Delta Method Temperature Prediction

George R. Kasica
Delta Method High Temperature Prediction

One of the methods that can be used to forecast the upcoming day's high temperature is a method known as the Delta Method. The name of the method comes from the fact that the main item that you are looking at to make your forecast is the change (known by the Greek letter Delta) in the forecast temperatures between several times and altitudes in the atmosphere and using this difference (or delta) to predict the upcoming day's high temperature. I will illustrate the exact process in detail below using the city of San Antonio, Texas during the period from February 5th to February 6th, 2007. Bur first, there are a couple of key restrictions that you must keep in mind in order to apply the delta method correctly and successfully:

The wind directions for both the current day and the following day (the one you are forecasting the high temperature for) should be similar. Note, this doesn't mean they have to be identical, but they should be close. For example on the current day say the wind is southeast and on the upcoming day it's forecast to be east or south, that is probably close enough to be similar. Now if it was southeast and it switched to something like northeast or southwest, that's a fairly dramatic change and the delta method would likely not work out.

More importantly probably than the wind direction, the amount of cloud cover, and hence incoming solar heating, should be similar during the daytime hours of each day. Again, it doesn't mean identical here, but close. If for example, day one has scattered clouds throughout the day and on the upcoming forecast day the clouds were predicted to be broken for the first few hours of the day and then become scattered that would likely work out successfully. Now if you had scattered clouds one day and the next was mostly overcast these clearly are not similar and the delta method would likely fail.

Now that we've qualified the conditions needed to apply the delta method, lets look at what conditions were at San Antonio on February 5th and what they were predicted to be on February 6th and then apply the delta method to obtain the high temperature for that day. Finally we'll compare the prediction to the actual high temperature for the day and see how well the method worked.

On February 5th we can see by looking at the METAR reports (Click here for a detailed Listing) the cloud cover for the day of the 5th showed basically a mostly sunny sky with just a few clouds in the sky as indicated by the blue highlighted items CLR for clear and FEW or BKN for Few or Broken cloud cover. Likewise looking at the wind directions highlighted in green we can see that for the beginning of the day the winds started out from a north to northwesterly direction before switching to a south to southwesterly direction after about the 1653Z report. Therefore what we will be looking at in the forecast will be for mostly clear skies and winds from something similar to a south to southwesterly direction.

Looking at the three forecast model outputs for San Antonio from February 5th at 12Z below we can see in the highlighted blue and green areas in fact that the cloud cover and wind directions are expected to be similar to today's values of clear and south to southwesterly respectively as shown by the CL or FW indications for the cloud cover and the wind directions from 17 to 23 depending on which of the three models you are looking at. Clearly the days are expected to be similar in terms of their cloud cover, and hence incoming solar radiation and their wind directions, so in this case it appears that the delta method should apply quite well.

 KSAT   GFS MOS GUIDANCE    2/05/2007  1200 UTC                      
 DT /FEB   5/FEB   6                /FEB   7                /FEB   8 
 HR   18 21 00 03 06 09 12 15 18 21 00 03 06 09 12 15 18 21 00 06 12  
 CLD  SC SC FW CL CL CL CL CL CL CL CL CL CL FW BK OV SC SC CL BK OV 
 WDR  18 19 16 17 18 22 21 19 19 19 18 17 17 20 20 18 19 18 16 17 20

KSAT   ETA MOS GUIDANCE    2/05/2007  1200 UTC                      
 DT /FEB   5/FEB   6                /FEB   7                /FEB   8 
 HR   18 21 00 03 06 09 12 15 18 21 00 03 06 09 12 15 18 21 00 06 12 
 CLD  BK FW CL CL SC CL OV CL BK FW CL CL CL CL CL CL CL CL CL CL CL 
 WDR  20 20 18 16 18 19 21 20 21 20 18 19 19 19 20 20 20 20 17 17 27

SAT  SC   NGM MOS GUIDANCE   2/05/07  1200 UTC                      
 DAY /FEB   5 /FEB   6                /FEB   7                /      
 HOUR   18 21 00 03 06 09 12 15 18 21 00 03 06 09 12 15 18 21 00     
 CLDS   BK CL CL CL CL CL CL CL CL CL CL CL CL CL CL SC SC SC SC     
 WDIR   20 18 15 16 17 23 00 00 20 19 17 17 17 18 17 19 19 19 17

The next item we need to apply the delta method to predict the upcoming day's high temperature is the high temperature reading for today. Looking at the METAR reports from above we can see that the high temperature occurred sometime in the 6 hours preceding the 2353Z report and that it was 21.7C or 71F. The last item that we need to utilize the delta method is a piece of data called grid interpolations, which are a computerized prediction for various locations of items such as temperature. These data items can be obtained from the Texas A&M meteorology department web site. Below are the relevant portions of the grid point interpolations for the ETA, NGM and GFS models:

ETA
---
Station: KSAT    Lat: 29.53  Lon: -98.47  Elev: 243  Closest grid pt: 25.1 km.
Initialization Time: 07-02-05 1200 UTC
PARAMETER/TIME      000    006    012    018    024    030    036    042    048
------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------
  DAY / HOUR        05/12  05/18  06/00  06/06  06/12  06/18  07/00  07/06  07/12
------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------
TEMPS
  2 M (F)               43     64     63     50     46     67     67     54     48
  850 MB (C)             5      5      6      6      5      6      7      9     13

NGM
---
Station: KSAT    Lat: 29.53  Lon: -98.47  Elev: 243  Closest grid pt: 25.1 km.
Initialization Time: 07-02-05 1200 UTC
PARAMETER/TIME      000    006    012    018    024    030    036    042    048
------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------
  DAY / HOUR        05/12  05/18  06/00  06/06  06/12  06/18  07/00  07/06  07/12
------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------
TEMPS
 1000 MB (C)            12     15     19     18     16     18     21     20     16
  850 MB (C)             5      5      6      6      6      7      9     11     11

AVN/GFS
-------
Station: KSAT    Lat: 29.53  Lon: -98.47  Elev: 243  Closest grid pt: 25.1 km.
Initialization Time: 07-02-05 1800 UTC
PARAMETER/TIME      000    006    012    018    024    030    036    042    048    054    060    066    072
------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
  DAY / HOUR        05/18  06/00  06/06  06/12  06/18  07/00  07/06  07/12  07/18  08/00  08/06  08/12  08/18
------------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------
TEMPS
  SFC (2 M) (F)         62     57     48     45     64     59     51     50     65     64     59     61     68
  850 MB (C)             5      6      6      6      8      9     12     12     12     13     13     14     15

In order to apply the delta method what we do is we compute the difference between the predicted temperatures from 0Z on February 6 to 0Z on February 7th which would include the day we are trying to forecast the high temperature for, February 6th. We then add that difference (or delta) to today's high temperature to obtain the prediction for tomorrow. The calculations are shown below for the various levels of the atmosphere, either 2 meters above the ground in the case of the ETA and GFS models or 850mb pressure level for all three (ETA, NGM and GFS) models.

ETA
850mb difference 06/00 6C to 07/00 7C = +1C (+2F)
2m difference 06/00 63F to 07/00 67F = +4F
High For 2/5/07 71F 
High Temp 850mb Delta 71 + 2 = 73F 
High Temp 2m Delta 71 + 4 = 75F

NGM
850mb difference 06/00 6C to 07/00 9C = +3C (+6F)
High For 2/5/07 71F 
High Temp 850mb Delta 71 + 6 = 77F

GFS
850mb difference 06/00 6C to 07/00 9C = +3C (+6F)
2m difference 06/00 57F to 07/00 59F = +2F
High For 2/5/07 71F 
High Temp 850mb Delta 71 + 6 = 77F 
High Temp 2m Delta 71 + 2 = 73F

To further refine our answers we could take a consensus or average of the various results we obtained above for each type of calculation:

850mb Consensus (73 + 77 + 77) / 3 = 75.6 say 77F
2m Consensus (75 + 73) / 2 = 74F

Lastly we could also take a consensus of both of the various methods the 850mb and the 2m results to give us one number:

Overall (77F + 74F) / 2 = 75.5 say 76F

Although this consensus is not strictly part of the delta method, it may allow us to get something of a better result by averaging the various methods
together.

As a result of our work above we have arrived at the forecast high temperature of around 76F for San Antonio, Texas for February 6th. Let's look at the relevant portion of the climatology report from the San Antonio National Weather Service Office and see how well we did:

...THE SAN ANTONIO CLIMATE SUMMARY FOR FEBRUARY 6 2007...
 
 CLIMATE NORMAL PERIOD 1971 TO 2000
 CLIMATE RECORD PERIOD 1886 TO 2006
 
 WEATHER ITEM   OBSERVED TIME   RECORD YEAR NORMAL DEPARTURE LAST     
                 VALUE   (LST)  VALUE       VALUE  FROM      YEAR     
                                                   NORMAL             
 ..................................................................
 TEMPERATURE (F)                                                     
  YESTERDAY                                                           
   MAXIMUM         71    410 PM  83    1925  65      6       69

The actual high temperature was 71F, about 5 degrees higher than our consensus number, but if we look at the various individual answers we obtained above, several of them were much closer in fact. The ETA 850mb result and GFS 2m result were just 2 degrees off at 73F, so clearly the method does work, and rather well, although the value of taking a consensus number is debatable based on just this one example.

Comparing our forecast highs to the expected numbers from the various MOS outputs above we see that they had predicted 69, 68 and 71(red highlighted items), far less accurate than our delta method, even with the questionable consensus numbers.

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