United States

Department

of Agriculture

Forest Service

Rocky Mountain

Research Station

General Technical

Report RMRS-GTR-165

January 2006

A Spline Model of

Climate for the

Western United States

Gerald E. Rehfeldt
Rocky Mountain Research Station

Publishing Services

Telephone (970) 498-1392

FAX (970) 498-1396

E-mail rschneider@fs.fed.us

Web site http://www.fs.fed.us/rm

Mailing Address Publications Distribution

Rocky Mountain Research Station

240 West Prospect Road

Fort Collins, CO 80526

Abstract __________________________________________

Monthly climate data of average, minimum, and maximum temperature and precipitation normalized

for the period 1961 through 1990 were accumulated from approximately 3,000 weather stations in the

Western United States and Southwestern Canada. About two-thirds of these observations were

available from the weather services of the two countries while the remaining third were added to the

normalized base from daily weather records of stations of short duration. Tests of the procedures used

to normalize these supplemental data showed that estimates on average were within 0.2 °C for

temperature variables and 2.7 mm for precipitation.

Weather data for the 48 monthlies were fit to geographic surfaces with thin plate splines.

Relationships between predicted values and observed monthlies for about 245 records withheld from

the modeling process produced values of R2 that averaged about 0.95 and ranged from 0.87 to 0.99.

The slope of the regression line for these relationships was essentially 1.0 for all 48 comparisons.

Predictions from the climate model can then be converted to variables of demonstrated importance in

plant geography, ecology, or physiology. As an illustration, algorithms are presented and justified for

estimating 18 variables derived from predicted values. These derived variables range from the

straightforward such as mean annual temperature or mean temperature in the coldest month to those

of degree-days >5 °C or freezing dates.

Applications of the model in plant biology are illustrated for (1) generating climate estimates for

locations specified by latitude, longitude, and elevation, (2) mapping climate variables, (3) separating

species distributions in climatic space, and (4) relating genetic variation among populations to climatic

gradients.

Keywords: thin plate splines, climate model, climate normals, climate surfaces, predicting climate,

mapping climate

Rehfeldt, Gerald L. 2006. A spline model of climate for the Western United States. Gen. Tech.

Rep. RMRS-GTR-165. Fort Collins, CO: U.S. Department of Agriculture, Forest Service, Rocky

Mountain Research Station. 21 p.

Author ___________________________________________

Gerald E. Rehfeldt is a retired Plant Geneticist with the Rocky Mountain Research Station,

Forestry Sciences Laboratory, Moscow, ID. He completed M.S. and Ph.D. degrees by 1968

at the University of Wisconsin, and spent the last 39 years studying genetic variation in Rocky

Mountain conifers.

Acknowledgment __________________________________

The development of this climate model and the preparation of this report were dependent

on contributions of many people from disparate disciplines. Contributors listed chronologically include Nadejda Tchebakova, Robert Monserud, Run-Peng Wei, William Wykoff,

Jeffrey Evans, Andrea Brunelle, Marcus Warwell, Andrew Hudak, Nicholas Crookston, and

Andreas Hamann.
Contents_____________________________________

Page

Introduction .........................................................................................................1

Climate Data and Their Normalization ................................................................1

Fitting of Thin Plate Splines to Monthly Climate Normals ...................................4

Deriving Climate Variables Relevant to Plant Biology.........................................8

Applications .......................................................................................................18

Conclusions.......................................................................................................20

References ........................................................................................................21

The use of trade or firm names in this publication is for reader information and does not

imply endorsement by the U.S. Department of Agriculture of any product or service

Federal Recycling Program Printed on Recycled Paper

USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 1

A Spline Model of Climate for the

Western United States

Gerald E. Rehfeldt

Introduction ______________________________________________________

Humans have been aware for centuries that climate is the primary factor controlling

the distribution of plants, but even today, plant-climate relationships are poorly understood (see Woodward 1987). To be sure, assessing the ecological relationships between

plants and climate has been hampered by a network of weather stations that largely reflect

agronomic interests and population centers. Ecological research, therefore, has been

impeded by an inability to predict climate for remote locations. Two recent developments

now alleviate this problem. First, concerns about anthropomorphic effects on climate

have generated large volumes of climate data. Second, the thin plate splines of

Hutchinson (1991, 2000) make possible the fitting of climate data to geographic surfaces.

Spline climate models can provide estimates of climate for specific points on the

geographic surface, identified by latitude, longitude, and elevation. Such estimates are

suited directly for assessing plant responses to climate.

This paper describes a spline climate model largely for the Western United States but

also Southwestern Canada, from longitudes between 102°W and 125°W and latitudes

between 31°N and 51°N. The model, which yields predictions of monthly temperature

and precipitation, is described and verified, algorithms that convert monthly values to

variables relevant to plant geography and physiology are presented and justified, and the

utility of the model in plant biology is illustrated. The description of the climate model

and the format of this paper follow that of McKenney and others (2001) for their climate

surfaces of Canada.

Climate Data and Their Normalization _________________________________

This model is based on the 1961 through 1990 monthly averages of daily minimum,

maximum, and average temperature and daily precipitation produced by the weather

services of the United States (U.S. Department of Commerce 1994) and Canada

(Environment Canada 1994). Because missing observations invariably exist within

weather records, adjustments are necessary for standardizing observations between

stations. These adjusted records are referred to as the monthly normals by the weather

services. For the area of interest herein, the number of stations represented in the

normalized data base were approximately 1,900 for the United States and 200 for

Southwestern Canada, with the number varying somewhat for each climate variable.

The weather stations contained in these data bases are referred to as standard stations,

and in order to qualify as a standard station there must be at least 20 years of record for

the 30-year period. To increase the representation of remote locations in this largely

agronomic data base, daily data from stations not represented in the data set of monthly

normals were normalized from raw daily data provided by EarthInfo, Inc. (1994). Raw

data were accepted if the station had at least seven complete years of observations for

precipitation and 5 years for temperature, and if the monthlies had less than four

missing observations.

The raw data were adjusted to the 1961 through 1990 normals by first selecting the

three standard stations that were the closest geographically to the station whose data
2 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006

were being normalized. The three closest standard stations were identified by sequential

screenings according to horizontal and vertical distances. The first screening was based

on a horizontal distance of 50 km and a vertical distance of 300 m. If this screening failed

to yield three stations within the established limits, a second screening was made using

a horizontal distance of 50 km and an altitudinal distance of 500 m. Likewise, a third and

fourth screenings were implemented when necessary using a horizontal distance of 80

km and, first, an altitudinal distance of 500 m, and, second, an unlimited altitudinal

distance. In approximately 80 percent of the cases, the closest three standard stations

were identified prior to the third level; less than 1 percent of the cases required level four

to locate three suitable standard stations.

Temperature normals were calculated by averaging deviations between monthly

means from data being normalized and those of the three geographically proximal

standard stations: Let N1, N2, and N3 be monthly normals for the three standard stations

closest geographically to station i. Then a normalized value of raw temperature data from

station i in month j becomes:

N

N

X X

n

N

X X

ij

j

ijk jk

jk

j

ijk

=

−  ( ) −















+ − ∑ − 1

1

2

2 jk

jk

j

ijk jk

n jk

N

X X

n

 ( )















+ −  ( ) −











∑ ∑

3

3





3

where Nij is an estimate of the 1961 through 1990 normal for station i in month j; Xijk is

raw data for station i in month j for year k common between station i and the standard

station; X1jk, X2jk, and X3jk are raw data for the standard stations; and n is the number of

years k in common between the two stations in month j.

Precipitation normals were calculated similarly by using ratios of the raw data to those

of a standard station:

N

N

X

X

N

X

X

N

ij

j

ij

j

j

ij

j =

×





 



 + ×





 



 + •

•

•

•

1

1

2

2

3

3

3

j

ij

j

X

X

×





 





•

•

where Xij•, is raw data for station i in month jsummed for all years k in common between

station i and the standard station; and X1j•, X2j•, and X3j• are raw data for the three standard

stations summed for the years in common between the two stations. Other variables are

defined above.

To test the adequacy of the normalization process, six standard stations of disparate

climate were removed from the normalized data base of eight States and were normalized

anew according to the procedures detailed above. In total, 48 observations were available

for comparing monthly normals of standard stations against those produced by the

normalizing procedures. On average, differences in monthly precipitation were within

2.7 mm, the standard deviation was 3.6 mm, and the largest difference was 21 mm, the

last of which was for Santiam Pass, Oregon, where precipitation averages 1,963 mm and

the error, therefore, is about 1 percent. For average, minimum, and maximum temperatures, the mean difference was 0.2 °C, the standard deviation was 0.14 °C, and the largest

difference was 1.1 °C. These comparisons thus provide a strong validation of the

procedures.

The procedures yielded an additional 1,200 weather stations with normalized monthlies. Data from stations that were either duplicates of or in proximity to standard stations

were discarded, the latter group of which was removed in order to break up clusters of

data points. This left approximately 1,000 additional stations available for the spline

models. These additional data were appended to the normals of the standard stations to

obtain a total of about 3,000 geographically diverse stations for the modeling of

precipitation monthlies (table 1, fig. 1) and 2,700 for temperature monthlies.
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 3

Table 1—Altitudinal range and number of weather stations within the

normalized precipitation database listed according to administrative

units

Number of Altitude (m)

Administrative unit weather stations Minimum Maximum

Alberta 27 717 1,439

Arizona 241 37 2,588

British Columbia 122 1 1,520

California 497 –59 3,801

Colorado 235 1,034 3,460

Idaho 167 303 2,226

Montana 297 585 2,274

Nebraska 26 1,010 1,479

Nevada 141 174 2,279

New Mexico 225 993 3,257

North Dakota 40 552 909

Oklahoma 3 1,226 1,326

Oregon 234 2 2,141

Saskatchewan 52 497 1,070

South Dakota 54 680 1,951

Texas 51 732 1,695

Utah 213 495 3,260

Washington 20 33 1,796

Wyoming 170 1,075 2,535

Figure 1—Location of weather stations

represented in the data base of

normalized monthly precipitation

available from national weather

services of the United States and

Canada (orange) and those appended

(blue) to this data base by procedures

described herein.
4 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006

Fitting Thin Plate Splines to Monthly Climate Normals ___________________

Thin plate splines of Hutchinson (2000) were fit to the twelve monthly values of four

climate variables in two stages. For both stages, the square root transformation was used

on precipitation data, but output is presented in the original units of measure. To ease

computation in large data sets, Hutchinson recommends the use of knots, which, in this

case, are stations located at regular geographic intervals. For the present analyses, 1,300

knots were used for the temperature splines and 1,600 for precipitation. In the first stage,

a data set was created for judging the effectiveness of the spline model. This data set

contained roughly equal contributions of data from stations (a) removed from the

normalized data in order to break up geographic clusters, (b) deleted because of

duplication, and (c) included to obtain a geographically representative sample. The total

number of stations represented in this data set was 242 for maximum, minimum, and

average temperature and 254 for precipitation. Hutchinson emphasizes the necessity of

using withheld data to judge the effectiveness of these nonparametric procedures. In the

second stage, the data withheld at level ‘c’ were returned to the normalized data base

before refitting the splines.

Relationships between the observed monthly normals of withheld data and those

predicted from the first stage of modeling are strong, with values of R2 ranging from 0.87

to 0.98 (table 2). Examples of the strongest and weakest relationships are illustrated in

figure 2. Table 2 also shows that the regression coefficient is near to 1.0 for all months

but that precipitation may be underestimated slightly when it is high. Table 2 and figure

2 thus demonstrate that the fit of the splines is quite good and is probably as good as can

be expected given the original distribution of the weather stations and the geographic

diversity of Western United States.

For the second and final stage of the model development, approximately one-third of

the withheld data were then returned to the data set of monthly normals, and the splines

Table 2—Statistics derived from the linear regression of observed monthly temperature and precipitation of data withheld from the

modeling on values predicted from the spline model. Regressions were of the form: Y = a + bX where Y is the observed

value from either 241 or 254 weather stations, and X is predicted from the spline model.

Month

Statistic 1 2 3 4 5 6 7 8 9 10 11 12

Mean average temperature

b 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.00

R2 0.97 0.98 0.98 0.98 0.97 0.96 0.95 0.95 0.96 0.96 0.97 0.97

s 1.05 0.91 0.78 0.75 0.80 0.94 1.02 1.04 1.00 0.94 0.86 0.96

Mean minimum temperature

b 1.01 1.02 1.03 1.03 1.02 1.02 1.00 1.00 1.01 1.03 1.02 1.01

R2 0.94 0.94 0.95 0.92 0.90 0.87 0.87 0.87 0.88 0.88 0.92 0.93

s 0.91 0.79 0.71 0.76 0.84 0.92 0.98 1.00 0.86 0.76 0.69 0.82

Mean maximum temperature

b 0.99 0.99 0.99 1.00 1.00 1.00 1.01 1.01 1.00 0.99 1.00 0.99

R2 0.98 0.98 0.99 0.98 0.97 0.97 0.96 0.95 0.97 0.98 0.99 0.98

s 1.66 1.52 1.28 1.28 1.40 1.63 1.74 1.79 1.82 1.76 1.56 1.62

Mean precipitation

b 1.04 1.04 1.03 1.02 1.03 1.02 0.97 1.03 1.07 1.10 1.04 1.03

R2 0.93 0.93 0.91 0.89 0.91 0.95 0.92 0.92 0.88 0.92 0.95 0.94

s 18.05 14.82 14.54 9.75 8.17 5.86 6.59 5.94 6.83 11.03 15.85 17.78

Note: s = standard deviation of the residuals
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 5

Figure 2—Predicted values of monthly means plotted against the actual illustrating one of the best fits (left)

and poorest fits (right) for 241 stations withheld from the normalized data base for temperature (upper)

variables and 251 stations for precipitation (lower). Statistics describing these relationships are in table 2.
6 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006

were fit anew. Hutchinson (2000) notes that among the diagnostic statistics, three in

particular are useful in judging the fit of the splines: the signal, the root mean square error

(RTMSE), and the root of the generalized cross validation statistic (RTGCV). The signal

is indicative of the degrees of freedom associated with the surface, which in a well-fitting

model should be approximately one-half of the number of observations (or knots);

RTGCV is a spatially averaged standard error, plus and minus two times of which is a

rough indication of the 95 percent confidence limits about a predicted point; and the

RTMSE is a measure of the standard error after the data error has been removed; it is,

therefore, an optimistic estimate of the surface error.

Table 3 presents diagnostic statistics for the second stage of the modeling. These

statistics show that the ratio of the signal to the number of knots is about 0.5 for the

temperature surfaces, but averages 0.64 for the precipitation surfaces. Hutchinson notes

that values larger than 0.5 would indicate either an inherently noisy system (large

variability) or a lack of adequate data. McKenney and others (2001) present ratios for

precipitation in excess of 0.7 and conclude, therefore, that the surfaces would be

improved if more data were available.

Table 3 also shows that values of the RTMSE are generally low. For precipitation

surfaces, this statistic, expressed as a percentage of the surface mean, ranged from 7

percent in the summer months to about 11 percent in the winter. McKenney and others

(2001) presented values as high as 28 percent of the mean and noted that winter

precipitation is a more spatially complex phenomenon. For temperature surfaces, the

standard errors are approximately 0.4 °C for the average and maximum temperature but

about 0.7 °C for minimum temperature. McKenney and others again note that larger

errors for minimum temperature are to be expected in mountainous terrain where cold

air drainages are prominent.

Errors of prediction expressed in the RTGCV suggest that predictions would be

surrounded by a confidence interval (α≈0.05) of about ±2 °C for average and maximum

temperature; ±3 °C for predictions of the minimum temperature; and about ±20 mm for

precipitation. Table 3 also shows that the ratio of the RTMSE to the RTGCV ranged from

0.42 to 0.47, indicating that the spline model overcame a substantial amount of noise in

the monthly normals (McKenney and others 2001). It is likely that some of this noise was

introduced by accepting data from stations with limited records (5 years for temperature

Table 3—Ratio of signal to the total number of knots, the root generalized cross validation (RTGCV), and the root mean square error

(RTMSE) for spline monthly temperature and precipitation surfaces. Number of knots was 1,400 for the temperature

variables and 1600 for precipitation.

Variable

Average temperature Minimum temperature Maximum temperature Precipitation

Month ratio RTGCV RTMSEa ratio RTGCV RTMSEa ratio RTGCV RTMSEa ratio RTGCV RTMSEa

1 0.58 1.00 0.46 0.56 1.59 0.72 0.60 0.84 0.39 0.64 15.2 7.16

2 0.55 0.90 0.41 0.54 1.47 0.66 0.55 0.78 0.35 0.64 12.7 6.00

3 0.49 0.74 0.33 0.50 1.25 0.54 0.49 0.75 0.32 0.65 12.5 5.95

4 0.48 0.73 0.32 0.48 1.24 0.54 0.48 0.82 0.36 0.68 8.4 4.03

5 0.50 0.79 0.35 0.49 1.32 0.58 0.49 0.91 0.40 0.63 6.8 3.22

6 0.52 0.91 0.41 0.51 1.49 0.66 0.51 1.03 0.46 0.60 5.4 2.50

7 0.52 1.00 0.45 0.54 1.58 0.71 0.50 1.19 0.52 0.56 5.3 2.39

8 0.51 1.00 0.41 0.53 1.63 0.73 0.50 1.15 0.50 0.62 5.2 2.42

9 0.48 0.94 0.39 0.50 1.68 0.74 0.48 0.99 0.43 0.64 5.4 2.57

10 0.47 0.89 0.37 0.49 1.68 0.74 0.51 0.81 0.36 0.62 8.3 3.87

11 0.51 0.83 0.38 0.50 1.49 0.66 0.53 0.69 0.31 0.64 13.1 6.20

12 0.57 0.97 0.45 0.55 1.58 0.72 0.61 0.80 0.37 0.64 14.5 6.87

a

Approximate untransformed value
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 7

and 7 years for precipitation) but most undoubtedly stems from the variable nature of the

climate, particularly in mountainous terrain.

Table 4 lists the eight stations with the largest RTMSE, calculated as a composite for

the 12 splines for each of the four variables. This table shows that for predictions of

average temperature, the stations with the largest composite residual were randomly

distributed geographically and, except for a coastal station from California, tended to be

from high elevation. Those with the largest residuals for the minimum temperature were

from high elevations where temperatures tend to be the coolest. The stations with the

worst fit for maximum temperature were from warm, maritime climates largely in

California, while those for monthly precipitation were all from northern coastal regions

where precipitation is high. Notice also that the source of the monthly normals had no

apparent effect on the size of the largest residuals. This suggests, therefore, that the

normalization procedures described herein did not unduly increase error variances.

Table 4—Eight stations and their locations with the highest aggregate root mean square residual

(RTMSR) for the 12 monthly surfaces of four climate variables.

Administrative

unit Station Latitude (°N) Longitude (°W) Altitude (m) RTMSR

Mean average temperature

Colorado 3113 39.95 105.83 2610 2.84

Arizona 4453* 34.75 112.12 1599 2.54

Montana 7750 38.03 114.18 1814 2.44

South Dakota 7227 44.13 103.73 1635 2.40

Montana 9072 41.66 115.79 1898 2.33

California 3191* 38.52 123.25 34 2.27

California 5356 38.70 119.78 1688 2.26

Utah 2057* 40.40 111.53 1607 2.18

Mean minimum temperature

Arizona 4453* 34.75 112.12 1599 5.26

Montana 7750 38.03 114.18 1814 4.70

Colorado 3113 39.95 105.83 2610 4.47

Utah 5607 41.22 112.64 1286 4.46

South Dakota 7227 44.13 103.73 1635 4.14

Nevada 8761* 39.30 119.63 1933 4.04

Utah 2696* 40.27 112.08 1488 3.99

New Mexico 3505 36.43 106.97 2263 3.98

Mean maximum temperature

California 7953* 34.00 118.50 4 4.50

California 9792 37.43 122.25 116 4.36

California 5866* 35.37 120.85 35 3.91

California 3191* 38.52 123.25 34 3.79

Oregon 4133 42.64 124.05 107 3.56

California 7767* 37.77 122.50 10 3.34

California 1484* 34.18 118.57 241 3.29

California 7714* 36.02 121.25 23 3.08

Mean precipitation

British Columbia 6330* 49.52 123.48 8 4.45

Washington 1760* 46.07 122.20 201 2.94

British Columbia 0590* 49.35 124.17 15 2.85

British Columbia 7200* 49.43 122.97 244 2.77

Washington 3333* 46.38 123.57 30 2.70

Washington 7538 48.08 123.10 57 2.54

California 0673 37.08 122.08 135 2.39

California 6154 34.95 119.68 658 2.29

*Data normalized by procedures described herein
8 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006

Deriving Climate Variables Relevant to Plant Biology ____________________

The spline climate surfaces predict 48 monthly values of temperature and precipitation. These values can be used to derive variables of demonstrable effectiveness in plant

geography, physiology, and ecology (see Tuhkanen 1980). The following 18 are a

sample:

1. mean annual temperature

2. average temperature in the coldest month

3. minimum temperature in the coldest month

4. average temperature in the warmest month

5. maximum temperature in the warmest month

6. annual precipitation

7. growing season precipitation, April through September

8. summer-winter temperature differential, variable #4 minus #2

9. degree-days >5 °C

10. degree-days <0 °C

11. minimum degree-days <0 °C

12. Julian date of the last freezing date of spring

13. Julian date of the first freezing date of autumn

14. length of the frost-free period

15. degree-days >5 °C accumulating within the frost-free period

16. Julian date when the sum of degree-days >5 °C reaches 100

17. annual moisture index, the ratio of variable #9 to #6

18. summer moisture index, the ratio of variable #15 to #7

Note that the first eight of these variables come directly from the monthlies, and four

others involve simple ratios or differences, but estimation of numbers 9 to 16 require

elaboration.

Estimation of Degree-Days

Degree-days are temperature sums above or below a threshold value. Three annual

degree-day sums are used as derived variables in the climate model: degree-days >5 °C

(DD5), degree-days <0 °C (DD0), and minimum degree-days <0 °C (MINDD0). The

first is widely accepted among plant geographers as a general indication of the warmth

of the growing season; the second is viewed as an indicator of the coldness of the winter;

and the third is included because many of the weather stations used herein from mild

climates record freezing temperatures but nonetheless have a DD0 equal to zero.

Degree-days are ordinarily calculated in three steps from daily temperatures:

1. Calculate for each day, i, the difference (Tdi) between the mean temperature of the

day (Ti) and the threshold temperature (Tt):

Tdi = Ti

 – Tt

where i is the Julian date.

2. Apply a condition that is dependent on whether DD5 or DD0 is being summed:

(a) if DD5 is the statistic of interest, all values of Tdi < 5 are equated to zero.

(b) if DD0 is the statistic of interest, all values of Tdi > 0 are equated to zero.

3. Summing Tdi across the period of interest. For an annual sum of DD5:

DD Tdi

i

5

1

365

=

=

∑
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 9

Notice, however, that these steps use daily temperatures while the spline climate

model predicts monthly temperatures. Obviously, Tdi could be calculated as above using

monthly average temperatures and multiplying by the number of days in the month to

estimate the degree-days that accumulated for that month. The problem, however, is that

in doing so, one ignores daily variations about the monthly mean. This would mean that

for the calculation of DD5, days with an average temperature > 5 °C would be ignored

if the monthly mean was < 5 °C. The resulting sum, therefore, would underestimate the

actual.

To provide an unbiased estimate of DD5 and DD0, regressions using monthly normals

of the National Climate Data Center (U.S. Department of Commerce 1994) and

Environment Canada (1994) were developed for predicting monthly degree-day sums

from monthly average temperatures. Because months have a different number of days,

the latter statistic was expressed as an average daily value by dividing the monthly sum

by the number of days in the month.

Degree-days > 5 °C—A complete set of monthly normals for average temperature and

DD5 were available for 2,143 weather stations for Western North America 1,868 stations

from the United States and 275 stations from Canada. The relationship between average

monthly temperature and degree-days >5 °C, expressed as the average daily accumulation,

is shown in figure 3, left. The figure illustrates that at temperatures >9.5 °C, the daily

Figure 3—Plots of average daily temperature monthly means against the number of degree-days > 5 °C

that accumulate per day during the month overlain with values predicted by the three-stage regression

model (left), and relationship between observed annual sums of DD5 with the predicted sum (right).
10 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006

accumulation is essentially equal to the amount by which the average temperature exceeds

5 °C; at mean daily temperatures below –13 °C, the daily contribution equals zero; and for

the interval of –13 °C to 9.5 °C, the daily contribution is exponential. Modeling this

response, therefore, could be done in three stages.

Stage 1: define γ as the daily contribution to the monthly sum of DD5, and equate γ

to zero for all temperatures < –13 °C.

Stage 2: fit the following power function to γ for temperatures between –13 °C and

+9.5 °C for each month:

γ ij

T c a be ij = + ( )′

where γ is daily degree-days >5 °C for station i in month j; a, b, and c are regression

coefficients; e is the base of the Napierian logarithms; and T' is the average temperature

for station i in month j transformed to a value between 0 and 1:

T T avg avg ′ = + ( ) 13 24 /

To provide a suitable number of degrees of freedom for the regression, observations

for June, July, and August were combined. As a result, the regressions were based on as

many as 1,891 observations in April and as few as 93 for the summer months.

All regressions were statistically significant (p < 0.01), accounting for an average of

95 percent of the variance in the dependent variable, a minimum of 92 percent (April),

and a maximum of 98 percent (January and September).

Regression coefficients for each month are:

ab c

January –4.3302 4.2501 3.8847

February –4.3866 4.2652 3.8238

March –4.2910 4.1651 3.5863

April –4.9051 4.0704 2.2671

May –4.2351 4.0838 3.8330

June –4.5055 4.2100 3.8902

July –4.5055 4.2100 3.8902

August –4.5055 4.2100 3.8902

September –4.2853 4.0123 3.3756

October –4.2737 4.0056 3.1198

November –4.2447 4.0844 3.5669

December –4.4445 4.3440 3.7888

Stage 3: fit the following linear regression for temperatures >9.5 °C for each month:

γ i = b0 + b1Ti + b2Mi

where γ is daily degree-days >5 °C for station i in month j; the b’s are regression

coefficients; and M is the mean annual temperature of station i. A minimum of 127

observations was available for these regressions (January), while the maximum was

2,118 (July and August). The smallest R2 was 0.996.

Regression coefficients for each month are:

b0 b1 b2

January –4.65118 0.96051 0.00807

February –4.67189 0.95861 0.01169

March –4.63736 0.90456 0.05482

April –4.55469 0.93601 0.03633

May –4.95205 1.01226 –0.01927

June –4.98636 0.99862 0.00046

July –4.99574 0.99926 0.00038
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 11

August –4.99671 0.99959 –0.00023

September –4.87564 1.00664 –0.01928

October –4.68604 1.02324 –0.04397

November –4.76599 0.96969 0.01103

December –4.31612 0.94154 0.00331

Note that these regressions predict daily degree-days from temperature by month. To

obtain the total for the month, the prediction must be multiplied by the number of days

in the month, nj

. The annual accumulation of DD5 is then the sum of the monthly

contributions:

DD nij j

j

5

1

12

= ( ) =

∑ γ

When this three-stage model is applied to the annual accumulation of DD5 for the

2,143 weather stations, a linear regression of observed values on the predicted produced

an R2 of 0.9994, an intercept of –15.5 DD5, and a regression coefficient of 1.0068 (fig.

3, right). The standard deviation of the residuals was 28.9 degree-days while the residuals

themselves ranged from –132 to +142.

Degree-days <0 °C—Negative degree-day monthly normals were available for

1,868 stations for the United States and 275 for Canada west of 100°. The relationship

between average monthly temperature and degree-days <0 °C expected to accumulate

each day is shown in figure 4, left. The figure shows that at average daily temperatures

>10 °C, the daily accumulation of negative degree-days is essentially zero, and that at

temperatures < –10 °C, the daily accumulation of degree-days is essentially equal to the

Figure 4—Plots of average daily temperature monthly means against the number of degree-days < 0 °C

that accumulate per day during the month overlain with values predicted by the three-stage regression

model (left), and relationship between observed annual sums of DD0 with the predicted sum (right).
12 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006

average temperature. This means, first, that bias in estimating DD0 would accrue when

10 °C > Tavg > –10 °C and, second, that a three-stage model would also be appropriate

for estimating DD0. As with DD5, therefore, regression models were used to fit mean

monthly temperatures to the average daily contribution to the monthly total of DD0.

Stage 1: define φ as the daily contribution to the monthly sum of DD0, and equate φ

to zero for all temperatures >10 °C.

Stage 2: fit the following function to daily for temperatures between –10 °C and

+10 °C for each month:

′ = + ( )′ φij

a bT c

e ij

where φ′

ij is daily degree-days <0 °C for station i; a, b, and c are regression coefficients;

e is the base of the Napierian logarithms; and T’ is the average temperature for station i

transformed as:

T T avg avg ′ = + ( ) 75 20 /

This transformation produced values between 3.25 and 4.25 for T ′ which provided

smoother transitions between stages than values between zero and 1.

The regression was based on slightly more than 21,000 observations, was statistically

significant (p < 0.01), and accounted for 97 percent of the variance in the dependent

variable. Regression coefficients for the best fitting model were:

a = 2.7862, b = -0.00000222, and c = 10.3641.

Despite the high degree of fit for this model, residuals were reduced further with the

following multiple regression:

φ φ ij ij i j ij = + b b bT bC 01 2 1 3 ′ + + + ˆ ( )

whereφˆ′

ij is the predictedvalue ofφij ; and T is the average temperature for station i in the

month following (j +1) month j, and Cis the average temperature in the coldest of the two

months adjacent to j.

Regression coefficients were: b0 = –0.09082, b1 = 1.00547, b2 = 0.01629, and b3 =

–0.01658. This model accounted for 98 percent of the variance in the dependent

variable, an increase of only 1 percent over the previous model. However, with more

than 20,000 degrees of freedom, a seemingly small increase in R2, produces a

substantial decrease in the residual mean square. Also, because the model predicts

daily contributions to the degree-day total, omitting a small correction for bias

produces errors that are magnified when daily contributions are summed to provide

annual estimates.

Stage 3: fit the following linear regression for temperatures < –10 °C:

φij ij ij ij =+ + + b bT bT bW 01 2

2

3

where W is the average temperature in the warmest of the 2 months adjacent to j.

The model accounted for 0.9969 of the variance in the dependent variable, and the

regression coefficients were: b0= 1.35535, b1= –0.88994, b2= 0.00143, and b3= –0.03380.

Note, again, that these calculations predict daily degree-days from temperature by

month. To obtain the total for the month, the prediction must be multiplied by the number

of days in the month, nj

. The annual accumulation of DD0 is then the sum of the monthly

contributions:

DD nij j

j

0

1

12

( ) φ

=

∑

When this three-staged model is applied to the annual accumulation of DD0 for the

2,143 weather stations, a linear regression of observed values on the predicted produced

an R2 of 0.9993, an intercept of 3.45 DD0, and a regression coefficient of 0.991 (fig. 4,
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 13

right). The standard deviation of the residuals was 53.5 degree-days, while the residuals

themselves ranged from –234 to +257.

Minimum degree-days < 0 °C—Predicting MINDD0 can be done with the algorithms presented above for DD0 but by using monthly mean minimum temperatures

instead of monthly average temperatures.

Estimation of Freezing Dates

Stepwise regression was used to develop algorithms for predicting the Julian date of

the last freeze of spring (SDAY), the first freeze of fall (FDAY), and, from them, the

length of the frost-free period (FFP) from monthly means of minimum temperatures (Mi

)

from 1,376 weather stations in Western North America (Koss and others 1988; Environment Canada 1994). Dependent variables included the first and second powers of the 12

monthly minimums plus interpolated estimates of the Julian dates when temperatures

reached one of three threshold values: –2, 5, and 11°C. By assuming that the monthly

mean temperature occurred on the median date, Julian dates of each threshold temperature could be interpolated for the spring (S–2, S5, and S11) and autumn (F–2, F5, and F11).

Also included as independent variables in the stepwise model were the 12 interactions

between the monthly mean temperatures and either S or F.

Weather station data showed that if the January mean minimum temperature was

>6.8 °C, then SDAY=0; and if the December minimum temperature was greater than

7.5 °C, then FDAY=365. The 35 observations for which these conditions were true were

removed from the data set, but as shown below, the conditions themselves were used as

conditional expressions in the estimation of the freezing dates.

The remaining observations were assorted into three groups for the regression

analyses. Group 1 comprised the 1,291 stations for which the minimum temperature in

either July or August was greater than 5.0 °C; group 2 included the 52 observations for

which the July or August temperature was less than 5.5 °C; and group 3 included the 151

observations for which the January or December temperature was greater than 2 °C.

Notice that in providing a sufficient number of observations for regression analyses

within these groups, some stations were represented in more than one group. Grouping

was necessary to accommodate the disparity in climate among the weather stations. The

interpolated dates appropriate to the first group were S5 and F5; S–2, and F–2 for the second

group; and S11, andF11 for the third group. Regressions were fit for SDAY and FDAY for

each group. The best fitting regression models were judged according to the significance

of the coefficients, reduction in the residual mean square, and the Mallows statistic.

As an example, the stepwise model for predicting SDAY for the first group of stations

was of the general form:

SDAY b b S b M b M b M b =+ + + + + 0 1 5 2 1 13 12 14 1

2 ..... ..... 25 122 M b SM b SM + + 26 5 1 37 5 1 .....

where M’s are monthly means of the month identified by the subscript for minimum

temperatue.

The best fitting regression models for this group produced values of R2 of 0.93 for

SDAY and 0.97 for FDAY; the corresponding values for analyses of the second group

were 0.73 and 0.78, and those for the third group were 0.78 and 0.79. The three stages

together produced a model that accounted for about 97 percent of the variance in SDAY

and FDAY. FFP was then calculated by subtracting SDAY from FDAY, and the

relationship between this estimate and the actual value accounted for 97 percent of the

variance in FFP of the 1376 stations. The relationships between the observed and

predicted values are shown in Figure 5 for the three variables as are the dispersion of

residuals against predicted values of FDAY.
14 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006

Figure 5—Plots of observed freezing data against those predicted by the three-stage regression models

for SDAY, the last frost of spring (upper left), FDAY, the first frost of autumn (upper right), and FFP, the

length of the frost-free period (lower left). At lower right is the relationship between predicted values of FDAY

and the residual from the regression model.
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 15

Models of the three groups of data together with the conditional expressions produced

the following general model for the freezing variables:

If M 7 8 or M 5.5 then: ≥

SDDAY S M = −1 08 93 2 08 . . + +5 2 11 12

5 6

1 9 3 85

30 28 0 92 1 80 1

+ −

= + − +

. .

.. .

M M

FDAY F M .

. :

. .

84

5 5

213 11 0 08

M9

then

SDAY M

If M or 7 8 M <

= − 10

2

9 27

8

2 65 0 04

211 97 5 75 9 2

− −

= + −

. . −

.. .

M SM

FDAY M 3 0 05 M FM 6 26 +

>

. −

If M 1 12 7 or M 2.0 and M > 11 then:

SDAY M S M S = 147 13 12 24 0 07 0 03 .. . . − −2 11 4 11 + M

FDAY M M M

6

2 2

2 = + 282 28 8 95 0 46 6 35 .. . . − + 12

IF JAN > 6.8 THEN SDAY = 0;

IF DEC >7.5 THEN FDAY = 365;

IF SDAY < 0 THEN SDAY = 0;

IF FDAY > 365 THEN FDAY = 365;

FFP = FDAY SDAY

IF SDA

−

Y > FDAY THEN FFP = 0

Estimating Temperature Sums for Specified Periods

The procedures described above for estimating degree-days use the daily contribution

to the total for each month. These contributions can be summed for each day to produce

an annual accumulation (fig. 6) from which estimates of degree-days within a specified

period can be made. Two examples follow.

Figure 6—Plots of observed (dots) and

predicted (lines) sums of degree-day > 5 °C

according to Julian date for four stations with

different summer temperature regimes. From

upper to lower: Death Valley, CA, Placerville,

CA, Priest River Experimental Forest, ID,

Barrow, AK.
16 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006

At low elevations (for example, 950 m) at the Priest River Experimental Forest in

northern Idaho, winter dormancy in most organisms (from trees to mosquitoes) has been

arrested by May 1. Climate normals from the Experimental Forest show that DD5 on May

1 averages 109. One can define a climate variable, therefore, as DD5100, the Julian date

on which degree-days > 5 °C reach 100. This value can be obtained directly by from the

dataset of daily degree-day sums.

Another example may involve the amount of summer warmth available for plant

growth and development. Such a variable can be defined as the number of degree-days

> 5 °C that accumulate during the frost-free period (DD5ffp). This value can be estimated

from the difference in degree-days between FDAY and SDAY.

Applications ______________________________________________________

The spline surfaces have many uses in biology. McKenney and others (2001), for

instance, have already used such surfaces to revise and remap plant hardiness zones for

Canada. The following examples provide a sample of additional applications.

Point Estimates

In ecological genetics, researchers deal with the ecological bases for genetic differentiation among populations. Because climate is the primary factor controlling differentiation, ecological genetic concepts must be couched in terms of climate. This would

require, therefore, estimates of climate for individual populations and test sites. For a

sample of 10 populations of Picea engelmannii, table 5 lists six climate variables that

described the climatic distribution of the species south of latitude 51 °N. The table shows

that across a latitudinal spread of about 17 ° and an altitudinal distribution of nearly 3,000

m, populations of this species occupy climates differing by as much as 10 °C in mean

annual temperature, 1,500 degree-days > 5 °C, 1700 degree-days < 0 °C, 700 mm in

annual precipitation, 11 °C in mean temperature in the coldest month, and 120 days in

the length of the frost-free season. Nonetheless, populations as geographically disparate

as north-central Washington (altitude = 762 m) and southern New Mexico (altitude =

2,622 m) can occupy remarkably similar climates.

Mapping Climate Surfaces

Hutchinson’s ANUSPLIN software (Hutchinson 2000) can output a grid file in that

point estimates are made for grids which are suitable for mapping. Figure 7 illustrates a

map of degree-days > 5 °C for western North America south of latitude 51 ° N that uses

a digital elevation model on a 1 km grid (0.0083333 degrees) (GLOBE 1999). There are

nearly 7 million pixels in this map, and for each, an estimate of degree-days > 5 °C has

been made with Hutchinson’s software. Predicted values range from –2,331 to 6,700.

Values less than zero arise from extrapolation, are absurd, and can be equated to zero.

Negative values as large as –500 occurred, for instance, in the ice fields of the Wind River

Range in Wyoming, but the largest negative values occurred on the volcanoes in the

Cascade Range.

Figure 8 shows a portion of the map of figure 7 at a scale exposing the underlying

pixels. The map can easily be confused for the elevation model that was used in its

construction, but nonetheless represents degree-days. The map includes the communities of Lewiston, ID, and Clarkston, WA, in the upper left plus the Grande

Ronde, Snake, Salmon, and Clearwater Drainages. The regions with the coolest

summers are in the Eagle Cap Wilderness toward the southwest. Values of RTGCV
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 17 Table 5—Point estimates of six climate variables for 10 climatically disparate populations of P. engelmannii. Location Climate variable Coldest month Length of Administrative Latitude Longitude Altitude Mean annual Degree-days Degree-days Mean annual minimum frost-free unit (°N) (°W) (m) temperature (°C) >5°C <0°C precipitation (mm) temperature (°C) period (days) Colorado 39.67 105.88 3598 –2.2 340 1932 789 –17.2 10 Utah 40.72 110.85 3178 –0.4 574 1582 741 –14.8 36 British Columbia 49.37 117.45 1997 0.3 599 1405 1157 –12.5 62 Montana 44.82 113.30 2226 1.8 934 1268 346 –16.3 33 Arizona 33.93 109.63 2927 4.0 997 668 839 –12.0 60 Idaho 44.67 116.13 1616 4.4 1288 829 686 –12.3 59 Montana 47.42 115.63 1159 5.7 1452 593 955 –8.4 93 Idaho 46.58 114.62 1204 5.8 1507 610 713 –9.6 77 Washington 48.93 118.88 762 6.7 1781 577 395 –9.2 115 New Mexico 32.80 105.70 2622 8.1 1783 209 694 –6.3 131
18 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006

Figure 7—Map of degree-days >5 °C ranging

from zero (light blue) to 6,700 (silver).

Figure 8—Map of degree-days > 5 °C using

shades of grey to code 19 classes ranging from

zero (light blue) to 3,700 (gray) for an area

surrounding Lewiston, ID, and Clarkston, WA,

(upper left). Major drainages counterclockwise

are the Grande Ronde, Snake, Salmon, and

Clearwater.
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 19

for the degree-day spline suggest that 95 percent confidence intervals about

predicted points are about ±45 degree-days.

Separating species in climatic space—Paleoecologists commonly use the

fossil pollen records of vegetation to infer climate. This can become problematic

because in many instances, distributions in climate space are unknown for taxa of

interest. For example, reconstructing the composition of plant communities at high

altitudinal sites in the Northern Rocky Mountains of the United States requires knowledge of the conditions under which either P. albicaulis or P. contorta can be dominate

(Brunelle and Whitlock 2003; Brunelle and others 2005). To address this problem, the

spline climate model was used to make point predictions for 277 locations known to be

inhabited by P. contorta and 74 locations inhabited by P. albicaulis. Although these two

species can be sympatric, it is not known at which of these locations they co-occur.

Eighteen derived variables were calculated for each population, and a principal component analysis was made for the total assemblage. The species separated nicely according

to the first and second principal components (fig. 9, left). The first component was

strongly influenced by the summer temperature and the second by the annual moisture

index. Consequently, the species could be separated nearly as well by these variables as

by the principal components (fig. 9, right). The conclusion would be that P. contorta

generally occurs at locations that are warmer and drier than those inhabited by P.

albicaulis. This suggests for paleoecological reconstructions that forests dominated by

P. albicaulis are cooler and wetter than those dominated by P. contorta. While this is an

intuitive conclusion, this spline model provides the first definite evidence of this

relationship.

Figure 9—Scatter plots of 277 populations of P. contorta and 74 populations of P. albicaulis

according to the first two principal components of 18 climate variables describing their provenance

(left), and to degree-days > 5 °C and annual moisture index of the provenance (right) (from Rehfeldt,

unpublished).

second principal component annual moisture index

first principal component

degree – days >5 °C

10 1900

1500

1100

700

300

7

4

1

–2

–5

–8

–3 0 3 6 9 01 2 345

P. albicaulis

P. contorta

P. albicaulis

P. contorta
20 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006

Figure 10—Mean date of the cessation of shoot elongation of 2-year Picea engelmannii populations

measured in greenhouse-shadehouse studies plotted against the provenance elevation (left) and

provenance degree-days < 0° (right) (from Rehfeldt, et al. 2004).

Evaluating genetic variation along climatic gradients—For years, forest geneticists have been using geographic variables to describe genetic differences among

populations as they exist across landscapes. The underlying assumption in these analyses

has been that the geographic variables were acting as surrogates for climate variables that

were difficult to observe. The spline climate model, however, offers an opportunity to

replace the surrogates with operative climate variables. Figure 10 illustrates genetic

variation among 215 populations of Picea engelmannii from the Western United States

in relation to the best fitting geographic variable (left) and the best fitting climate variable

(right). The geographic surrogate, elevation, is a weak (R2 = 0.07) but nonetheless a

statistically significant (p < 0.05) predictor of variation among populations. By contrast,

the winter temperature of the provenance, degree-days < 0 oC, is a strong predictor

(R2 = 0.62) of genetic differentiation. In this example, geographic variables were poor

surrogates for climate and led, therefore, to a fallacious view of the species’ genetic

structure.

Conclusions ______________________________________________________

Climate models developed from thin plate splines provide geographic surfaces that

can be addressed to obtain point estimates of climate. Errors of prediction of these

estimates are available, and the estimates can be converted to variables known to be

relevant in biology. Spline climate models have unlimited applications in assessing plant

responses to climate.
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-165. 2006 21

References _______________________________________________________

Brunelle, A.; Whitlock, C. 2003. Postglacial fire, vegetation, and climate history in the Clearwater Range,

Northern Idaho, USA. Quaternary Research. 60: 307-318.

Brunelle, A.; Whitlock, C.; Bartlein, P.J.; Kipfmueller, K. 2005. Holocene fire and vegetation along

environmental gradients in the Northern Rocky Mountains. Quaternary Science Reviews. 24: 2281-2300.

EarthInfo, Inc. 1994. Database Guide. Earthinfo, Inc. Boulder, CO.

Environment Canada. 1994. Canadian Monthly Climate Data and 1961-1990. Normals. Prairie and Northern

Region Commercial Weather Services Division, Edmonton, Alberta.

GLOBE Task Team. 1999. The Global Land One-kilometer Base Elevation (GLOBE) Digital Elevation

Model, Version 1.0. National Oceanic and Atmospheric Administration, National Geophysical Data

Center, 325 Broadway, Boulder, Colorado 80303, U.S.A.

Hutchinson, M. F. 1991. Continent wide data assimilation using thin plate smoothing splines. Pages 104-

113, In: J.D. Jasper, ed. Data assimilation systems. BMRC Research Report 27, Bureau of Meteorology,

Melbourne, Australia.

Hutchinson, M. F. 2000. ANUSPLIN Version 4.1 User’s Guide. Australian National University, Centre for

Resource and Environmental Studies, Canberra.

Koss, W.J.; Owenby, J.R.; Steurer, P.M. Devoyd, S.E. 1988. Freeze/frost data. Climatography of the U. S.

No. 20. Supplement No. 1. National Oceanic and Atmospheric Administration, National Climatic Data

Center, Ashville, North Carolina.

McKinney, D.W.; Hutchinson, M.F.; Kesteven, J.L.; and Venier, L.A. 2001. Canada’s plant hardiness zones

revisited using modern climate interpolation techniques. Canadian Journal of Plant Science. 81: 129-143.

Rehfeldt, G.E.; Tchebakova, N.M.; Parfenova, E. 2004. Genetic responses to climate and climate change in

conifers of the temperate and boreal forests. Recent Advances in Genetics and Breeding. 1:113-130.

Tuhkanen, S. 1980. Climatic parameters and indices in plant geography. Acta Phytogeographica Suecica.

67:1-105.

United States Department of Commerce. 1994. U.S. Divisional and Station Climatic Data and Normals.

Volume 1. National Oceanic and Atmospheric Administration, National Climatic Data Center, Ashville,

North Carolina.

Woodward, F.I. 1987. Climate and plant distribution. Cambridge University Press, London.


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The Rocky Mountain Research Station develops scientific information and technology to improve management, protection, and use of

the forests and rangelands. Research is designed to meet the needs

of National Forest managers, Federal and State agencies, public and

private organizations, academic institutions, industry, and individuals.

Studies accelerate solutions to problems involving ecosystems,

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multiple use economics, wildlife and fish habitat, and forest insects

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may be found worldwide.

Research Locations

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