Figure
1
An essential ingredient of the variational method is a satellite forward radiance model. The forward model produces a simulated radiance based on temperature, moisture, and ozone profiles along with the temperature of the surface or cloud top, and the pressure of that radiating surface (i.e., surface pressure or cloud top pressure whichever applies). Also needed are the zenith angle, used to determine the air mass path and optical depth between the radiator and the satellite. The forward model used for this work was obtained from NESDIS. The forward model coefficients used for this study were vintage late 1995.
In order to apply the forward model appropriately, a determination of clear and cloudy fields-of-view (FOV) need to be determined. The LAPS cloud analysis is used to identify clear and cloudy LAPS grid points. The analysis as presented here is only working from FOVs classified as clear. Cloudy FOVs probably can be used, but this is an early attempt at this technique, so a conservative approach was chosen. Later research may focus on using a combination of both clear and cloudy FOVs in the algorithm.
The first step in the algorithm is to assure all the data needed for proper execution are present. These include channel radiances derived from AWIPS imagery, the LAPS cloud analysis output, the LAPS surface temperature output, and LAPS 3-D temperatures. The forward model also requires an ozone profile along with moisture and temperature profiles above 100 hPa. These are gotten from climatology since LAPS extends only to 100 hPa. The entire ozone profile is provided by the forward model since LAPS does not analyze this parameter.
Next, the forward model is run to verify "clear" LAPS gridpoints, where clear is defined as those points in which both the modeled and measured GOES image radiances in channel 4 (11 micron) agree to with 2K. This step uses the LAPS thermal and as yet unmodified moisture profiles. Disparity in the channel 4 brightness temperature comparison indicates that the LAPS thermal profile is too far off or perhaps it is really cloudy where the LAPS cloud analysis is indicating it is clear. (It doesn't have to be totally cloudy for a disparity to exist, it can be partially cloudy and this will still be detectable in this difference test.) This is a conservative test; it really goes beyond simple cloud detection though that is a likely cause of differences, the forward model check is very sensitive and in many ways eliminates any thermal profiles that subsequent variational technique will find difficult to deal with. We are basically saying that we will not worry about moisture adjustment unless the thermal profiles are reasonable.
At this point, all gridpoints offering promise of moisture adjustment have been identified. If the domain is totally cloudy, the GOES adjustment is discontinued and returns unmodified moisture values which are passed to the final QC step. Assuming some gridpoints have been classified as clear, the next step is a variational adjustment at those locations. The functional evaluated at each gridpoint has the form,
5 3where the goal is to determine the optimum set of three coefficients. Each coefficient, ck is a scaling factor for the moisture corresponding to three atmospheric layers (k). The layers range from surface to 700 hPa (k=1), 700 - 500 hPa (k=2), and above 500 hPa (k=3). The forward model radiance (R) is a function of LAPS temperature (t), ozone climatology profile (o), and LAPS mixing ratio (w). The moisture profile is scaled with the appropriate coefficient (c). The observed radiance derived from AWIPS image data is designated as Roi where subscript i indicates the imager channel number.
J = sum [Roi -R(t,o,cw)i]2 + sum (1-ck)2
i=3 k=1
The first term in the functional maximizes agreement between the forward model and observed radiance at the expense of only modifying the water vapor profile. The second term adds stability and gives more weight to solutions in which the coefficients departure from unity (no change to the initial profile) is minimized. The stability term was discovered to be necessary since without it some very good radiance matches were solved but with unreasonable coefficients.
Note that differences in all three channels are minimized in this technique, not only the moisture channel. Thus, any improvement in the "dirty window," channel 5, will also contribute to the solution. A variational technique is used to minimize this function and typically requires three to 10 iterations to converge. A limit of 50 iterations was set as the maximum number to attempt. If limit was reached, that particular gridpoint was excluded and treated as cloudy.
Once the coefficients are determined, Laplaces equation is solved for interior points for which coefficients have not been determined. Then the entire domain is averaged using a spatial invariant filter; simply averaging the values in a 3x3 gridpoint window, assigning that average to the window's central grid location.
When the coefficients have been determined, they are applied to the specific humidity field at each pressure level for which they are designated. The modified specific humidity field is then advanced to the final analysis step. In this August 1996 release, the coefficient adjustment is limited to above 500 hPa only.
To date, this algorithm has demonstrated a 50-70% improvement in the LAPS upper-level RMS error (comparing against Denver RAOB data) [click on figure 1].