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6−4−1 



 

Section 6-4 

 

 

Computation of Mean Areal Temperature 

 

Introduction 



 

Temperature time series are needed for model calibration whenever snow is to be included in the

  computations.    Temperature time series are also needed when the frozen ground options are to 

be used in the Sacramento Model and possibly for some of the other NWSRFS operations such a

s the Consumptive Use irrigation model to estimate evaporation.    As was pointed out in chapter

  6, the snow model is very sensitive to temperature.    Small differences in temperatures can hav

e a significant effect on the timing of snowmelt and will alter the model’s determination of whet

her precipitation is in the form of rain or snow.    Thus, it is very important that the MAT estimat

es are as unbiased as possible with regard to what actually occurred in nature.    Bias can easily o

ccur if one is not careful when extrapolating low elevation temperature measurements to high ele

vation portions of the basin.    Such bias can seriously affect the simulation results and the proper

  determination of snow model parameter values.   

 

At this stage of the process it is assumed that all the temperature data are available in the proper f



orm and that the data have been checked for consistency and adjusted if necessary.    It is also as

sumed that the areas where MAT estimates are to be generated have been determined based on th

e definition of the flow points needed for calibration and the possible subdivision of the drainage

  areas into elevation zones or subareas specified by other means.    It is also assumed that it has 

been determined as to whether mountainous or non-mountainous area procedures should be used 

to compute areal estimates.    Generally the mountainous area procedures need to be used for tem

perature only when there are definite variations in elevation over the river basin. 

 

Limitation of Current NWSRFS Historical MAT Program 



 

The current NWSRFS historical data MAT program uses only daily maximum and minimum tem

perature data to compute 6 hour mean areal values.    In order to generate 6 hour estimates from 

daily max/min data, the program must make assumptions as to when during the day the max and 

min values occur and the shape of the diurnal variation in temperature.    The program assumes t

hat the max temperature occurs in the early afternoon and the min temperature occurs around 6 a.

m..    The observation time of the max/min values is used to determine the day to which each val

ue is assigned.    If a station has a p.m. observation time, it is assumed that the max and min occu

r on the day that the observation is taken.    If a station has an a.m. observation time, it is assume

d that the recorded max temperature occurred the previous day and the min temperature occurred

  that morning.    Thus, it is important to specify changes in observation time whenever there is a 

change from afternoon to morning or vice-versa for a station.    The diurnal temperature pattern t

hat is used in the MAT program is based on typical spring time patterns at the Central Sierra Sno

w Laboratory near Donner Pass in California and the NOAA/ARS snow research station near Da

nville, Vermont. 

As would be expected, the daily temperature pattern produced by the MAT program is in error w




 

 

6−4−2 



henever the actual max and min temperatures occur at times considerably different from those as

sumed.    However, errors also occur just due to the use of daily max/min values.    For an a.m. o

bserving station, rather than the recorded minimum temperature occurring that morning as assum

ed by the program, the recorded value could have occurred the previous morning.    For a p.m. ob

serving station, rather than the recorded maximum temperature occurring that afternoon as assum

ed by the program, it could have occurred the previous afternoon.    Thus, even though the max a

nd min values occur at the times assumed by the program, they may be assigned to the wrong da

y.    Both of these problems, i.e. values occurring at times other than assumed and values being a

ssigned to the wrong day, are illustrated in Figures 6-4-1 and 6-4-2.    In Figure 6-4-1 not only ar

e the 6 hour temperatures in error on days 5 and 6 due to the max actually occurring in the early 

morning hours, but some of the 6 hour values on days 2 and 3 are also off because the min record

ed on the 3

rd

 actually occurred on the morning of the 2



nd

.    Similarly in Figure 6-4-2, the 6 hour 

MAT values on days 5 and 6 are in error for the same reason as the previous figure and some of t

he 6-hour values on day 4 are off because the max recorded on the afternoon of the 4

th

 actually o



ccurred on the afternoon of the 3

rd

.    Both these problems could be eliminated by using instantan



eous temperature data to determine the diurnal pattern and the time of occurrence of the max and

  min values.    Such an enhancement would significantly reduce problems of mistyping the form 

of precipitation and would improve computations of the timing of snowmelt in many regions of t

he country.    Instantaneous temperatures are used in the operational MAT program.   

 

 

 



 

 



 

 

6−4−3 



 

 

 



 

 

 



 

 

 



 

 

 



 

 

 



 

Figure 6-4-1.    Sample MAT computations using only max and min

  temperatures and an a.m. observation time. 

 

 



 

 

 



 

 

 




 

 

6−4−4 



 

 

 



 

 

 



 

 

 



 

 

 



 

Figure 6-4-2.    Sample MAT computations using only max and mi

n temperatures and a p.m. observation time. 

 

 



Estimation of Missing Max/Min Temperatures 

 

Unlike with precipitation, mean monthly temperatures are used to adjust estimator stations to be 



consistent with the station with the missing values in non-mountainous, as well as mountainous a

reas.    Missing temperature values, either maximum or minimum, are estimated in NWSRFS by 

the equation: 

 

           



                  (6-4-1) 

 

                                                                                                                                            _ 



where:    T = max or min temperature value (T signifies mean value), 

                          x = station being estimated, 

              i = estimator station, 

             n = number of estimator stations, and 

            w = weight applied to each estimator; computed as: 

 

           



                        (6-4-2) 

 

 



where:  d = distance, 

                    ΔE = elevation difference, and 




 

 

6−4−5 



                        F

e

 = elevation weighting factor. 



 

In non-mountainous areas the elevation weighting factor is zero.    In mountainous areas F

e

 is use


d so that elevation, as well as distance, is used to determine which estimator stations are used.   

This is important since elevation has such a dominant effect on temperature in the mountains.   

The "TEMPCK" option in the MAT program can be used to investigate the effect of different F

e

 



values on the accuracy of the estimates for a selected station.    Normally, default values of F

e

 are



  used.    The default values suggested are 20 (miles/1000 feet) for English units and 100 (kilome

ters/1000 meters) for metric units.    As with precipitation, the station with the largest weight and

  an observed data value in each quadrant is used as an estimator.    The quadrants are again base

d on the HRAP 4 kilometer grid system.    Also, as with precipitation, estimated values are not us

ed to determine missing values. 

 

The TEMPCK option in the MAT program allows the user to select various values of F



e

 to estim

ate all the maximum and minimum temperatures at a specified station and then statistically comp

are the estimated and observed values.    The TEMPCK option was used in the White Mountains 

of New Hampshire and the Mogollan Rim area of Arizona to investigate how the accuracy of the

  estimates varied with F

e

.    A value of F



e

 that gave nearly the best statistical result in both regio

ns was selected as the default.    It should be noted that for any given station the optimal value of

  F


e

 depends on the locations and elevations of the surrounding stations in the network.    The mo

st significant changes in the accuracy of the estimates occur when the use of a different F

e

 causes



  a change in the estimator station in one or more quadrants.    If the closest stations in terms of di

stance are also the nearest in elevation, changing F

e

 doesn’t have much affect on the accuracy of 



the estimates.    It is recommended that the default value of F

e

 be used unless the TEMPCK optio



n clearly shows that another value would improve the estimation of missing data.    It is likely th

at the TEMPCK option would only be worthwhile trying for a station with a lot of missing data t

hat had a large weight for an important area.. 

 

Once all the missing max and min values have been estimated for all stations, then the assumed t



ypical diurnal temperature variation is applied in order to compute 6 hour mean temperature valu

es at all the stations.    If all stations have missing values for max or min temperatures on a given

  day, several resulting MAT values will be set to missing. 

 

 



 

Computation of Station Mean Temperature 

 

Unlike precipitation, where the PXPP program computes monthly mean values in a consistent m



anner for the entire period of record, there currently is no preliminary processing program for te

mperature in NWSRFS.    The initial mean monthly max and min values are generally computed 

directly from the observed data for the station and then altered when consistency corrections are 

applied as mentioned in Section 6-2 (under the section titled “Guidelines for Making Consistenc

y Corrections”), however, the mean values are still based only on the period of observed data for 

the station.    Inconsistencies exist when different stations use different periods of record to comp




 

 

6−4−6 



ute the mean monthly values.    Since there is no preliminary processing program for temperature

, the stations used should have relatively long periods of record (at least 70% of the historical dat

a period as mentioned in Chapter 3) so that the means computed from the data are close to the m

ean for the entire MAT computational period.      However, in mountainous areas there are situati

ons when parts of the region, especially high elevation areas, do not have any stations with that l

ength of data and thus, stations with relatively short periods of record need to be included in the 

analysis.    For stations with relatively short periods of record, the monthly means should be com

puted in a manner similar to that used by the PXPP program for precipitation.    This procedure i

nvolves the use of a base station with a long, consistent record.    The elevation of the base statio

n should also be as close as possible to the elevations of the stations with short periods of record.

    For temperature data this procedure must be done manually.    The steps in the process are as f

ollows: 


 

1. Select a base station and get its mean monthly max and min values for the MAT computati

onal period from a MAT run for the entire period. 

 

2. For each station with a short period of record, make an MAT run for the period that the sta



tion has data and get the mean monthly values for both that station and the base station for thi

s shorter period.    Compute the difference in mean monthly max and min temperatures betwe

en each station and the base station for the period when both stations have data. 

 

3. For each station with a short record, apply the difference computed from the period when 



both it and the base station have data to the mean values obtained in step 1 for the base statio

n.    This results in an estimate of the mean values for the entire period for the stations with s

hort records.    These estimates will be the initial mean monthly max and min values to use fo

r these stations. 

 

4. As consistency corrections are applied and new mean values are computed, do not substitu



te the updated means computed by the MAT program for the stations with short records as th

e updated mean values computed by the program are based solely on the period of observed 

data for each station.    If consistency corrections are applied to the stations with short record

s, the mean monthly values for those stations will need to be manually adjusted based on the 

magnitude of the correction and the portion of the observed data period over which it is appli

ed. 


Table 6-4-1 contains an example of the process for computing the appropriate means for a station

  with a short record.    In this example MAT values are to be generated for the historical period f

rom 1949 to 1998.    Station B has a consistent and essentially complete (i.e. very little missing d

ata) record for the entire historical period.    The table shows the mean monthly max and min val

ues computed for this station for the 1949-1998 period.    Station X is a high elevation station tha

t was not available until 1985.    Station B is the highest elevation station with a long record in th

e vicinity of Station X.    By running the MAT program for the 1985-1998 period, mean monthly

  values can be generated for both stations for this portion of the historical record.    These values

  are shown in the table (note that the means for Station B for 1985-1998 are generally higher tha

n those for the entire historical period indicating that the later years were somewhat warmer).    T




 

 

6−4−7 



he average difference in temperatures between Stations X and B are then computed based on the 

period when both had data (note that in order to get a good estimate of the average difference bet

ween the stations, the record for both stations should be as complete as possible during the overl

apping period -- the program computes the mean of all observed values for each station, whereas 

in reality the difference should be computed using only those months when both stations have no

  missing data) .    These differences are then added to the means for Station B for the entire histo

rical period in order to estimate the means for Station X that are appropriate for use during the 19

49-1998 period. 

 

 

Sta./Yrs. 



 

value 


 

Jan 


 

Feb 


 

Mar 


 

Apr 


 

May 


 

Jun 


 

Jul 


 

Aug 


 

Sep 


 

Oct 


 

Nov 


 

Dec 


 

max 


 

30.3 


 

 31.


 

 38.



 

 45.



 

 52.



 

 60.



 

 68.



 

 69.



 

 54.



 

 42.



 

 35.



 

 31.



3

 

 



Sta. B 

49-98 


 

min 


 

    2.


 

      5



.6 

 

 11.



 

 21.



 

 28.



 

 31.



 

 35.



 

 36.



 

 30.



 

 20.



 

 10.



 

      6



.4

 

 



max 

 

33.6 



 

 34.


 

 41.



 

 47.



 

 54.



 

 63.



 

 70.



 

 70.



 

 57.



 

 45.



 

 38.



 

 33.



8

 

 



Sta. B 

85-98 


 

min 


 

    5.


 

      9



.5 

 

 12.



 

 24.



 

 30.



 

 33.



 

 37.



 

 35.



 

 33.



 

 24.



 

 14.



 

      8



.9

 

 



max 

 

26.8 



 

 27.


 

 34.



 

 39.



 

 45.



 

 53.



 

 60.



 

 60.



 

 47.



 

 37.



 

 30.



 

 27.



4

 

 



Sta. X 

85-98 


 

min 


 

    9.


 

 13.



 

 15.



4  

 

 25.



 

 30.



 

 31.



 

 34.



 

 32.



 

 31.



 

 23.



 

 16.



 

 12.



3

 

 



max 

 

 -6.8 



 

 -7.4 


 

 -7.2 


 

    -8.


 

 -9.2 



 

 -9.8 


 

-10.4 


 

-10.1 


 

 -9.6 


 

 -8.4 


 

 -7.3


   

 

    -6.



4

 

 



  Diff. 

 (X-B) 


 85-98 

 

min 



 

    3.


 

    4.



 

      2



.6 

 

      1



.2 

 

 -0.4 



 

 -2.4 


 

    -3.


 

    -3.



 

 -1.9 



 

 -0.3 


 

    1.


 

      3



.4

 

 



max 

 

 23.



 

 24.



 

 31.



 

 37.



 

 43.



 

 50.



 

 58.



 

 59.



 

 45.



 

 34.



 

 28.



 

 24.



9

 

 



Sta. X 

49-98 


 

min 


 

      6


.5 

 

      9



.8 

 

 13.



 

 22.



 

 28.



 

 29.



 

 32.



 

 33.



 

 28.



 

 20.



 

 12.



 

      9



.8

 

 



  Table 6-4-1.    Example of Computing Mean Temperatures for a Station with a Short Record. 

 

This procedure should certainly be used for any stations with data for less than 25 percent of the 



computational period.    It should be considered for all stations with data for less than about 70% 

of the computational period.    Ideally a preliminary processing program should be written for te

mperature data so that the proper means for all stations for the entire historical period will be co

mputed correctly and automatically. 

 Non-Mountainous 

Areas 


 

Determination of Station Weights and Generation of MAT Time Series 




 

 

6−4−8 



 

For non-mountainous areas the stations to be weighted and the weight assigned to each station ar

e based on the location of the station relative to the boundaries of the area for which MAT is bein

g computed.    In the NWSRFS historical data MAT program, grid point weighting is used in non

-mountainous areas.    Basin boundaries are specified by latitude/longitude points.    The 4 km H

RAP grid is overlain over each MAT area and 1.0/d weights are determined for each grid point w

ithin the area based on the closest station in each quadrant around the point.    The weights for all

  the grid points in the area are then normalized so that the sum of the station weights is 1.0.    Th

ese weights are then applied to the 6 hour mean temperature values that have been computed for 

each station to get the 6 hour MAT time series for each area. 

 

 Mountainous 



Areas 

 

Introduction 



 

In a mountainous area the average temperatures vary considerably over each MAT area with mos

t of the variation due to differences in elevation.    Thus, the estimation of the mean areal tempera

ture in mountainous regions is based on the typical relationship between temperature and elevati

on.    The temperature-elevation relationship serves the same function for the recommended mou

ntainous area temperature procedure as an isohyetal analysis does for the precipitation procedure.

    Such relationships are generally regional in nature, thus one relationship can be used for a num

ber of MAT areas to estimate the average temperature at the mean elevation of the area. 

 

Temperature-Elevation Analysis 



 

The temperature-elevation analysis should be done on a regional basis, not for individual watersh

eds.    The regions for which temperature-elevation relationships are developed need to be carefu

lly selected based on climatic factors.    Typically for temperature the major factors are latitude a

nd distance from major water bodies, especially oceans.    Other climatic factors may also need t

o be considered.    It is up to the user to determine the regions over which the relationship of tem

perature with elevation can be considered the same. 

 

The typical relationship between temperature and elevation varies seasonally due to differences i



n heating caused primarily by differences in the length of days and the amount of solar energy.   

Also the variations of maximum and minimum temperature with elevation are not typically the sa

me.    The lapse rate during the middle of the day when the max temperature generally occurs is 

generally steeper than the lapse rate in the early morning when temperatures are typically at the 

minimum values.    Thus separate temperature-elevation relationships are developed for max and 

min temperatures on a monthly basis to reflect these variations. 

 

In order to develop the relationships, the mean max and min temperatures for all stations in the re



gion (after consistency corrections have been made) are plotted versus elevation for each month 

of the year.    In NWSRFS, the TAPLOT program can be used to generate these plots.    TAPLO

T uses the letters of the alphabet to plot each station, thus a maximum of 26 stations are allowed.



 

 

6−4−9 



    If there are more than 26 stations in a region, some of the stations that represent similar elevati

ons can be removed before generating the plots.    Once the plots are generated, lines can be draw

n for each month to represent how max and min temperatures typically vary with elevation.    Wh

en drawing the lines there are several factors that should be considered: 

 

1. Though one straight line is normally drawn, the rate of change of temperature with elevati



on can vary in some regions.    Also during certain months in some areas, lapse conditions m

ay persist at higher elevations, while there is an inversion at the lower elevations.    This is es

pecially true during winter months in interior Alaska where even max temperatures can show

  inversions at low elevations during months with little or no sunlight. 

 

2. There may be some stations that experience local effects, like cold air drainage, and are th



us not representative of the elevation where they are located 

 

3. The lines drawn should represent physically realistic lapse and inversion conditions.    For 



example, lapse rates are typically in the range of about 0.3 to 0.9 

°C/100 m for most areas, th

ough they can be less, especially for min temperatures, in the winter at northern latitudes.   

A typical lapse rate used in many snowmelt modeling studies throughout the world is 0.6

°C/

100 m (3.3



°F/1000 ft) for mean daily temperature (the max temperature lapse rate would be g

reater than this value and the min temperature lapse rate would be smaller).    It can be helpfu

l to draw a 0.6

°C/100m lapse rate line on each plot to assist in making sure that the relationsh

ip selected is physically realistic. 

 

4. The seasonal variation in lapse rates should exhibit a smooth transition from month to mon



th.    Typically lapse rates are greatest in the summer and smallest in the winter.    In some re

gions minimum temperature lapse rates will show a double peak.    In such regions the min la

pse rate will peak in the spring and again in the fall when humidity is lower and it is easier to

  cool the atmosphere and decrease during the humid summer months.    It is a good idea to c

ompute the lapse rates from the temperature versus elevation relationships that are developed

  and plot the rates on a monthly basis to make sure the seasonal changes are reasonable and t

hat there are not abrupt variations from one month to the next. 

 

The temperature versus elevation relationships are primarily used to extrapolate temperatures fro



m lower to higher elevations due to a lack of high elevation data in most basins.    Since most sno

w accumulation, and thus the majority of the runoff, comes from higher elevations, it is critical th

at the extrapolation is physically realistic.    Improper lapse rates can result in biased MAT value

s being generated at higher elevations.    Biased temperatures will result in unreasonable snow m

odel parameter values and poor simulations of the snow accumulation and ablation process. 

Figure 6-4-3 shows max and min temperature versus elevation plots for the month of March for t

he Merrimack River basin in Massachusetts and New Hampshire.    This river basin was divided 

into 2 regions as far as temperature versus elevation relationships.    This was partly due to differ

ences in latitude from south to north in the basin and partly due to the influence of the ocean on t

he southern portion of the basin.    During the summer months the relationships were essentially t

he same, as the cooling effects of the ocean in the south offset the increased latitude of the northe



 

 

6−4−10 



rn portion.    In the winter, temperatures were significantly cooler at a given elevation in the nort

h than in the south where the ocean had a moderating effect.    The lapse rates for max and min te

mperatures for each month were the same for both regions, only the intercepts varied.    It should 

be noted that in addition to the max and min temperature versus elevation lines for the two region

s, a line showing the 0.6

°C/100m lapse rate is also drawn on the plots to assist in making sure tha

t the lapse rates are realistic. 

 

 



 

 

 



 

 

 



 

 

 



 

 

 



 

 

 



Fi

gu

re 



6-

4-

3.



    Temperature-elevation plots for March for the Merimack River 

basin. 


 

 

Figure 6-4-4 shows a plot of the seasonal variation in lapse rates for the Upper Missouri River ba



sin.    This figure shows the lapse rates computed from the initially drawn temperature versus ele

vation lines and adjusted lapse rates.    These adjustments were made so that the seasonal lapse r

ate pattern made a reasonably smooth transition from one month to the next.    The adjusted lapse

  rates were used to slightly modify the initial temperature versus elevation lines.    It is very imp

ortant to compute and plot the lapse rates for each month as a check that the values are realistic a

nd that the transition from month to month is reasonable. 

 

Determination of Station Weights 



 

The procedure recommended for temperature computations in mountainous areas for use with N

WSRFS involves establishing a synthetic or "dummy" station at the mean elevation of each MA



 

 

6−4−11 



T area and giving all the weight to the synthetic station.    A synthetic station is one with no 

  observed data; i.e. all the data values are estimated from surrounding real stations.    In order to 

define a synthetic station, mean monthly max and min temperatures and a location are needed.   

The mean monthly max and min temperatures are picked off the regional temperature versus elev

ation plots for the mean elevation of the MAT area that the synthetic station represents.    The loc

ation of the synthetic station is subjective, but should be selected so that the best possible estimat

or stations will be used to generate the synthetic data values.    The location, elevation, and comp

leteness of data for potential estimator stations are the major factors to consider when locating th

e synthetic station.    Ideally the elevation of the estimator stations should be as similar as possibl

e to the elevation of the synthetic station.    Typically the synthetic station should be located so t

hat several good estimators will be used to generate the data values. 

 

 



Figure 6-4-4.    Seasonal variation in lapse rates for the Upper Missouri River basin. 

 

Computation of MAT 



 

The final step in the historical temperature analysis process in a mountainous area is to input the 

synthetic station information and the station weights and run the calibration MAT program to co

mpute the time series for each MAT area for the entire period of record.    For each MAT area, th

e synthetic station associated with that area is given a predetermined weight of 1.0 and all other s

tations, real and synthetic, are given zero weight.    Besides generating 6-hour MAT time series, t

he program will produce summary tables for each zone showing the mean temperatures for each 

month and year, as well as the overall mean for the period of record. 




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