## Introduction

Data products derived from the National Oceanic and Atmospheric Administration (NOAA) Geostationary Operational Environmental Satellites (GOES) not only have applications in operational meteorology but also have potential applications in climate research. With the continuation of the GOES series of satellites and the availability of previous GOES data, long-term climatologies of land and sea surface temperature [skin temperature (ST)] and precipitable water (PW) over much of the Western Hemisphere are possible. The monitoring of climate variations and trends would be a direct application of these climatologies. GOES high-temporal resolution, lacking on polar orbiting sensors, also makes it a potential source of data for diurnal variation studies of ST and PW and associated model parameterizations. Also, because of GOES temporal and spatial resolution, its retrieved products can be a useful complement and validation for other measurements in field experiments that provide data for evaluating and validating algorithms and instruments. The potential for GOES-derived data products to be useful in climate research is evident. However, an understanding of the accuracy of GOES retrieval algorithms and the limitations of retrieval methodologies is lacking. Further study is needed to understand the usefulness of these data products.

In this paper a split-window retrieval algorithm that retrieves ST and PW from GOES infrared (IR) radiance measurements is investigated. The retrieval technique used in this study is the Physical Split Window (PSW) technique (Jedlovec 1987) that was applied to Visible and Infrared Spin Scan Radiometer (VISSR) Atmospheric Sounder (VAS) data in a case study by Guillory et al. (1993) and performed well in an algorithm comparison study by Knabb and Fuelberg (1997). The purpose of this evaluation was to determine the expected errors in retrieving ST and PW as a result of the limitation of the algorithm itself due to approximations and constraints imposed by its formulation. Retrieval errors resulting from this source will be referred to here as algorithm performance errors.

Errors in retrieved quantities from GOES measurements can be the result of several error sources, such as 1) algorithm performance errors, 2) errors in the observed radiances such as instrument random noise, 3) surface emissivity estimation errors, and 4) cloud contamination. Comparisons of retrieved quantities with ground truth data can give an estimate of the expected retrieval algorithm accuracy associated with the combined effect of these errors sources. However, it is useful to understand the relative importance of the individual sources of error and their sensitivity to different environmental conditions. From this understanding an optimal application of the algorithm can be determined in order to improve the accuracy of the retrieved data products. It is sometimes difficult, though, to separate out the contributions of the various error sources and perform sensitivity studies when ground truth data is limited. In the case of determining PW retrieval accuracy, ground truth data is routinely available from radiosondes twice a day. However, the temporal resolution of the radiosondes may not be adequate to evaluate diurnal trends in the retrieval errors and provide insight into the algorithm diurnal performance. Moreover, ground truth data for ST is not available routinely and thus, the evaluation of ST retrieval performance using satellite observations is limited.

To investigate the PSW algorithm performance under various conditions without the influence of other error sources, a controlled experiment was performed using simulated GOES radiances. Since rawinsonde data is too limited in temporal and spatial extent to be used as ground truth for this particular investigation, model data was used as a surrogate for the real atmosphere, thereby providing the necessary ground truth. This approach provides an excellent dataset for the experiment because of the continuity of that data in the model. The modeled data used in this study was generated by the Limited Area Mesoscale Prediction System (LAMPS) described by Chang et al. (1981) and Perkey (1976). The LAMPS model geophysical parameters served three purposes in the investigation: 1) to develop synthetic VAS and *GOES-8* split-window channel observations, 2) to act as ground truth for retrieval verification, and 3) to provide a first-guess field for the retrieval technique.

The evaluation applied a retrieval methodology similar to that which might be used in actual applications of the algorithm for climate analysis studies. Several algorithm performance issues are addressed, including the 1) impact of the time of retrieval (diurnal effects), 2) sensitivity to the first-guess field (required by the algorithm), and 3) sensitivity to the split-window channel characteristics of the *GOES-7* VISSR VAS and the *GOES-8*/*9* imager and sounder. Since GOES temporal resolution makes it ideal for studying ST and PW diurnal trends, an evaluation of the diurnal performance of the retrieval algorithm will provide an assessment of its potential to discern these trends. The algorithm’s sensitivity to the first-guess field is also an issue because first-guess information for climate studies may be limited to the availability of archived traditional datasets, such as rawinsondes provided at 1200 and 0000 UTC, and model analyses available two to four times a day. Performing retrievals that are not coincident in time with the first-guess field may degrade retrieval performance and create biases in the retrieved parameters. Also important is the understanding of retrieval performance differences associated with using observations from different GOES sensors. Since *GOES-7* and *GOES-8*/*9* have slightly different split-window channel characteristics and spatial resolution, retrieval performance may be different for each of these sensors or channel combinations. An understanding of these performance differences is needed to discern artificial trends that may appear in climatologies as a result of using retrieval products from more than one GOES sensor or channel combination.

## Background

### Sensor characteristics

In April 1994 the first in a new series of geostationary satellites (*GOES-8*) was launched into orbit. This new series of satellites that carry advanced instrumentation are making significant advances in short-term weather prediction and climate research (Menzel and Purdom 1994). The new satellites have significant advantages over their predecessors, the GOES VAS series. First, the three-axes stabilized platform allows for more effective viewing of the earth and its atmospheric processes. Second, the satellites employ separate multispectral scanners for imaging and sounding functions, thereby allowing for simultaneous operations of these functions. Each instrument has an increased number of spectral channels, which allows for additional meteorological applications. Third, the sensors provide better spatial resolution over the GOES VAS. Detector and filter improvements and the three-axes stabilized observing mode yield better radiometric performance. The improved radiometric performance along with the higher spatial resolution should have a positive impact on the retrieval of surface and atmospheric parameters.

The *GOES-8* imager has five spectral channels: one visible and four infrared channels—two of which are spectrally positioned in the longwave IR window region. The horizontal resolution of the imager is 1 and 4 km at nadir for the visible and IR window channels, respectively. The water vapor channel (6.7 *μ*m) resolution is 8 km at nadir. In the routine operation mode, the imager provides full-disk coverage every 3 h and coverage of the contiguous United States (CONUS) every 15 min (Menzel and Purdom 1994). The *GOES-8* sounder has 18 IR channels plus a low-resolution visible channel. Three of the channels are in the longwave IR window region. In the routine mode, the coverage of the sounder is primarily of the CONUS and adjacent ocean areas every hour at a horizontal resolution of 10 km at nadir.

The split-window retrieval technique being evaluated in this study utilizes the atmospheric longwave window channels of the GOES instruments, which are weakly sensitive to differential absorption of water vapor in the 10–13-*μ*m range. Figure 1 presents a visual comparison of the *GOES-8* imager and sounder channel bandwidths and their position with respect to the transmittance spectrum in the thermal IR window region of the atmosphere. The split-window bandwidths for VAS and the Advanced Very High Resolution Radiometer (AVHRR) on board the NOAA polar orbiting satellites are shown for comparison. For a typical atmosphere the window region is moderately transparent to upwelling radiation. Water vapor is a weak absorbing constituent throughout the region and increases in strength with increasing wavelength. In the absence of water vapor, transmittance of the column is near unity except for the ozone band at 9.6 *μ*m and carbon dioxide absorption at 12.7 *μ*m and beyond. The *GOES-8* imager window channels (10.7, 12.0 *μ*m) span the window region sensing the differential absorption characteristics of water vapor. This is the water vapor signal that is to be recovered in the split-window retrieval process. The imager channels are similar to those of AVHRR with the 12-*μ*m channel being shifted to slightly longer wavelengths. The three *GOES-8* sounder window channels (11.0, 12.0, and 12.7 *μ*m) provide narrower bandwidths than those of the *GOES-8* imager and VAS. The position of the 11- and 12.7-*μ*m channels are near those of VAS but are shifted to longer wavelengths when compared to the imager channels. The 12.0-*μ*m sounder channel has a similar width and does not overlap the 11- and 12.7-*μ*m channels.

The varying positions and widths of these channels make significant differences in the observed brightness temperatures. The effect of these brightness temperature differences on the ability to reliably retrieve ST and PW is evaluated in this paper. The evaluation focuses on the *GOES-7* VAS channels, typically available in the multispectral imaging mode, and the *GOES-8* imager channels. The interest in exploring the *GOES-8* imager assets, as opposed to the sounder’s, is that its full-disk images provide coverage of much of the Western Hemisphere, whereas the sounder’s coverage, as stated above, is much smaller and is usually limited to the CONUS. Utilizing the imager data necessitates the use of a split-window retrieval technique.

### Algorithm description

The PSW retrieval technique was originally developed to retrieve ST and PW from aircraft data. Details of this early formulation may be found in Jedlovec (1987). An initial application of the technique to GOES VAS data was done in a case study by Guillory et al. (1993). This study showed significant promise for the PSW technique to retrieve total PW over local and regional areas with a constant first guess fixed in space and time. The rms retrieval errors for PW were typically less than 20% of the total PW amount. This performance is similar to the performance of other retrieval algorithms for VAS data (Chesters et al. 1983, 1987; Kleespies and McMillin 1990). In a recent study by Knabb and Fuelberg (1997), the PW retrieval performance of three algorithms were compared. These included the PSW technique, the Chesters technique (Chesters et al. 1983, 1987), and the operational simultaneous physical algorithm (Hayden 1988) of the National Environmental Satellite, Data and Information Service. The study entailed performing PW retrievals using simulated *GOES-7* VAS radiances calculated from 84 radiosonde profiles obtained over several case days from 20 sites over North America. The PSW technique performed as well or better than the other techniques in many of the sensitivity analysis cases. Overall, the PSW technique demonstrated a smaller bias and standard deviation in retrieval errors than the others. Also, the correlation of the retrievals with the observed data was higher.

In this study a more generalized PSW algorithm is used. It incorporates the basic technique used above but expands its application to handle various sets of split-window observations from aircraft and satellite platforms. The generalized PSW algorithm used here incorporates a spatially varied multiple first-guess methodology (described later) and the ability to use two or more IR window channel observations employing a matrix solution. These changes allow the algorithm to be applied to regional and hemispheric studies where temperature and moisture vary considerably in space and to new satellite sensors such as *GOES-8*/*9* with varying split-window capabilities. The technique is similar to that used by Hayden et al. (1996) to produce derived product images (DPI) from the *GOES-8* imager. However, the PSW technique uses only the longwave window channel observations and does not apply a bias correction to the observed radiance data.

The PSW technique is derived from a perturbation form of the radiative transfer equation that is simplified through parameterization to retrieve bulk layer parameters rather than profile information. The technique uses at least two split-window channel observations to simultaneously solve for perturbations of PW and ST from an initial guess value. The physical approach requires a priori information, which includes estimates of temperature and mixing ratio profiles, PW, and ST. From the temperature and mixing ratio profiles, channel transmittances and brightness temperatures are obtained and used in the solution equations.

*δT*

_{s}and total precipitable water

*δU*

_{ps}from a mean or first-guess state, assuming an emissivity of unity can be written as

*δI*

_{I}

*CδT*

_{s}

*D*

_{D}

*δU*

_{ps}

*δI*is the difference between the radiance observed at the satellite and the radiance calculated from a first-guess temperature and moisture profile representing the observed scene. The derivation of (1) is given in the appendix. The coefficients

*C*and

*D*are defined aswhere

*τ*and

*B*are the transmittance and Planck function, respectively, for a wavelength domain associated with an IR sensor window channel. The quantities

*T*and

*p*are the temperature and pressure, respectively. The overbar denotes a first-guess quantity and the subscript“ps” denotes that the atmospheric parameter is evaluated at the surface pressure. Thus,

*T*

_{ps}is the guess surface air temperature, while

*T*

_{s}is the guess skin temperature. The quantities ε

_{D}and ε

_{I}are residuals [see Eqs. (A15) and (A16)] involving functions of unknown perturbations or departures between the first-guess and observed atmospheric temperature

*T*(

*p*) and vertical moisture structure, and also functions of the cross products of

*δT*

_{s}and

*δU*

_{ps}and higher-order terms of these perturbations. The derivation of (1) is the same as in Guillory et al. (1993), except here the assumptions and approximations made in the derivation are explicitly defined in the terms ε

_{I}and ε

_{D}so as to demonstrate their effects on the algorithm’s retrieval performance.

_{I}and ε

_{D}are significantly smaller than

*δI*and the coefficient

*D,*respectively, and thus are set equal to zero. The resulting equation becomes linear in the two unknowns

*δT*

_{s}and

*δU*

_{ps}. Solutions for

*δT*

_{s}and

*δU*

_{ps}involve formulating (1) for two different IR window sensor channels and then solving simultaneously resulting inwhere the subscripts

*λ*1 and

*λ*2 denote two different IR sensor window channels. The retrieved skin temperature

*T*

_{s}and total precipitable water

*U*

_{ps}is obtained from (3) by

To ensure that ε_{I} and ε_{D} in (1) are small, the following constraints must be placed on the first-guess temperature and moisture profiles. 1) The first-guess temperature profile is the same or very close to the observed profile, 2) the vertical moisture structure of the first-guess and observed profiles are the same, and 3) the magnitudes of *δT*_{s} and *δU*_{ps} are such that their products and higher values are small. When these assumptions are not strictly met, significant errors in the linearization of (1) occur, and ε_{I} and ε_{D} manifest themselves as errors in the solutions of *δT*_{s} and *δU*_{ps}. Thus, one objective of the paper is to quantify the effects of ε_{I} and ε_{D} on the retrieval performance of the algorithm and demonstrate their sensitivity to the first-guess quantities and sensor channel characteristics under various meteorological conditions.

Also note that there is an additional potential source of error in the application of the perturbation equation that is not accounted for in the terms ε_{D} and ε_{I} defined in (A15). The derivation of the perturbation equation assumes that the first-guess profile has the same number of pressure levels or atmospheric depth as the observed location profile. If the elevation of the terrain spatially varies at the retrieval location, it is possible that a first-guess profile representative of a location some distance away may significantly differ in elevation. If the first-guess profile is not adjusted for this terrain difference, an error in the retrieval may occur. In this analysis, the first-guess profile was not adjusted to account for this terrain elevation difference. The reason for not including this terrain adjustment in the current analysis is that earlier versions of the PSW algorithm did not include this adjustment and it was desirable to use a version consistent with that used in the analysis of Guillory et al. (1993) and Knabb and Fuelberg (1997).

## Methodology

### Case study description

This investigation used a case study that was representative of summertime conditions over the east-central United States. The case study consisted of two days, 17–18 June 1986, of the Cooperative Huntsville Meteorological Experiment (Dodge et al. 1986). Surface conditions over the region on the morning of 17 June were characterized by a weak east–west cold front extending from Virginia into Missouri, which advanced southward during the day. Convergence of moisture-rich air ahead of the front, along with a region of localized instability, led to isolated intense thunderstorm activity over the Tennessee River valley. At the upper levels, anticyclonic flow dominated the central United States with a closed cyclonic circulation located over northwest Texas. Dry conditions were associated with the anticyclone, with dry air stretching along the Great Lakes and also over the eastern plain states. These conditions provided an interesting case because of the significant variation in PW and the somewhat limited amount of cloud cover present. A thorough description of the meteorological conditions for this case study is given by Fuelberg et al. (1991).

### Model description

LAMPS is a 15-level hydrostatic primitive equation model with options for different grid spacings. A terrain-following sigma–height coordinate system is used with good vertical resolution in the lowest 2 km (five atmospheric layers). The model’s prognostic variables include wind, specific humidity, potential temperature, and rain and cloud water. Vertical velocity is diagnosed from the model continuity equation with the time rate of change of pressure (*dp*/*dt*) set to zero at the top of the model (16 km). The diurnal cycle of the soil and sea surface temperature is not specifically forecasted but is parameterized based on seasonal amplitudes that vary over land and water. Horizontal finite differencing is fourth-order accurate, and time differencing is second-order leapfrog. Diffusion terms are applied to the prognostic equations to eliminate small-scale features that cannot be handled numerically. Time-varying lateral boundary conditions are used. Precipitation is partitioned into resolvable and convective modes with the latter being parameterized by a one-dimensional plume model described by Kreitzberg and Perkey (1976). The utility of LAMPS to accurately predict mesoscale features leading to convection has been well documented (Perkey 1980; Kalb 1985).

LAMPS was initialized with rawinsonde data from 1200 UTC 17 June 1986 and run to produce a 48-h simulation. Gridded fields of forecasted parameters for the first 24 h of the model run were used in this study. The spacing of the gridpoint data was 35 km and covered a 83 × 107 (2905 km × 3745 km) grid domain over the eastern two-thirds of North America. The PW values used for verification were calculated from the model gridpoint profiles of temperature and mixing ratio. The ST values were those prescribed by the model. Figure 2 shows the average ST and PW for the region over the 24-h period calculated from LAMPS; also shown is the model-calculated surface air temperature (top panel). Figure 2 shows that the domain-averaged PW field was fairly constant with time, showing little diurnal variability and a slight decrease over the 24-h period. The domain-averaged ST and surface air temperature does show a substantial diurnal variation with the ST, at times, dropping below the surface air temperature referred to here as a surface temperature inversion. This behavior demonstrates the realistic prediction of the surface–atmosphere temperature contrast by LAMPS. Figures 3 and 4 (top panels) show the LAMPS-calculated ST and PW fields used as ground truth. Clouds generated by LAMPS are also shown and depicted as white.

### Simulated radiances

The synthetic radiance fields for the instrument channels were derived using the profiles of temperature and mixing ratio, along with the surface pressure, temperature, and moisture at each LAMPS model grid point. The LAMPS profile information was interpolated to 40 pressure levels with the highest vertical resolution in the lowest 300 mb for accurate transmittance estimation. A spectroscopy algorithm (Weinreb et al. 1981) was used to generate atmospheric transmittance values for 40 layers from the top of the atmosphere to the surface, as defined by the LAMPS surface pressure values at each grid point. VAS and *GOES-8* imager and sounder split-window channel transmittances were obtained by convoluting the monochromatic transmittance profiles with instrument spectral response information. This was done at each level, for each model grid point, and for every time used in the study. The spectral transmittance profiles were then used in the radiative transfer equations to generate fields of synthetic radiances at a 35-km resolution for each sensor. Surface emission was represented by the Planck function that used the surface radiating temperature specified by the LAMPS model and an emissivity of unity. The unit emissivity assumption, although not always valid in nature, provides consistency with the retrieval process since the retrievals are made with this assumption as well.

The radiance fields (images) were simulated “noise-free”; that is, random instrument noise was not added to the simulated radiance data because of the concern that it would mask algorithm and sensor performance characteristics. The simulated radiance fields in the VAS split-window channels compared well with observed fields (not shown) and provided additional confidence in LAMPS forecast fields. Radiance differences occurred mainly in cloudy regions since the cloud forecast processes in LAMPS are somewhat limited. Subtle but observable differences occurred in the cloud-free regions that were due to the specification of the surface emission (land or sea surface temperature) in the model.

### Retrieval methodology

The ST and PW retrievals were made from the simulated IR images using a field of view of 35 km (one pixel) with a spacing of 70 km. This spacing produced, on average, about 2000 retrievals over the domain, depending on cloud cover. The approach approximated the resolution and spacing of retrievals typically produced in the operational and research environments (Hayden et al. 1996; Rao et al. 1998, in press). Retrievals were made using *GOES-8* imager split-window channels 4 (10.7 *μ*m) and 5 (12.0 *μ*m), and VAS split-window channels 7 (12.7 *μ*m) and 8 (11.2 *μ*m). Retrievals were also made for combinations of the *GOES-8* sounder window channels 6 (12.6 *μ*m), 7 (12.0 *μ*m), and 8 (11.0 *μ*m). To examine the diurnal effects on retrieval performance, retrievals were made every 2 h. Three retrieval cases were considered for the evaluation. Each case used a different type of first-guess selection that affected the retrieval performance. This was done to evaluate the algorithm sensitivity to various first-guess selections that may be used in actual retrieval scenarios. All cases obtained guess temperature and water vapor profiles from the LAMPS-gridded data that were used to create the simulated radiances but with substantially reduced spatial resolution in the form of a 2.5° × 2.5° grid. The surface air temperature of the guess profile was used as the guess for ST in the algorithm. The algorithm used the nearest guess gridpoint information at the retrieval location. The guess grid resolution was chosen to be consistent with global 2.5° gridded model analyses and reanalyses products (e.g., National Centers for Environmental Prediction, European Centre for Medium-Range Weather Forecasts, etc. analyses) that are readily available.

Case 1 retrievals were generated using a first-guess field in which the time of the first guess was the same as the retrieval time. In practice, this case would occur when radiosonde or model analyses are used as a first-guess field, and retrievals are made at the time the radiosonde or model analyses are valid. When retrieval times are not coincident with those of the first-guess fields, first-guess fields can be obtained by interpolating between the valid times. It was expected that by providing guess information valid at a time coincident with the retrieval time, better retrieval performance would result than when the guess and retrieval times differed. For this case, guess data was assumed available for all times of the day, thereby providing the best temporal guess possible for the purpose of determining the lower bound on the retrieval errors as a function of time of day. Also, by using a guess coincident in time with the retrieval, biases associated with retrieval sensitivities to the first-guess field are minimized allowing for possible diurnal trends in the algorithm performance to be discerned.

Cases 2 and 3’s retrieval methodology was the same as case 1 except the guess field was selected from the 0000 and 1200 UTC LAMPS model data and held constant (in time) for each retrieval time. These cases simulate retrieval conditions when radiosondes or model analysis data are used as a first-guess field to make retrievals at times when the guess and retrievals are not coincident in time. Since radiosonde and model analysis data are typically available at 0000 and 1200 UTC, these times were used as the guess fields for cases 2 and 3, respectively. The first objective for using a temporally constant guess was to identify trends in retrieval performance due to a degraded guess, which results when the retrieval time differs significantly from the time of the guess field. A second objective was to demonstrate the sensitivity of the retrieval performance to first-guess fields that characterize nighttime and daytime skin–air temperature contrast conditions.

Since the PSW technique does not retrieve emissivity, an a priori knowledge of the surface emissivity is required. In practice, usually an emissivity near 1.0 or an estimated value that is associated with the type of surface being observed is used for each channel (Hayden 1988). For this study a guess emissivity of 1.0 was used, which was the same emissivity used in simulating the observations. Thus, the retrieval performance presented below will not include effects due to incorrect emissivity assumptions. It is desirable to present algorithm and sensor performance characteristics unmasked by random noise and emissivity errors so as to establish a lower bound on the retrieval errors and discern any diurnal trends in the algorithm’s performance.

## Results

Statistics of the retrieval performance are presented for each of the three cases associated with the different first-guess field conditions. The retrieval performance is determined by the statistics of the retrieval errors. Here, retrieval error is defined as the model truth ST or PW minus the retrieved value. Four statistical values are presented: 1) the mean error (ME) or bias, which is the average retrieval error over the field; 2) mean absolute error (MAE), which is the average of the absolute value of the retrieval error; 3) standard deviation of the retrieval errors (SDE); and 4) the correlation coefficient (COR) of the true value of the parameter with the retrieved value. A second set of statistics is also calculated for the first-guess field used in the retrievals. The guess error is defined as the true value of the parameter minus the first-guess grid value used in the retrieval and is a measure of the quality of the first-guess field. Statistics as described above are obtained for the guess error. Retrieval performance can thus be measured with respect to the improvement over the guess field. A relative PW error (percentage) is also presented and defined as the retrieval error divided by the average true PW value over the field. This error is given in parentheses after the stated error in millimeters.

### Case 1 ST retrievals

The first case presented is for the condition where the time of the guess profiles is coincident in time with the retrievals. For this case it is assumed that guess data is available for all times for the purpose of determining the lower bound on the retrieval errors as a function of time a day. Examples of the *GOES-8* imager retrievals of ST and PW for this case are presented in Figs. 3 and 4. Considering the ST retrievals first, a comparison of the case 1 results (Fig. 3, middle panels) with the LAMPS model-calculated (true) values (Fig. 3, top panels) shows that the 0000 and 1200 UTC 18 June retrievals capture the ST fields very well.

The statistics for the ST retrieval errors for 0000 and 1200 UTC 18 June as well as other times beginning at 1200 UTC 17 June are presented in Fig. 5. In each panel the guess error statistic (read from the right axis) is also provided. Recalling that the guess grid surface air temperature was used as the guess skin temperature in the algorithm, the guess ME (top panel) is fairly large ranging from ±6 K. Note that the guess ME is essentially the difference between the model surface air and skin temperatures seen in Fig. 2. The retrieval ME (Fig. 5), however, is less than 0.1 K for the *GOES-8* imager and VAS, which is a considerable improvement over the guess for all retrieval times. The retrieval MAE and SDE having values of less than 0.5 K overall also show considerable improvement over the guess statistics that have values of almost an order of magnitude greater. The algorithm is able to remove the large biases in the guess ST values and improve upon the guess SDE by about 85%–95%. The retrieval COR (0.99) also exhibits a significant improvement over the guess COR (0.86–0.94). As suspected from the images depicted in Fig. 3, none of the statistics exhibit any significant trends associated with time of day. The slight trend of an increase in the retrieval MAE and SDE at 2000 and 0800 UTC appears to correspond with the maximum difference between the mean surface air temperature (mean guess ST) and the mean true ST (Fig. 2) as exhibited by the guess ME. Significant differences, however, are seen between the VAS and *GOES-8* imager retrieval MAE and SDE. For these statistics, the *GOES-8* imager shows an improvement over VAS by 0.1–0.3 K. These differences are associated with the relative spectral position of the channel bands and their relative insensitivity to atmospheric moisture (Fig. 1), which will be demonstrated in section 4c.

### Case 1 PW retrievals

An example of the *GOES-8* imager retrieval of PW for this case is presented in Fig. 4 (middle panels). Comparing case 1 PW results with LAMPS model–calculated (true) values (top panels), it is seen that the 0000 UTC retrievals capture the PW fields very well. However, at 1200 UTC the PW retrievals appear to be noisy as compared to the true PW. Also, the PW gradients are not as well depicted, suggesting less accurate retrievals. Again, the simulated radiances contain no random errors, and thus the apparent difference in PW retrieval performance between 1200 and 0000 UTC is strictly a characteristic of the PSW algorithm.

Figure 6 gives statistics for the retrieval errors for PW at these times as well as intermediate times. For each of the error statistics the guess value is also shown, as was the case for the ST results. A striking feature of these statistics is that the PW retrieval errors, unlike the ST values, show a significant diurnal trend except for the ME. The guess error statistics, however, are fairly consistent over time. The PW guess ME, having values less than ±0.2 mm, indicates that the guess exhibit very little bias over the field. The guess MAE and SDE are fairly constant with an SDE of 3.3 mm (14%). The COR is high with a value of about 0.96 for most of the retrieval times. For this case, the statistics suggest that a fairly good and consistent guess was provided to the algorithm.

The PW retrieval MAE, SDE, and the COR in Fig. 6 indicate a range of retrieval performance from improvement over the guess PW to retrievals that are worse than the guess. Performance differences associated with the VAS and *GOES-8* imager sensors are also seen. The retrieval ME is seen to be fairly constant for the various retrieval times with values varying a few tenths of a millimeter about zero for both VAS and the *GOES-8* imager. The differences seen in the retrieval ME between the sensors (on the order of 0.3 mm) is believed to be slight biases in the forward radiative transfer model calculations of brightness temperatures associated with the parameterizations of the spectral response functions of the sensors. Thus, the ME differences are internal biases and not considered to be physical effects associated with sensor characteristics. The retrieval MAE and SDE are seen to have maximum values at 1200 UTC 17 June and 1000 UTC 18 June. At 1200 UTC 17 June, values of the retrieval MAE and SDE are 3 mm (12%) and 4.5 mm (18%), respectively, while the COR is at a minimum of about 0.92. At these times, the retrieval SDE values are worse than the guess SDE values by as much as 0.6–2.0 mm, 18%–62% with respect to the guess SDE, depending on the sensor and time. The best performance occurs between 1600 and 0000 UTC, which corresponds to about late morning to late afternoon local time. During this time, the retrieval SDE is about 2.0 mm (8%) with an improvement over the guess SDE of about 1.3 mm or about 40% (*GOES-8* imager). Thus, the performance is seen to degrade as night approaches, and from approximately 0400 to 1300 UTC, the retrieval performance is worse than the guess PW.

Differences in performance between VAS and the *GOES-8* imager are also seen in Fig. 6 over the 24-h period. As stated above, the difference in the ME is believed to be related to the radiative transfer model and therefore not significant. However, the differences in retrieval MAE, SDE, and COR suggest that real performance differences exist between the sensors. Note that the performance differences between the sensors are a function of time of day. Between 1400 and 0400 UTC the *GOES-8* imager appears to perform slightly better than the VAS channel pair, as indicated by the difference in SDE of about 0.3 mm or 13% with respect to VAS. This improvement is also seen in the COR for this time period. Late at night and in the early morning hours, however, the SDE of the *GOES-8* imager is worse than that of VAS by as much as 0.5 mm or 12% with respect to VAS. The difference in performance at night is also exemplified in the COR values.

### Discussion of case 1 results

From the results above it is seen that the ST retrieval improves upon the first-guess values on the order of 85%–95%, depending upon sensor channel pair, and shows ST errors of less than 0.5 K. The PW results show an improvement over the guess by about 40% with a retrieval SDE error of about 2 mm (8%) during the day. At night, the retrieval performance degrades and the retrievals become significantly worse than the guess values. It is interesting that the PW results show a diurnal trend, while the ST results do not. A cause for the degradation of the PW retrievals during the night is suggested by the temperature plots in Fig. 2. From Fig. 2 (top panel), note that the ST plot is almost an inverse correlation to the PW SDE and MAE plots of Fig. 5. The PW errors are at a maximum at 1200 and 1000 UTC, which corresponds to the times of the strongest surface temperature inversions (ST less than surface air temperature). To understand how temperature inversions can affect the retrieval of PW, and also why the ST retrievals are unaffected, the ST and PW perturbation equation applied in the retrieval algorithm will need to be examined.

The error statistics for case 1 show the effects of a nonzero ε_{I} and ε_{D} in (1) (section 2b) when they are assumed to be zero. Specifically, the residual quantity ε_{I}, as seen from (A15) and (A16), is a sum of functions involving the unknown departures of the first guess from the observed vertical moisture structure, air temperature profile, and the cross product of the departures of ST and PW and higher-order terms. The residual quantity ε_{D} is a function of departures in surface air temperature and the vertical derivative of the departures of the temperature profile. To ensure that ε_{I} and ε_{D} are small, these departures involving the first guess will need to be minimized. When these conditions are not strictly met, significant errors in the linearization of (1) occurs, and ε_{I} and ε_{D} manifest themselves as errors in the solutions for the departures in ST (*δT*_{s}) and PW (*δU*_{ps}). It is obvious in (1) that the effects of ε_{I} and ε_{D} will depend on their relative magnitudes as compared to *δI* and *D,* respectively. Thus, ε_{I} and ε_{D} do not have to be large in order to affect the solutions of *δT*_{s} and *δU*_{ps}. For example, if *D* → 0, then the error (ε_{D}) in the *δU*_{ps} coefficient *D* becomes very significant. This is the case when the perturbation equation is applied under a temperature inversion condition.

*D*is dependent not only on an atmospheric term given by the integral but also on the difference between the guess ST

*T*

_{s}and the surface air temperature

*T*

_{ps}. It is possible for

*D*→ 0 whenNoting that ∂

*τ*

*U*is negative and that the integral in (5) is positive, the condition

*D*→ 0 occurs when

*T*

_{s}<

*T*

_{ps}, that is, when a surface temperature inversion is present. Moreover, the integral itself gets smaller when inversions are present due to the change in sign of ∂

*T*

*p*)/∂

*p*at the inversion layers. For an isothermal atmosphere the integral is zero. Thus, for retrieval cases when inversions are present the coefficient

*D*has the potential of becoming very small and causing the residual quantities, assumed to be zero in (1), to become significant.

The discussion to this point pertains to the effects of temperature inversions in the guess profile and not in the observed profile. To reduce the values of ε_{I} and ε_{D} in (1), the guess profile is expected to characterize the observed temperature profile with the guess ST providing a reasonable guess of the observed ST. This implies that when there are temperature inversions present in the observed scene, the guess should characterize these conditions. But, in doing this, the coefficient *D* derived from the guess profile tends towards zero, resulting in the errors of *D* to become significant. Thus, by providing a good guess temperature profile and ST value during conditions when temperature inversions are present in the observed field, retrieval performance will be degraded.

The physical interpretation of the coefficient *D* tending toward zero when inversions are present is that information about the perturbation of PW is not present in the radiance difference between the observed and first-guess state. This occurs because the change in radiance at the top of the atmosphere, due to a perturbation in PW, is the result of two processes, as described by the terms of the coefficient *D* in (5). These processes involve 1) changes in IR radiation emitted by atmospheric water vapor and 2) changes in the attenuation of the radiation emitted from the land–sea surface. The effects of these two processes oppose each other. For example, an increase in PW causes an increase in radiation from the atmosphere, while the same increase in PW causes an increase in attenuation of the surface radiation. When the skin temperature is lower than the surface air temperature, the two effects have the potential of being nearly equal but opposite and the net change in the observed radiance is near zero. Thus, no information is provided about the perturbation in PW. Since the perturbation equation is applied to two different channels, it is assured that for at least one channel the coefficient *D* will not be zero. However, when *D* is near zero, the signal, in terms of perturbed radiation between the observed and perturbed state (first guess), will also be small and its value will be on the order of the approximations made in linearizing the perturbation equation.

To constrain *D* from approaching zero when temperature inversions are present, a judicious choice of the guess ST can be used. It is desirable to choose a guess ST close enough to the true value to provide a reasonable guess without incurring significant errors in the skin temperature retrieval but large enough to constrain *D* from going to zero. In this analysis *T*_{s} was chosen to be equal to *T*_{ps}, which reduces *D* to the integral in (5). To demonstrate the sensitivity of the PW retrievals to the guess ST, case 1 analysis was repeated for 0000 and 1000 UTC using different guess ST values. These results are presented in Table 1. At 0000 UTC the PW and ST retrieval errors, as might be expected, get larger as *T*_{s} differs from the true ST (note in Fig. 2 that the average guess ST error is ∼3 K for *T*_{s} = *T*_{ps}; ∼−7 K for *T*_{s} = *T*_{ps} + 10 K, etc.). At 1000 UTC (when surface temperature inversions are present), the ST and PW SDE show an anomalous increase when the true ST is used as a guess. Moreover, the PW retrieval SDE improves as the guess ST error gets larger or *T*_{s} increasingly differs from the true ST. Noting the coefficient *D,* also in Table 1, along with the behavior of the PW retrieval SDE, the correlation between the two is consistent with the above discussions concerning the effects of temperature inversions.

Though the PW SDE improves for the 1000 UTC case when the *D* coefficient is constrained from going near zero, the PW SDE is still larger than the guess PW SDE (3.2 mm, see Fig. 6). The fact remains that when inversions are present, the perturbation in radiance caused by differences in the guess and observed PW is small and thus, only modest improvements, if any, can be made over the guess PW with this retrieval technique. In the 0000 UTC case, PW retrievals are not improved by causing the *D* coefficient to become larger. This is because the guess ST errors affect the value of the residual ε_{I}, which as shown below, affects the ST and PW retrievals. Thus any improvement in the relative magnitude between ε_{D} and the coefficient *D* is compensated by an increase in ε_{I}. This was not the case at 1000 UTC since the effect of the near-zero *D* was initially the dominate source of the retrieval error until the guess ST error became large and prevented any additional improvement in the PW retrieval.

*D*is negative and the magnitude of the clean channel value is smaller than that of the dirty channel. The reverse is true for the coefficient

*C.*In Table 2 average values of the perturbation equation parameters are given for 0000 and 1000 UTC. However, when deep surface inversions are present it is not only possible for

*D*to tend toward zero but, as can be seen in (5), it can become positive and increase in the positive direction. Since the clean channel

*D*value is smaller than the dirty channel value, it tends toward zero more quickly and has the potential to become greater in the positive direction for deepening surface temperature inversions. Thus, the ratio of the

*D*coefficients approach the condition in (6). This situation can be prevented by setting

*T*

_{s}=

*T*

_{ps}as before.

_{I}and ε

_{D}are considered. Let the quantities

*δT*

_{s}and δ

*U*

_{ps}, as defined in (4), refer to the retrieved perturbations obtained from (1) in which it was assumed that ε

_{D}and ε

_{I}were zero. Let the primed quantities

*δT*

^{′}

_{s}

*δU*

^{′}

_{ps}

_{D}and ε

_{I}are not assumed zero and their values known. Also defineThe approximate effects of a nonzero ε

_{I}and ε

_{D}on the solutions of

*δT*

_{s}and

*δU*

_{ps}can be determined by solving for the unprimed quantities

*D*and

*δI*in (7) then substituting into (3). One obtainswhere it was assumed that the average quantities

*γ*

*α*

*δT*

_{s}and

*δU*

_{ps}is solely due to

*α*

*γ*

*δT*

^{′}

_{s}

*T*

^{′}

_{s}

*T*

_{s}and

*δU*

^{′}

_{ps}

*U*

^{′}

_{ps}

*U*

_{ps}, where

*T*

^{′}

_{s}

*U*

^{′}

_{ps}

*δT*

_{s}and

*δU*

_{ps}is obtained from (8) asNote that from the definition in (4) that

*δT*

^{′}

_{s}

*δT*

_{s}=

*T*

^{′}

_{s}

*T*

_{s}, the ST retrieval error, and

*δU*

^{′}

_{ps}

*δU*

_{ps}=

*U*

^{′}

_{ps}

*U*

_{ps}, the PW retrieval error. From (10) it is seen that the error in

*δT*

_{s}is not a function of

*α*

*D*tending toward zero seen in the PW retrievals are not present in the ST retrievals. However, the quantity

*γ*

*δT*

_{s}and

*δU*

_{ps}retrievals. Also note that the error is proportional to the observed perturbation or guess error. The error calculation in (8) is considered approximate because of the assumption in (9).

*α*

*γ*

*GOES-8*imager channels. This effect can be demonstrated by calculating

*γ*

*ε*

_{I}≪

*δI*,Substituting in (11) the expression for

*ε*

_{I}obtained from the first term in Q2 of (A16), and also the expression for

*δI*′, one obtains after rearrangingTable 2 provides average values for the derivatives in the numerator of (12) as well as averages for the coefficients

*C*and

*D*for each channel for the

*GOES-8*imager and VAS. The values for the derivatives are fairly constant between sensor channels, but the coefficient values differ significantly for some channels and between sensors. The differences between the coefficients are a result of the separation and spectral position of the sensor channels. The significance of these differences can be seen by substituting the 0000 UTC mean values for each sensor channel pair from Table 2 into (12) assuming

*D*≈

*D*′. If values for the guess SDE obtained from Figs. 5 and 6 are also substituted into (10) for

*δT*

_{s}′ and

*δU*

_{ps}′, the result isHere,

*γ*

*γ*

*C*and

*D.*Note from (2) that the differences in

*C*between channels are dominated by

*τ*

_{ps}.

Using the values of *γ**γ**GOES-8* imager and VAS, respectively. Thus, for channels that are spectrally positioned in the more absorbing portion of the window region, larger values for *γ**GOES-8* imager and VAS, respectively. The above analysis of *γ*_{I} [see Eq. (A16)]. This term appears to be a significant error source for the conditions of case 1. However, ε_{I} contains many terms that can interact with one another by subtracting or adding their influences, yielding a combined error effect that may be different from that of a single term.

The difference in PW retrieval performance between the *GOES-8* imager and VAS is explained in a similar manner as given for the ST case. The smaller coefficient *D* for the *GOES-8* imager, seen in Table 2, yields a larger value of *α* at retrieval times containing temperature inversions. Thus, the *GOES-8* imager retrieval performance is worse than VAS. Moreover, the cleaner (less absorbing) the channel pair, the worse the retrieval performance will be when surface temperature inversions are present. When surface temperature inversions are not present (e.g., 0000 UTC), the differential PW retrieval performance between the *GOES-8* imager and VAS appears to be due to the influence of *γ.* Thus, the *GOES-8* imager shows a slight improvement in performance.

Since the values of ε_{I} and ε_{D} are dependent upon how well the first-guess temperature and vertical moisture structure satisfies the approximations and constraints imposed on the derivation of the perturbation equation, it is expected that *γ* and *α* will be dependent on first-guess selection methodologies. This is demonstrated in cases 2 and 3 where the first-guess field is not coincident in time with the retrievals and thus are less characteristic of the observed temperature and moisture vertical structure and surface conditions.

### Case 2 ST retrievals

Case 2 results are presented below, first for ST and then for PW. The retrieval methodology used in this case is the same as case 1 except the guess field is selected from the 0000 UTC LAMPS model data and held constant for each retrieval time. An example of the *GOES-8* imager ST retrieval for case 2 is shown in Fig. 3 (bottom right panel) for 1200 UTC 18 June. Case 2 retrievals at 0000 UTC are the same as for case 1 at 0000 UTC since the guess and the retrieval at this time are coincident. The 1200 UTC ST retrievals are very similar to the LAMPS model true values (top panels) and case 1 retrievals (center panels). The ST gradients are captured very well with only small differences occurring over the northwest and southeast regions.

The ST retrieval statistics for this case are presented in Fig. 7. The statistic at 0000 UTC is the same as presented in case 1. The retrieval error statistics as well as the guess error statistics worsen for times moving away from 0000 UTC. As in case 1, the guess ST errors vary with time and are correlated with the difference between the surface air temperature at 0000 UTC and the ST at the time of retrieval. In general, the guess error is larger for case 2, as might be expected. The guess ME ranges from about −10 K at 1200 UTC and 0800 UTC to about 6 K at 2000 UTC 17 June. The effect of these large guess errors at these times are also seen in the guess MAE and SDE.

The ST retrieval errors (Fig. 7) show almost an order of magnitude improvement over the guess statistic, as was seen in case 1 (Fig. 5). However, the retrieval ME (bias) for case 2 is seen to be slightly worse than case 1, ranging from −0.1 to 0.2 K with the bias being negative at 1200 UTC 17 June. Here, a negative bias indicates an ST retrieval being too warm. The trends in the retrieval MAE, SDE, and COR indicate that the retrieval performance is affected by using a noncoincident guess/retrieval time. The retrieval MAE and SDE show an increase from 0000 to 1200 UTC of about a factor of 2. These errors are also about a factor of 2 larger than those of case 1 at 1200 UTC. The values for the retrieval MAE are less than 0.8 K, and the retrieval SDE values are less than 1.2 K for all hours. However, the *GOES-8* imager shows significant improvement over that of VAS with a retrieval MAE and SDE of less than 0.3 and 0.4 K, respectively. The error trends in Fig. 7 demonstrate that as the first guess departs from the temperature and moisture structure characteristics of the observed profile, the residual ε_{I}, and thus *γ,* increases and causes poorer ST retrieval performance. Even though ST retrieval performance is degraded by using a noncoincident guess/retrieval time, the performance, as indicated by the statistics, is still quite good, especially for the *GOES-8* imager.

### Case 2 PW retrievals

An example of a *GOES-8* imager PW retrieval for case 2 is shown in Fig. 4 (bottom right panel) for 1200 UTC 18 June. The case 2 PW retrievals do not compare well with the true values (top panel) and case 1 retrievals (center panel). Although the presence of clouds makes it difficult to compare the images, the case 2 PW gradients differ significantly from the true gradients, as seen over the Tennessee and Dakota regions. Recall that the case 1 retrievals at 1200 UTC 18 June were influenced by the presence of temperature inversions. Case 2 demonstrates that when inversions are present in the observations, retrieval performance does not improve when a guess profile that is essentially free of inversions is used (albeit a noncoincident retrieval/guess time).

Figure 8 gives the statistics for the PW retrieval errors for case 2. The statistics for the guess and the retrieval errors increasingly worsen for times moving away from 0000 UTC. The guess ME (top panel) shows a bias over the field ranging from about 0.0 to 1.0 mm mostly from 0000 to 1200 UTC 18 June. The small guess ME is a result of the mean PW over the domain remaining fairly constant with time, as seen in Fig. 2. On the other hand, the retrieval ME is significant on the order of −2.0 mm, implying that, on average, the retrievals are moist biased.

The retrieved PW MAE and SDE show substantial increases as the time of retrieval differs from 0000 UTC. These increases are on the order of two to three times the value for 0000 UTC. The COR as well shows substantial degradation in retrieval performance at 1200 UTC as compared to 0000 UTC 17 June. During the optimal retrieval times (1600–0000 UTC), the retrieval SDE (*GOES-8* imager) increases from about 2 mm (8%) at 0000 UTC to about 3 mm (13%) at 1600 UTC. The retrieval ME during the optimal retrieval times also varies from near zero at 0000 UTC to about 2 mm (8%) at 1600 UTC. It is interesting that, for this case, the PW retrieval errors at most times are better than the guess errors except for the ME. Even at night the retrievals are an improvement over or about equal to the guess error. However, the retrieval errors at night are not considered good for this case. At night, the retrieval errors for case 2 (Fig. 8) are a little worse than those for case 1 (Fig. 6), as demonstrated by the increase in retrieval SDE and MAE. The degradation in the case 2 retrievals over that of case 1 is seen to be about 2 mm at 1200 UTC and about 0.8 mm at 1000 UTC 17 June.

This case demonstrates the sensitivity of the retrieval performance to a noncoincident guess/retrieval time and the effects of using an increasingly degraded first-guess field. As the first-guess temperature profile and moisture structure departs from the observed values, the residuals ε_{I} and ε_{D} increase. The increase of ε_{I} and ε_{D} is propagated into the retrieval solutions in terms of *α* and *γ* given in (10) and thus the retrieval performance is degraded. It is interesting that between 1800 and 0000 UTC that the retrieval performance is almost a constant improvement over the first-guess error. The ratios of the VAS and *GOES-8* imager retrieval SDE to their respective guess error values are approximately 0.7 and 0.6, implying a 30% and 40% improvement over the first guess, respectively. This same improvement in VAS retrievals was also seen by Knabb and Fuelberg (1997). Their algorithm comparison study examined 84 retrievals associated with the same number of radiosondes using a noncoincident guess/retrieval time. One retrieval time was considered (0000 UTC), and they used a first guess from the average of radiosonde profiles 12 and 24 h earlier. The overall retrieval SDE reported in their study was 4.97 mm with a first-guess SDE of 6.99 mm, which resulted in a 29% improvement by the retrievals. Their reported retrieval error is larger than those given in this study because of the difference in the quality of the first-guess data as evident in their first-guess SDE. They also report sensitivities in the retrievals to first-guess departures in moisture structure, air temperature, and low-level temperature inversions. These findings are also consistent with the effects of ε_{I} and ε_{D} presented here.

This case also illustrates that using a first guess free of inversions does not improve the algorithm performance at retrieval times having inversions in the observations. As seen in Fig. 8, retrieval performance degrades when inversions are present in the observed field. This is especially evident between 0000 and 1200 UTC 18 June. Since the first guess is valid at 0000 UTC and held constant, the cause of this effect is different from that in case 1 where the coefficient D was demonstrated to tend toward zero when inversions where present in the first guess. The increase in error for case 2 during temperature inversions is still due to *α* in (10) increasing, but the cause, in this case, is ε_{D} becoming large instead of *D* → 0. As seen in (A15), ε_{D} has a dependence on *δT*_{ps}, the departure of the guess surface air from the observed (true) value. An indication of the magnitude of these departures for case 2 is seen in Fig. 2 by comparing the mean surface air temperature value at 0000 UTC (guess *T*_{ps}) with those at the retrieval times. These departures are seen to be generally correlated with the retrieval error statistics in Fig. 8, especially with the retrieval ME.

Figure 8 also shows that differences in retrieval performance exist between the sensors. These differences are similar to that observed in case 1. The effect of using a poor guess appears to be the same for both sensors, except at night when the statistics for both sensors show less of a difference than for case 1. This is seen at 1000 UTC where the retrieval SDE difference between the sensors for case 1 is about 0.6 mm or 12% (w.r.t. VAS) as opposed to a case 2 difference of 0.2 mm or 4%. The retrieval ME differences between the sensors show little change from case 1. During the optimal retrieval period the *GOES-8* imager continues to demonstrate better performance. At these times, retrieval SDE differences between the sensors are seen to be on the order of 0.3 mm or about 13% (w.r.t. VAS).

### Case 3 ST retrievals

The retrieval methodology for case 3 was the same as in case 2 except the guess field is selected from the 1200 UTC LAMPS model data field. Since the analysis spans a 24-h period containing two 1200 UTC times, the nearest 1200 UTC data with respect to the retrieval time was used as the guess field. An example of the *GOES-8* imager ST retrievals for case 3 at a 0000 UTC retrieval time is shown in Fig. 3 (bottom left panel). For this example the first-guess field from 12 h earlier (1200 UTC 17 June) was used. Case 3 retrievals at 1200 UTC are the same as case 1 retrievals at 1200 UTC since the guess and retrieval times are coincident. Figure 3 shows that the case 3 retrievals at 0000 UTC capture the overall temperature gradients fairly well as compared to the model true values (top panel). Good agreement is seen over the northeast region with some differences seen over the Dakotas and Montana. Differences are also seen over the Arkansas and Oklahoma region.

The ST retrieval statistics are presented in Fig. 9. Due to the use of two guess fields the statistical results are more complex and less symmetric over time as observed in the other cases. The guess ST (surface air at 1200 UTC) varies greatly from the true ST, as exemplified by the variation in the guess ST ME. The 1200 UTC 17 June ST guess has the largest departures from the true ST with a maximum guess ME and MAE on the order of 12 K.

Because of the different characteristics exhibited by the two guess fields, ST retrieval errors associated with retrieval times before 0000 UTC are substantially different from those after 0000 UTC. As in previous cases, the ST retrievals show essentially no sensitivity to temperature inversions in the guess field or the observed field, which is not the case for PW retrievals. In this case, as before, the ST retrieval errors are an order of magnitude less than the ST guess errors. Overall, the magnitudes of the retrieval errors for this case are the same as for case 2. In general, the largest errors occur at times when the guess ST has the largest deviation from the true ST. Retrievals from the VAS sensor, as in the other cases, are consistently worse than those from the *GOES-8* imager. Maximum ST retrieval ME and SDE values for VAS are about 0.5 and 1.1 K, respectively, which are about double that observed for the imager.

### Case 3 PW retrievals

An example of *GOES-8* imager PW retrievals for case 3, which used a first-guess field of 1200 UTC 17 June, is shown in Fig. 4 (bottom left panel). In this example, the general PW gradient is not captured very well by the retrievals, and there are significant departures from the model true PW values (top left panel). The retrievals over Indiana and Montana show a dry bias, while retrievals over Oklahoma and Kansas show considerable variability, making the moderate moisture pattern over that region indistinguishable.

The error statistics for case 3 are given in Fig. 10. For this case, the guess and retrieval errors increase as the retrieval time approaches 0000 UTC. Since two different guess fields were used, a discontinuity is seen in the error statistic at 0000 UTC. Moreover, the 1200 UTC 17 June guess produces retrievals that never improve over the guess field values even though the PW guess error statistics for each of the two guess fields are seen to be fairly similar. The retrieval errors in this case are the effects of the increasing nature of ε_{I} and ε_{D}, as before, and also the effects of the first guess containing temperature inversions causing the coefficient *D,* as seen in case 1, to be near zero. The difference in the retrieval errors associated with the two 1200 UTC first-guess fields are the result of each first guess having different surface and profile parameter characteristics. They are also applied over different retrieval times having different environmental conditions.

An important difference between the two guess fields is that they each contain a different amount of guess grid points having air temperature inversions between the first two standard levels near the surface, even though both guess fields are at 1200 UTC. The 17 June guess has 50% of its grid points containing inversions, while the 18 June guess has 20%. Although the depth of the inversion is also a factor, it is expected from earlier discussions of the coefficient *D* that the 17 June guess will give the worst retrieval performance. An indication of the appropriateness of the first-guess field and the variation of ε_{D} can be obtained, as in case 2, by comparing the guess mean surface air temperature in Fig. 2 (*T*_{ps} at 1200 UTC) with those at each retrieval time. The mean surface air temperature departures *δT*_{ps} for retrieval times associated with the 17 June guess are larger than those corresponding to the 18 June guess. Moreover, *δT*_{ps} correlates well with the retrieval ME in Fig. 10. The other retrieval statistics also have a similar trend due to these departures.

It is interesting to note that the case 3 PW retrieval errors at 0000 UTC (first-guess field from 1200 UTC) are nearly the same as the case 2 retrieval errors at 1200 UTC, which used a 0000 UTC guess. This implies that the effect of temperature inversions in the PW retrieval process is the same whether the inversions are in the guess field or the observed field. The effect of retrieving at night (inversions in the observations) can be seen at 0400–0800 UTC (Fig. 10) where the SDE and MAE show an increase. Again, this is due to an increase in ε_{D} and thus *α* in (10).

Comparing PW retrieval errors from cases 2 and 3 also shows that the guess time closest to the retrieval time usually gives the best retrieval performance. At 0600 UTC (halfway between 0000 and 1200 UTC) the retrieval errors are about the same for cases 2 and 3. However, at 1800 the 0000 UTC guess provides better retrieval performance. This is because of the larger number of inversions in the 1200 UTC 17 June guess. It is not until 1400 UTC that the retrieval performance associated with using the 0000 UTC guess is worse than the performance associated with the 1200 UTC 17 June guess. As with the other cases, the *GOES-8* imager shows a slight degradation in PW retrieval performance over that of VAS when inversions are present in the guess field or the observations.

**GOES-8 **sounder performance

**GOES-8**

The results presented above did not include the performance of the *GOES-8* sounder. Retrievals using the *GOES-8* sounder channel pair 8 (11.0 *μ*m) and 6 (12.7 *μ*m) produced almost identical results as the VAS channels 8 (11.2 *μ*m) and 7 (12.7 *μ*m). The sounder pair 8 and 7 (12.0 *μ*m) produced results very similar to the imager channels 4 (10.7 *μ*m) and 5 (12.0 *μ*m). This is not too surprising because of the similarity in spectral positions of the dirty channels (more absorbing channels, see Fig. 1) among the sensors. Retrieval performance seemed to be less sensitive to the locations of the clean channels (VAS: 11.2 *μ*m, sounder: 11.0 *μ*m, imager: 10.7 *μ*m). This is probably due to the water vapor absorption being relatively constant for this spectral region as compared to the region of longer wavelengths associated with the dirty channels (Fig. 1). A summary of the retrieval performance for the *GOES-8* sounder channels is presented in Table 3 along with values from the other sensor/channel cases for comparison. The case 1 values provided are the minimum and maximum retrieval errors observed for this case which occurred at 0000 and 1200 UTC 17 June, respectively. The maximum retrieval errors for cases 2 and 3 are also given, which occurred at retrieval times of 1200 and 0000 UTC, respectively. The minimum retrieval errors for cases 2 and 3 are equal to the case 1 errors corresponding to 0000 and 1200 17 June UTC, respectively, and thus were not repeated in the table. A comparison of the ST results associated with each of the channel combinations indicates that the *GOES-8* imager and the sounder channel pair 8 and 7 provide significantly better retrievals. The PW performance is also better when no inversions are present (0000 UTC).

## Summary and conclusions

The performance of a physical split-window technique to retrieve ST and PW from GOES observations was investigated. Retrievals were made from simulated channel radiances using combinations of the VAS and *GOES-8* imager and sounder split-window channels. The retrievals were every 2 h in order to examine the diurnal effects on retrieval performance. Three retrieval cases are considered, each using a different type of first-guess selection that affected the retrieval performance. In all cases the first-guess profiles are obtained from the model (truth) profiles and provided to the algorithm unperturbed and with a reduced resolution in the form of a 2.5° × 2.5° grid. The first-guess ST was set equal to the surface air temperature of the guess profile.

The case 1 retrievals used a first guess that was coincident in time with the retrieval time, thereby providing the best possible temporal first-guess field. This case illustrates the diurnal performance of the retrieval algorithm. The ST retrieval performance of the PSW algorithm showed almost an order of magnitude improvement over the guess ST errors. No diurnal performance trends were seen in the ST retrieval errors even though strong diurnal trends were present in the guess errors. The retrieval performance indicated bias errors of less than 0.1 K for all the sensors and SDE values of about 0.2 K for the *GOES-8* imager and 0.3–0.5 K for VAS. The ST retrieval SDE was an improvement over the guess SDE by about 85%–95% where the surface air temperature was used as the first-guess ST.

PW retrieval performance for case 1 showed improvement over the guess PW values except at night and in the early morning hours. Even though no significant diurnal trends were seen in the guess PW errors, the PSW algorithm showed strong diurnal performance trends correlated with the presence of lower-level temperature inversions. During the optimal retrieval times, from about 1000 to 1800 LST, the retrieval performance was essentially constant with bias errors of less than 0.2 mm and an SDE value of about 2.0 mm or about 8% of the average PW field value. This was an improvement of about 1.3 mm or about 40% over the guess SDE value (*GOES-8* imager). The *GOES-8* imager performed slightly better than VAS during the optimal retrieval times by about 13%, but slightly worse than VAS by about 12% when inversions were present. For this case, the lower-bound PW retrieval errors are those given for the optimal retrieval times. At other times, the retrieval error is dependent on the number of temperature inversions in the domain and first-guess field and thus will vary with atmospheric conditions and region.

Cases 2 and 3’s retrieval methodology was the same as case 1, except the guess field was 0000 and 1200 UTC, respectively, and held constant (in time) for each retrieval time. These cases illustrate the effects of the first-guess field with and without temperature inversions. They also identify trends in retrieval performance due to a poor first guess, which occurs when the retrieval time differs significantly from the time of the first-guess field. The ST retrieval errors for case 2 (0000 UTC guess) and case 3 (1200 UTC guess) demonstrated a sensitivity to the quality of the first-guess field. The ST retrieval performance was degraded for retrieval times moving away from the time of the first-guess field. The retrieval ME became more significant with maximum values between ±0.5 K. For retrieval times 12 h away from the time of the first-guess field, the retrieval SDE approximately doubled to 0.5 and 1.1 K for the *GOES-8* imager and VAS, respectively. However, the retrieval SDE is still almost an order of magnitude improvement over the guess SDE value with the *GOES-8* imager performing significantly better than VAS. No significant diurnal trends were observed in the ST retrieval performance in any of the cases.

The PW retrieval performance for cases 2 and 3 is similar in that they degrade as the retrieval time differs from the time of the first-guess field. When the first-guess field has no temperature inversions (case 2), the retrieval performance error for most retrieval times is better than the guess error statistics. Moreover, for retrieval times when inversions are also not present in the observations, the improvement over the guess is significant and fairly constant with a value of 30% and 40% for VAS and the *GOES-8* imager, respectively. This improvement over the guess SDE for VAS was similar to that reported by Knabb and Fuelberg (1997) in their algorithm comparison study. Conversely, when a significant number of temperature inversions are present in the first-guess field (case 3), the retrieval performance at most retrieval times is worse than the first-guess errors. Whether the inversions are in the first guess or the observations, PW retrieval performance is degraded. The PW retrieval performance over the domain is correlated to the overall number of temperature inversions in the first-guess field. When temperature inversions are present in either the observations or in the first-guess field, results show that the *GOES-8* imager retrievals are slightly worse than VAS.

The algorithm performance errors presented in this study are the result of the limitations inherent in the algorithm as a result of the approximations and constraints imposed in its formulation. Contributing error sources that would be present when the algorithm is applied to observed data were not included in this evaluation. Error sources, such as instrument random noise, emissivity assumption errors, and cloud contamination, provide additional errors that can greatly affect the accuracy of a retrieval. The results presented above are considered lower-bound errors, demonstrating the sensitivity of the algorithm to various conditions. From these results some important conclusions can be made about the application of the algorithm to observed data.

The results suggest that care will be needed in applying the PSW algorithm to discern natural diurnal variations in PW. This is because of the likelihood that temperature inversions will be present in the observations or the first-guess field in these types of studies. As seen in the retrieval MAE and SDE above, the presence of temperature inversions may degrade algorithm performance to the point where retrievals, at times, may not be any better and may even be worse than the guess PW. However, since the retrieval ME over the field essentially shows no sensitivity to temperature inversions, this may suggest that the mean PW retrieval over a region may be appropriate for monitoring regional diurnal PW cycles. Unfortunately, the retrieval ME does show a sensitivity to the first-guess field quality, especially if the first-guess field is biased as a result of not being coincident in time with the retrievals. Since it is difficult to provide an observed first-guess field that is coincident in time with retrievals made at regular time intervals sufficient to sample the diurnal cycle (except for model forecasts or interpolation), the quality of first-guess fields obtained for this purpose may not be consistent over the day. Therefore, biases in the retrievals are expected to result and make it difficult to discern the natural diurnal PW cycle. This conclusion is independent of any effects from random instrument noise and emissivity errors. On the other hand, results indicate that the PSW algorithm does show promise in capturing the diurnal cycle of ST since the algorithm exhibits no diurnal performance trends associated with temperature inversions. Although algorithm performance fluctuations due to variations in the first-guess field are seen, these fluctuations are approximately an order of magnitude smaller than the natural diurnal variation of ST even when a constant guess field is used.

The case study results also suggest that climatologies of ST and PW from GOES are possible, but a judicious selection of retrieval times and first-guess field will be needed. Climatologies of ST will be weakly affected by the first-guess sensitivities. First-guess conditions, such as time of the first guess, will need to be consistent over the period of record. Retrieval times that are coincident with available guess data are desirable. Climatologies of PW may be restricted to obtaining PW values between the optimal retrieval times when no temperature inversions are present. This may present limitations when attempting to avoid those times of the day having increased cloudiness or when forming daily composite values to reduce cloud masking. The PW climatologies will also be sensitive to the first-guess selection. The first-guess field will need to be consistent and near coincident with the retrieval time as stated above. The results also indicate that significant retrieval performance differences exist between the VAS and *GOES-8* split-window channel pairs. When constructing climatologies spanning both sensors or analyzing datasets from both sensors, these performance differences will need to be considered.

Finally, when applying the PSW algorithm in comparison studies with other instruments or retrieval algorithms, the retrieval and first-guess constraints, as given above, will need to be considered when setting up the comparison conditions.

The results presented above can provide insight into the application of the PSW algorithm to retrieve ST and PW from GOES for use in climate research, such as diurnal variability studies, construction of climatologies, and instrument comparison studies. Furthermore, the trends described by the results may be characteristic of split-window routines in general, and thus may also have implications for operational retrievals from the *GOES-8*/*9* as well.

Many people have provided valuable assistance through the course of this work. Special recognition is given to a former colleague, Grant Carlson, who played a key role in the early stages of this work. LAMPS data was provided by Diane Samuelson (NASA); technical information on LAMPS was provided by Mike Kalb (USRA), Don Perkey (UAH), and Kevin Doty (UAH). This research was funded by NASA’s Earth Science Enterprise. The faithful support of this research activity by Dr. Ramesh Kakar is sincerely appreciated.

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# APPENDIX

## Derivation of the Perturbation Equation in the Physical Split-Window Algorithm

*I*observed at the satellite, assuming a nonscattering, plane-parallel atmosphere and an emissivity of unity, can be expressed aswhere

*B*and

*τ*are the Planck function and atmospheric transmittance, respectively, for a wavelength domain associated with a sensor spectral channel. The variables

*T*and

*p*are the temperature and pressure, respectively. The subscript “s” denotes the surface and “ps” denotes that the atmospheric parameter is evaluated at the surface pressure. Thus,

*T*

_{s}is the surface or skin temperature and

*T*

_{ps}is the surface air temperature.

*τ*

*τ*

*δτ,*

*δ*represents the perturbation. Substituting (A2) into (A1) producesThe last integral on the right-hand side of (A3) can be further evaluated by integration by parts to yieldwhere

*δτ*(

*p*) at

*p*= 0 is evaluated as zero. The perturbation in transmittance

*δτ*in (A3) and (A4) can also be expanded aswhere

*U*is the precipitable water and

*r*(

*δU*) are higher-order terms in the perturbation

*δU.*Substituting the result of (A4) and (A5) into (A3) and also expanding the temperature as a perturbation by

*T*

*T*

*δT,*

*r*(

*δT*) are higher-order terms in

*δT,*one obtainsEquation (A8) can be rearranged into the form

*I*

*I*

*δI,*

*I*

*δI*is the perturbation in radiance due to a perturbation in skin temperature and atmospheric temperature as well as precipitable water given byThe function

*R*

_{1}[

*r*(

*δU*),

*r*(

*δT*

_{s})] in (A11) is a residual quantity involving the higher-order terms of

*δU*and

*δT*

_{s}given byThe first integral on the rhs of (A11) can be approximated by assuming that the guess and observed moisture profile have the same moisture structure, that is,resulting inwhere

*R*

_{2}[

*δU*/

*U*] is a residual quantity present when the condition in (A13) is not strictly met. Inserting the result of (A14) into (A11) and rearranging terms, the result iswhere

*δT*

_{ps}and

*δT*(

*p*) are near zero as well as higher-order terms in these parameters causing

*Q*

_{1}and the integral in

*Q*

_{2}in (A15) and (A16) to vanish. 2) The moisture structure of the first guess is the same as the observed profile causing

*R*

_{2}in (A15) to vanish. 3) The products of

*δT*

_{s}and

*δU*

_{ps}, as well as the higher-order terms in these parameters, are small, causing

*R*

_{1}and the first term in

*Q*

_{2}to vanish. With the

*Q*and

*R*terms in (A15) set to zero, the resulting perturbation equation becomes linear in the unknowns

*δT*

_{s}and

*δU*

_{ps}. A solution for

*δT*

_{s}and

*δU*

_{ps}is obtained by formulating (A15) for two or more different IR sensor spectral window channels denoted by

*λ*1,

*λ*2, . . . ,

*λn,*where

*n*represents the total number of window channels resulting inThe

*C*and

*D*coefficients for each channel are calculated using (A16), and

*δI*is determined from the difference between the observed radiance and the radiance from the first-guess profile calculated using (A10). The resulting values for the perturbations are obtained by solving (A17) simultaneously. Equation (A17) is analogous to Eqs. (A12) and (A13) in Guillory et al. (1993); however, the coefficient

*D*derived by Guillory includes only the integral term given in (A16). This is because Guillory applied the condition that

*T*

_{s}=

*T*

_{ps}, thus causing the first term of

*D*given in (A16) to vanish.

Sensitivity of precipitable water (PW) and skin temperature (ST) retrieval errors and the perturbation equation coefficient *D* to the guess ST used in the retrieval algorithm. Values are for retrieval times of 0000 and 1000 UTC using the *GOES-8* imager channels 4 and 5. Here, *T*_{ps} is surface air temperature.

Perturbation equation parameters for the *GOES-8* imager VAS. Here, *B* is the Planck function for spectral channel, [xe3][pr-12]*τ*_{ps} is the guess profile total transmittance, and *T*_{ps} is the surface air temperature.

Summary of algorithm retrieval errors for *GOES-7* VAS channels 7 and 8, *GOES-8* imager channels 4 and 5, and *GOES-8* sounder channel pairs 6, 7 and 7, 8. Case 1 is time of first-guess field, same as retrieval; case 2 is the time of first-guess field, 0000 UTC;and case 3 is the time of first guess field, 1200 UTC.