the first-guess humidities at level j before and after taking the single-height, single-channel algorithm into account; h* and R* are the height and relative humidity, respectively, returned by the single-channel, single-height algorithm; h_j is the height at level j; R(h*) is the relative humidity at h* interpolated from the R_j; and the value of 2.0 km was chosen as an appropriate width equal to the typical spacing between the three levels returned by the single-height, single-channel algorithm. The effect of this is to give a first-guess profile that has an RH close to the values from the single-channel, single-height algorithm at the appropriate levels, but that approaches the climatological RH away from those levels.

e. Clouds

Cloud liquid water is added to the model atmosphere where the retrieved humidity is sufficiently high. If at any iteration, the new humidity profile has any levels for which RH > 95% and that do not already have cloud liquid water, cloud liquid water is added to the most humid of those levels. (Therefore, only one additional level is considered to be saturated per iteration.) When cloud liquid water is added to a level, it is considered to be saturated and the statistical constraint for that level is changed to RH = 100% ± 1%. Whenever a new level is considered to be saturated or a saturated level has a retrieved RH > 101%, 3 mg cm⁻² of cloud liquid water are added to the model atmosphere. If a saturated level drops below 99% RH, 1 mg cm⁻² cloud liquid water is removed. This cloud liquid water is spread evenly from 0.5 km below the lowest saturated level to 0.5 km above the highest saturated level. Thus, this algorithm implicitly retrieves a cloud top and a cloud base; the quality of these retrievals is not tested here.

a. SSM/I brightness temperatures

We can use the retrieval algorithm described in section 4 for any combination of channels. SSM/I observations are made within a few minutes of SSM/T-2 observations from the same satellite, so we can reasonably use measurements from both sensors together. Because of the different scan geometries, SSM/I and SSM/T-2 pixels will not correspond exactly. As a result, any attempt to introduce SSM/I observations into the SSM/T-2 dataset will effectively downgrade the horizontal resolution of the retrievals by increasing the area corresponding to a set of observations. It is important to minimize this effect. Retrievals were performed for each SSM/T-2 pixel, using interpolated SSM/I brightness temperatures at that point. For the 19- and 22-GHz channels of the SSM/I, which have field of view sizes on the order of the 183-GHz channels’ fields of view, the nearest observation is used. For the 37-GHz channels, a Gaussian-weighted average of all SSM/I observations centered within 50 km of the SSM/T-2 pixel is used.

A comparison experiment was done by performing six sets of retrievals. The first set was done with the three water vapor line channels centered at the 183- and 150-GHz channels. The next five sets were performed using these four channels plus one other channel from the SSM/I (19V, 19H, 22V, 37V, 37H). Results from this experiment are presented in section 7a.

b. TIWV pseudochannel

The possible inputs to the retrieval algorithm need not be limited to observed brightness temperatures. One possible parameter is the total integrated water vapor. The TIWV is defined as the height of water that would result if all the water vapor in a column of air were condensed into liquid. It may be expressed in units of length or mass per unit area. The TIWV can be calculated from SSM/I observations (Alishouse et al. 1990;Manning 1997). If we can define an SF for relative humidity with respect to TIWV, we can treat the TIWV as another channel in the retrieval formalism.

It is readily shown that the SF for TIWV is simply equal to the saturation density of water vapor so that the change in TIWV due to a given change in the relative humidity profile is given by

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where g^TIWV_R(h) = ρ_sat(h). The relative humidity SF with respect to TIWV, as well as four of the SSM/T-2 frequencies, is plotted in Fig. 1.

The retrieval algorithm was run using four SSM/T-2 channels (183 ± 1, 3, 7; 150 GHz) in combination with each of the five lower-frequency SSM/I channels (i.e., five separate times). The retrieval algorithm was also run for three sets of channel combinations as shown in Table 1. Results from this experiment are also presented in section 7a.

c. Transformation of data

It has been shown that if an SF with respect to absorption coefficient, γ, is known, it can be transformed into an SF with respect to relative humidity. The SFs can be transformed into any variable with which we can express the moisture profile. Let us define L = ln(RH) + 1. This transformation, which is approximately equivalent to expressing the water vapor content in terms of dewpoint depression, was chosen to reduce the nonlinearities present in the SFs with respect to relative humidity. The offset of one was chosen for ease of programming, as the upper limit of the parameter remains one (i.e., 100% RH). Rewriting the right-hand side of Eq. (2.1), we have

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we see that the SF with respect to L is

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where g_L(h) satisfies

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The reduction of nonlinearities can be seen by examining Fig. 3. Figure 3a shows the SFs for RH for the 183 ± 1-GHz channel for the U.S. standard atmosphere with three RH profiles (each constant with height). Figure 3b shows the same SFs for L. An increase in RH from 30% to 90% causes approximately a factor of 5 increase in the magnitude of the SF for RH, but only about a 20% increase in the magnitude of the SF for L. The SFs for L are more nearly constant for changes in RH because of the factor of RH(h) in Eq. (5.4), since the relative humidity SFs increase for decreasing humidity (i.e., a unit change in water vapor has a larger effect on the T_B at lower humidities).

In addition to reducing nonlinearities in the retrieval process, using the parameter L has the advantage of never returning a negative relative humidity. Retrievals were run in terms of both R and L; results are presented in section 7b.

a. Physical error modeling

Two methods were used to evaluate the accuracy of the retrieved humidity profiles. The first method is to use the normalized T_B error C, defined in section 4b. This value is a measure of how closely the observed brightness temperatures match the computed brightness temperatures of the retrieved humidity field. This variable is a good indicator of the physical consistency of the model, that is, of how well the retrieved humidity field reproduces the observed radiation field.

The physical error method has two shortcomings. First, it is only as good as the model. Errors in the spectroscopy can lead to errors in the predicted radiation field for a given humidity profile, and therefore cause the wrong humidity profile to be retrieved. Second, more than one humidity profile may yield a radiation field close to the observed one. There is no guarantee that the retrieved humidity profile is correct, even if the retrieved brightness temperatures match the observations exactly. This is a consequence of trying to solve for a continuously varying field given only a finite set of measurements.

b. Ground truth comparisons

The second method of validation is to compare the humidity fields to so-called “ground truth,” or another measurement of the humidity fields. Two sources of ground truth were used: radiosonde observations (obtained from the National Center for Atmospheric Research) and ECMWF analyses. This method has the advantage of providing an independent measurement of the humidity profile for comparison. However, the discrepancies in timescales and space scales between satellite measurements and either radiosonde measurements or analyses must be kept in mind.

It must also be noted that there is some error in the humidities reported by the radiosondes. In particular, upper-level radiosonde measurements of humidities are not always reliable. Data from many different types of radiosondes were used in this study. These instruments have been shown to have both random and bias differences when flown on the same balloon (Falcone et al. 1992). ECMWF analyses may be inaccurate not only due to measurement errors, but also due to sparsity of data in certain areas.

These validation methods can be used to verify that the retrieved humidity profiles are physically consistent with the observed brightness temperatures and that they are quantitatively similar to profiles obtained from more conventional measurements. We can have confidence in the retrievals if both of these conditions are met.

c. Case studies

While statistical results from a large number of cases are essential in assessing an algorithm’s performance, we may also gain insight into its performance by looking at individual cases. In the comparisons below, case studies are used as examples to illustrate specific features of each algorithm and to supplement the quantitative results.

a. SSM/T-2 data with SSM/I data

Retrievals were performed using SSM/T-2 data alone (the 183 ±1-, ±3-, ±7-, and 150-GHz channels), and by separately adding each of the five lower-frequency channels from the SSM/I. The same dataset of 1271 observations (chosen for their matches with radiosonde observations; the radiosonde locations are shown in Fig. 4) was used in each case. The yield and average normalized brightness temperature error (for all retrievals) for each set of channels are shown in Table 2. Once again, the yield is the percentage of SSM/T-2 pixels for which a good retrieval (C < 1) is found. It can be seen that the yield is not improved by the addition of any single SSM/I channel. The normalized brightness temperature error is slightly improved only by the addition of the horizontally polarized 37-GHz channel, possibly due to wind speed effects. The fact that the yield remains above 92% in each case is evidence that the SSM/T-2 and SSM/I measurements are consistent.

Next, the addition of the SSM/I-derived TIWV was considered. Retrievals were performed using three sets of channels: 1) the four standard SSM/T-2 channels (the three 183-GHz channels and the 150-GHz channel); 2) these channels plus the TIWV; and 3) the three 183-GHz channels and the TIWV, omitting the 150-GHz channel. This experiment used a dataset of 11 658 observations in the Pacific Ocean from approximately 1500 to 1700 UTC on 31 January 1993. Table 3 shows the yield, average normalized brightness temperature error, and average number of iterations needed to get a normalized brightness temperature error less than unity for each case.

From these results, it can be seen that the both the yield and the normalized brightness temperature error are improved by adding the TIWV pseudochannel. There is little difference between these values in cases II and III, which differ only in that case II includes the 150-GHz channel. The SFs of the 150-GHz and TIWV channels have similar shapes, with both having a maximum at the surface, so they contribute similar information. In fact, case II requires more iterations to reach a satisfactory answer, probably because the algorithm is trying to include information from these two channels, which may not be in complete agreement. The combination of channels used in case III is preferred over that in case II for two reasons. Using more channels increases the capture cross section for errors (i.e., the likelihood that the retrieval will be faulty due to bad data). Also, using case III caused the algorithm to converge faster. For these reasons, only cases I and III will be examined in more detail. Case I will be referred to as the SSM/T-2-only retrieval and case III will be referred to as the combined SSM/T-2 and SSM/I retrieval.

Figure 5 shows retrieved relative humidity profiles using SSM/T-2 data by themselves and with SSM/I data, along with ground truth profiles from the ECMWF analyses and radiosondes. The radiosonde measurements at 0.5 km and below are clearly in error. The figure shows an extremely dry atmosphere aloft with a shallow humid layer. The SSM/T-2-only retrieval was unable to resolve the humid layer, but when the SSM/I TIWV was added, the low-level humidity was resolved.

Figures 6 and 7 show the retrieved TIWV for four atmospheric layers over part of the Pacific Ocean for the SSM/T-2 and combined retrievals, respectively. The same product from the ECMWF analyses is shown in Fig. 8. In each of these figures, the ITCZ can be seen at about 5°N and the South Pacific convergence zone (SPCZ) at about 20°S. There is an extratropical cyclone spanning both swaths centered at about 50°S, another in the left swath centered at about 35°N, and another cyclone with a larger comma cloud centered over the Gulf of Alaska. The retrieved TIWV field shows more detail than the ECMWF field, particularly in resolving the cyclones. This might be expected because the ECMWF analysis of these regions is based on very sparse data, and also because of the 2.5° × 2.5° resolution of the ECMWF archived data (illustrated by the rectangles in Fig. 8).

In each plot of retrieved RH, a black area indicates that the normalized rms difference of retrieved and observed brightness temperatures was greater than unity. As can be seen by looking at the values for “yield” in Table 3, these areas are more extensive for the SSM/T-2-only retrieval. The locations of these pixels are in areas where precipitation and large amounts of cloud liquid water are likely: in the convective areas of the ITCZ and SPCZ and in the extratropical cyclone north of Hawaii. The difficulty in modeling these features accounts for the failure to converge in these locations. A clear shrinking of these areas can be seen going from Fig. 6 to Fig. 7, particularly in the ITCZ, SPCZ, and the cyclone just north of Hawaii. This demonstrates that the addition of the TIWV pseudochannel enables accurate retrievals further into areas of heavy cloud and perhaps light precipitation. Additionally, the combined retrieval appears to be smoother and more coherent, suggesting that it may be more accurate than the SSM/T-2-only retrieval.

Isolated cases are found where the SSM/T-2 only retrieval converges and the combined retrieval does not. Operationally, one method could be tried whenever the other fails in order to maximize the yield.

Figures 9a–d show the retrieved TIWV versus the TIWV as reported by radiosondes for four atmospheric layers and two sets of channels. In each figure, the top panel is for the retrieval using only the SSM/T-2, while the bottom panel is for the combined retrieval with the SSM/I TIWV instead of the 150-GHz channel. The same data are summarized statistically in Tables 4 and 5. Only data from retrievals with a normalized brightness temperature error of less than one (and the corresponding observations) are used. Since the set of these retrievals that are considered to be good varies slightly depending on the retrieval method, the statistics for the radiosondes are slightly different between the two tables. The fact that, in both cases, the rms difference between the retrieved and radiosonde TIWV is less than the standard deviation of either sample is evidence that the two variables are correlated. By comparing rms differences and correlation coefficients for the two cases, it can be seen that the combined SSM/T-2 an SSM/I retrieval gives better results than the SSM/T-2 with a greater improvement for levels closer to the surface. This can be verified visually by referring to Figs. 9a–d.

Tables 6 and 7 are analogous to Tables 4 and 5, but the ECMWF analysis layer integrated water vapor (IWV) has replaced the radiosonde layer IWV as the ground truth. Once again, rms differences and correlations are improved by adding the SSM/I TIWV, especially for lower layers.

b. Relative humidity variable transformation

Retrievals were performed on the same dataset from 31 January 1993 used above, with the standard retrieval method (solving for RH) and also solving for L using the transformation L = ln(RH) + 1. The channels used were the three 183-GHz channels and the SSM/I TIWV (case III).

Statistical results from this study are presented in Table 8. The retrieved TIWV for four layers is shown in Fig. 10. Qualitatively the retrieved RH fields look very similar to the combined retrieval when solving directly for RH. The areas where the retrieval fails to converge (in the ITCZ, SPCZ, and the cyclone north of Hawaii) are now even smaller (as compared to that in Fig. 7). As Table 8 shows, the normalized brightness temperature error has also decreased. This demonstrates that a data transformation can increase the yield over the retrievals performed in the previous section (at the expense of making the computations slightly more complex). Tables 9 and 10 show how well the retrieved profiles matched radiosonde and ECMWF data, and scatterplots comparing retrieved layer IWV with radiosonde layer IWV are shown in Fig. 11. By comparing rms differences from Table 9 and Table 5, there appears to be significant improvement for the TIWV and for the surface to 850-mb layer, slight improvement from 850 to 500 mb, and degradation for the 500–200-mb layer. Similar comparison of Table 10 with Table 7 rms differences shows degradation for all layers. From these ground truth comparisons, it does not appear that the data transformation enhances the retrieval algorithm’s accuracy. However, the yield is clearly increased. An operational algorithm could use the standard retrieval method for most cases, and use the data transformation when the standard method fails to converge in order to maximize both the yield and the accuracy of the retrievals.

a. Conclusions

The addition of SSM/I data to the water vapor profile retrieval method of Al-Khalaf (1995) for retrievals over ocean was investigated. It was found that by adding SSM/I channels individually, no improvement could be made in the normalized brightness temperature error. Next, a pseudochannel consisting of Manning’s (1997) TIWV from SSM/I was added to the mix. Its addition was found to improve the normalized brightness temperature error. When using the TIWV pseudochannel, the 150-GHz SSM/T-2 channel could be removed from the computations with little impact on performance (and with the benefits of simplifying computation and reducing the capture cross section for errors).

The combined retrieval consisting of the three 183-GHz channels from the SSM/T-2 plus the SSM/I TIWV pseudochannel was further compared to the SSM/T-2 only retrieval using the three 183-GHz channels plus the 150-GHz SSM/T-2 channel. The combined retrieval performed better in terms of yield and in terms of normalized brightness temperature error. The RH from the combined retrieval also correlated more closely with ECMWF analyses and radiosonde measurements for the layers from the surface to 850 mb, from 850 to 700 mb, and from 700 to 500 mb. The combined retrieval also compared favorably to the NESDIS operational algorithm.

The pixels where the retrieval fails to converge are nearly always in an area of precipitation or high cloudiness. Consequently, the increase in the yield can be viewed as an enhanced ability to view into such areas. This is a benefit because it is precisely such regions where better observations are needed to forecast severe weather, whether simply to initialize a numerical weather prediction model or to gather data to increase our knowledge about storm systems.

Another benefit of using the TIWV pseudochannel is that the 150-GHz channel is not needed. This channel failed on the F-11 SSM/T-2 in June 1993, while the other channels continued to operate until at least March 1995. Humidity profile data for climatological studies of this time period can still be obtained by using the SSM/I TIWV instead.

A further improvement in the yield and the normalized brightness temperature error was made by using the transformation L = ln(RH) + 1. This transformation made the SFs more nearly linear. This method allowed retrievals even further into areas of heavy cloud.

b. Recommendations for future work

If a similar method to this were used to retrieve three-dimensional water vapor fields on an operational basis, it would be desirable to achieve the maximum yield possible. The yield could be increased by taking all the cases that failed to converge using the standard method, and trying other approaches such as changing the first guess (perhaps to a very dry or very humid situation). The data transformation L = ln(RH) + 1, which was found to improve the yield but slightly degrade the retrieval accuracy, could be used in cases where retrieving in terms of RH failed to converge.

The SSM/I could be used to provide constraints other than TIWV. Because the 37-GHz channels on that instrument are very sensitive to cloud liquid water, SSM/I data could be used as a check for clouds and also as a constraint on the amount of cloud liquid water. Cloud liquid water retrieval algorithms using SSM/I have been derived by Alishouse et al. (1990) and Manning (1997).