United States 

D

Agriculture 

epartment of Correlation and Prediction

Forest Service of Snow Water Equivalent Pacific Southwest 

Research Station 

Research Paper from Snow Sensors 

PSW-RP-211 

Bruce J. McGurk David L. Azuma 
McGurk, Bruce J.; Azuma, David L. 1992. Correlation and prediction of snow water equivalent 

from snow sensors. Res. Paper PS W-RP-211. Berkeley, CA: Pacific Southwest Research Station, 

Forest Service, U.S. Department of Agriculture; 13 p. 

Since 1982, under an agreement between the California Department of Water Resources and the 

USDA Forest Service, snow sensors have been installed and operated in Forest Service-administered 

wilderness areas in the Sierra Nevada of California. The sensors are to be removed by 2005 because 

of the premise that sufficient data will have been collected to allow "correlation" and, by implication, 

prediction of wilderness snow data by nonwilderness sensors that are typically at a lower elevation. 

Because analysis of snow water equivalent (SWE) data from these wilderness sensors would not be 

possible until just before they are due to be removed, "surrogate pairs" of high- and low-elevation 

snow sensors were selected to determine whether correlation and prediction might be achieved. 

Surrogate pairs of sensors with between 5 and 15 years of concurrent data were selected, and 

correlation and regression were used to examine the statistical feasibility of SWE prediction after 

"removal" of the wilderness sensors. Of the 10 pairs analyzed, two pairs achieved a correlation 

coefficient of 0.95 or greater. Four more had a correlation of 0.94 for the accumulation period after 

the snow season was split into accumulation and melt periods. Standard errors of estimate for the 

better fits ranged from 15 to 25 percent of the mean April 1 snow water equivalent at the high-elevation 

sensor. With the best sensor pairs, standard errors of 10 percent were achieved. If this prediction error 

is acceptable to water supply forecasters, sensor operation through 2005 in the wilderness may 

produce predictive relationships that are useful after the wilderness sensors are removed. 

Retrieval Terms: snow sensor, water supply forecasting, snow water equivalent prediction, snow 

pillow 

The Authors: 

BRUCE J. McGURK and DAVID L. AZUMA are hydrologist and mathematical statistician, 

respectively, in the Pacific Southwest Research Station's Cumulative Watershed Effects and Inland 

Fisheries Research Unit, headquartered in Berkeley, California. 

Acknowledgments: 

We thank David M. Hart of the Department of Water Resources' California Cooperative Snow 

Surveys for his technical assistance during this study. 

Publisher: 

Pacific Southwest Research Station 

P.O. Box 245, Berkeley, California 94701 

February 1992 
Correlation and Prediction of 

Snow Water Equivalent from 

Snow Sensors 

Bruce J. McGurk David L. Azuma 

Contents 

In Brief ............................................................................................................................................ ii

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

Correlation versus Prediction ................................................................................................. 1

Length of Record .................................................................................................................. 1 

Correlation and Regression Methodology and Results .............................................................. 2 

Characteristics of Data from Snow Sensors ........................................................................ 2

Selection of Surrogate Sensor Pairs ....................................................................................... 2

Processing of Sensor Records ............................................................................................. 4

Record Reduction ........................................................................................................... 4

Record Smoothing ......................................................................................................... 4

Record Division-Accumulation and Ablation .............................................................. 7

Meltout at Low-Elevation Sensors ................................................................................. 8

Multiple Regression ....................................................................................................... 9

Discussion ....................................................................................................................................... 9 

Current and Future Uses ........................................................................................................ 9

Regression Problems ........................................................................................................... 10

Correlation versus Regression .............................................................................................10 

Confidence Intervals ...........................................................................................................10 

Record Reduction and Smoothing ....................................................................................... 10

Accumulation and Ablation Division .................................................................................. 11 

Multiple Regression ............................................................................................................ 12

Rain-Snow Line ................................................................................................................... 12

Conclusions ................................................................................................................................... 13 

References ....................................................................................................................................13
In Brief... 

McGurk, Bruce J.; Azuma, David L. 1992. Correlation and 

prediction of snow water equivalent from snow sensors. 

Res. Paper PSW-RP-211 Berkeley, CA: Pacific Southwest 

Research Station, Forest Service, U.S. Department of Agriculture; 13 p. 

Retrieval Terms: snow sensor, water supply forecasting, snow 

water equivalent prediction, snow pillow 

In 1982 the USDA Forest Service agreed to allow the 

California Department of Water Resources (DWR) to temporarily install fifteen snow sensors in some wilderness areas 

administered by the Forest Service (FS), in the Sierra Nevada. 

The intent of the temporary installation was to correlate and, by 

implication, allow development of prediction equations that 

would allow water supply forecasters to make predictions of 

snow water equivalent (SWE) at the wilderness sites on the basis 

of data collected by sensors in nonwilderness areas. Removal is 

mandated because, according to the FS, installations such as 

snow sensors do not comply with the uses of wilderness intended 

by the Wilderness Act (PL 88-572). 

This analysis assesses the feasibility of predicting the SWE 

at high-elevation snow sensors from low-elevation sensors 

using least squares regression. Because of the short record of 

most wilderness sensors, ten pairs of sensors with longer records 

were selected as surrogates for the wilderness-nonwilderness 

pairs. Of the ten pairs, only two pairs met the initial criterion of 

a correlation coefficient of at least 0.95. Six of the ten had a 

correlation coefficient of less than 0.75. Record smoothing and 

deletion of unchanging values yielded little improvement. Improved results were obtained by dividing the records for the ten 

ii 

pairs into accumulation and ablation periods. April 1 was used 

as the division date. Six pairs then yielded correlation coefficients of 0.94 or better for the accumulation period, but only one 

pair reached that level during ablation. 

The selection of the ten sensor pairs was based on minimizeing horizontal and elevational difference and eliminating sensors that had frequent technical problems or short records. 

Colocation within a drainage basin was also a goal. Elevational 

difference proved to be an important attribute for pairing, and 

distance and basin were less important. This analysis underscored the difficulty in a priori selection of pairs. Future 

analyses should define pairs by calculating correlation coefficients for between five and 15 sensors within 500 m elevation 

and a large distance such as 75 km. The 500-m difference in 

elevation may minimize the co-occurrence of rain at the low 

sensor and snow at the high sensor. 

The lowest standard errors (SEs) of the regressions for the 

sensor pairs ranged from 15 to 25 percent of the high-elevation 

mean April 1 SWE. Other sensor pairs yielded SEs of 30 to 55 

percent of the mean April 1 SWE. Although improved sensor 

selection and more sophisticated prediction methods would 

probably yield better pairings and lower SEs, even the best SEs 

obtained in this analysis were 10 percent of the high-elevation 

sensor's long-term April 1 mean SWE. Confidence intervals (95 

percent) around predictions based on these "best" equations 

would be at least +20 percent of the April 1 mean SWE. If this 

prediction error is acceptable to water supply forecasters, then 

data collection through 2000 or 2005 in the wilderness areas 

may produce predictive relationships that are useful after the 

wilderness sensors are removed. 

USDA Forest Service Res. Paper PSW-RP-211.1992. 
Introduction 

I n 1982 the USDA Forest Service (FS) agreed to allow the 

California Department of Water Resources (DWR) to temporarily install fifteen snow sensors1

 in some wilderness areas 

administered by the Forest Service, in the Sierra Nevada. The 

intent of the temporary installation of sensors in wilderness was to 

allow "correlation" (defined below) between wilderness and 

nonwilderness sensors. If such a correlation could be established, 

then the FS believed that forecasters could predict future water 

supply on the basis of data collected at sensors in the nonwilderness 

areas. Under the terms of a Special-Use Permit signed by the FS 

and the California Department of Water Resources (DWR) in 

December 1982, each sensor was to be allowed a 10-year correlation period. If correlation is not successfully achieved after 10 

years, a 5-year extension may be allowed. Removal is mandated 

because, according to the FS, installations such as snow sensors do 

not comply with the uses of wilderness intended by the Wilderness 

Act (PL 88-572). In that the last sensors were installed in 1990, they all 

must be removed by 2005 or sooner under the current agreement. 

The Environmental Assessment (EA) that was prepared by the 

FS recognized the need to include both wet and dry April-to-July 

runoff years (AJRO) in the correlation period by stipulating that 

AJROs from both the upper and lower deciles should be represented. Two wilderness sensors that were installed before the 1983 

season do have upper and lower decile seasons because of the wet 

1983 and the dry 1,987 seasons. In the history of the basins of 

concern, there is less than a 35 percent chance that both upper and 

lower decile AJROs will occur during a 10-year period. The 

chance increases to 60 percent during a 15-year period. 

The EA set a standard of 0.95 as a successful correlation level. 

The only reference to analytical methods to be used during this 

temporary installation and correlation exercise was stated in the 

Special-Use Permit: "Standard statistical techniques acceptable to 

the FS will be used in making this determination." However, when 

researchers at the Pacific Southwest Research Station and DWR 

began development of a statistical methodology for predicting 

snow water equivalent (SWE) with sensors in 1988 (Brandow and 

Azuma 1988), they recognized two problems in the Permit between the DWR and FS. First, achieving an "acceptable" correlation of 0.95 did not necessarily mean that a low-elevation 

(nonwilderness) record could successfully predict the SWE at a 

wilderness sensor. Second, time periods covered by existing 

wilderness sensor records were too short to test whether an 

acceptable level of correlation could be achieved. 

1The snow sensor network provides water supply forecasters in California with 

current snow water equivalent information at 110 sites in California's northern 

coastal and inland mountains. The sensors supply part of the information needed 

to predict annual runoff to the state's reservoirs and rivers. Fifteen sensors are 

located in USDA Forest Service-administered wilderness areas. Each sensor 

consists of a thin stainless steel tank with a surface area of 1.8 m2 that is covered 

with a few centimeters of soil, a buried radio transmitter, and a solar panel and 

antenna mounted on a 6-m tall mast. 

USDA Forest Service Res. Paper PSW-RP-211. 1992. 

Correlation versus Prediction 

The correlation coefficient "r" is a measure of the linear 

relationship between two variables, such as SWE from snow 

sensors X and Y (Haan 1977): 

where xi and yi are the observed SWEs, x and y are the means of 

the observed SWEs, and n is the number of daily SWE pairs of 

observations. 

Although correlation is a measure of a linear relationship, 

regression is a more suitable technique for predicting how one 

variable changes given a change in another variable. Regression 

fits are often evaluated by R2

, the coefficient of determination, 

which is a measure of the ability of the regression line to explain 

variations of the dependent variable (Haan 1977): 

where ŷi are the estimated values. 

The pertinent practical question is how accurately can a daily 

SWE at a high-elevation, wilderness sensor be predicted on the 

basis of the SWE at a low-elevation, nonwilderness sensor? 

Evaluation of the goodness of fit of the regression is typically 

based on the standard error of estimate of the regression equation 

(SE) and on the coefficient of determination, R2 (Haan 1977): 

Because of the concern for error around the prediction, this 

report will focus on SE, but will report R2 and the correlation 

coefficient, r, as well. 

Length of Record 

The second issue relates to the length of record of the 

wilderness sensors. Most of these sensors were not installed 

until after 1982, and some were not completed until 1990. By the 

time sufficient record is obtained and analyses performed, the 

10-year deadline will have passed. A solution to this problem is 

to select "surrogate sensor pairs" of high- and low-elevation 

sensors that currently have adequate records and perform the 

analysis on them (Brandow and Azuma 1988). Brandow and 

1 
Azuma selected nine pairs of sensors on the basis of proximity, 

elevational difference, degree of correlation of their co-located 

snow courses, and representation from the Trinity basin and the 

north, central, and south Sierra Nevada. Correlations of 0.95 

were attained with some surrogate sensor pairs, but the SEs 

ranged from 10 to 30 percent of the peak predicted SWE. The 

authors concluded that this magnitude of error would adversely 

affect the accuracy of the forecast of the current water supply, 

especially in some of the southern Sierra Nevadan basins with 

extensive wilderness (Brandow and Azuma 1988). 

It is presumed that the results from the analysis of the 

surrogate pairs apply to the wilderness-nonwilderness sensors. 

The only difference between surrogate pairs and wildernessnonwilderness pairs is the wilderness designation of the highelevation wilderness sensor. If good predictive relationships 

were obtained using surrogate pairs, good results might be 

achievable with the 10 or 15 years of data that would be available 

from the no wilderness-wilderness sensors before the wilderness sensors were removed from FS-administered sites. Conversely, if poor predictions were obtained, it would indicate that 

the use of predicted data for the wilderness sites would not be 

appropriate. 

This paper reports on the feasibility of predicting SWE at high 

elevations from SWE obtained by sensors at low elevations in the 

Sierra Nevada using least squares regression and provides guidelines for selection of no wilderness-wilderness sensor pairs. 

Correlation and Regression 

Methodology and Results 

Characteristics of Data from Snow 

Sensors 

A snow sensor measures the weight and hence the SWE of 

the snowpack via the pressure in the fluid-filled tanks. A 

pressure transducer produces a voltage that is measured, converted to a digital value, and transmitted to a central archiving 

site at daily or more frequent intervals. Various technical and 

physical problems contribute to errors in the data stream (see 

Brandow and Azuma 1988; McGurk 1986; and Suits 1985 for 

further discussion). The two major types of errors that affect 

SWE prediction are missing data and short-term fluctuations 

(chatter or flutter). Data maybe missing for long periods because 

of sensor leaks or major system failures, or for short periods 

because of temporary telemetry problems. In either case, the 

missing data hinder the development of a prediction equation or 

prevent prediction for varying periods. 

Chatter refers to the approximately 1- to 5-cm variations in 

SWE that occur between daily readings and that cannot be 

attributed to either precipitation or melt. An example of chatter 

can be seen on April 17 for the 1986 plot at the Central Sierra 

Snow Laboratory (CSSL) (fig. 1). The 5-cm spike in the sensor 

trace is not echoed by an independent measurement from an 

isotopic profiling snow gauge (Kattelmann and others 1983), 

and less than 1 cm of precipitation was recorded at CSSL during 

that time. Smoothing algorithms can dampen these oscillations, 

but chatter nevertheless adds noise to the data. Inherent accuracy 

of the transducers is in the same range as the chatter and varies 

on the basis of transducer manufacturer and the range measured 

by the transducer. 

The typical seasonal plot of SWE vs. time from a snow sensor 

in the mid-to-high elevation zone of the Sierra Nevada has an 

accumulation period, a peak SWE around 1 April, and an 

ablation (melt) period that ends when the snow is gone (fig. 1). 

The steep portions are storm (rising SWE) or melt (falling SWE) 

events, and the horizontal intervals are clear weather periods 

with little melt. The sensor's advantages include its ability to 

respond to new snow or melt within hours and to report these 

data at regular intervals or upon demand. Midwinter rains on the 

snowpack are not generally detectable except at nonwilderness 

sites that also have storage precipitation gauges. The disadvantages of sensors are their complexity, their need for seasonal 

maintenance, their inaccessibility (both geographic and position 

under several meters of snow), and the data reduction and 

archiving requirements of the telemetered SWEs. Sensors are, in 

general, reasonably accurate, efficient tools to repeatedly measure SWE nondestructively in remote locations (Kattelmann 

and others 1983; McGurk 1986). The profiling snow gauge 

SWEs in figure 1, in contrast, are probably accurate to within ± 

0.5 cm but require a nuclear source and detector, plus line power 

and an operator (Kattelmann and others 1983). 

Selection of Surrogate Sensor Pairs 

The surrogate sensor pair technique used by Brandow and 

Azuma (1988) was also used in this analysis. Length of record, 

geographic proximity, and difference in elevation between 

members of the pair were the basis of the pairings. A list of 32 

sensors that were deemed to be the most reliable and accurate 

with respect to on-site SWE sampling was supplied by DWR 

(Hart 1989). Length of record was a primary criterion, and then 

proximity and elevational difference were considered, resulting 

in a selection of seventeen sensors that were sorted into a list of 

ten pairs (table 1). All but three of the sensors were in operation 

by 1971, so most of the pairs had a potential 15 years of record. 

Most of the pairs are in the same river basin and are less than 30 

km apart (fig. 2). The mean elevational difference between the 

chosen sensor pairs is 586 m, but ranges from 215 m to 1,460 m. 

Seven of the ten pairs are between 300 m and 675 m apart in 

elevation. The ten pairs ranged from 8 to 34 km apart, and had 

a mean proximity of 24.3 km. 

2 USDA Forest Service Res. Paper PSW-RP-211. 1992. 
Figure 1-Snow water equivalent in water year 1986 from a snow sensor and an isotopic profiling snow gauge at the Central Sierra Snow Laboratory 

near Soda Springs, California. 

Table 1-Elevation differences, geographic proximity, and number of years with at least thirty data values 

for the ten snow sensor pairs used in a surrogate pair analysis 

Sensor name Elevation Years name Elevation Years Elev. 

difference 

Proximity 

m no. m no. m

Blue Canyon 1615 12 Central Sierra 

Snow Laboratory 

2100 485 23 

Robbs Saddle 1800 14 Schneiders 2665 865 34 

Bloods Creek 2195 8 Mud Lake 2410 15 215 18 

Poison Ridge 2100 7 Green Mountain 2410 15 310 25 

Graveyard 5 Green Mountain 2410 310 8 

Poison Ridge 2100 7 Kaiser Pass 2775 16 675 31 

Green Mountain 2410 15 Mammoth Pass 2895 12 485 20 

West Woodchuck 2680 17 State Lakes 3140 26 460 29 

Giant Forest 2030 16 Upper Tyndall 3490 18 1460 32 

Quaking Aspen 2195 6 Pascoes 15 595 22 

Sensor 

km

10 

15 

2100 15 

2790 

USDA Forest Service Res. Paper PSW-RP-211. 1992. 3 
Figure 2-Location of surrogate sensor pairs in California's Sierra Nevada. 

Processing of Sensor Records 

Sensor data were supplied by DWR on magnetic tape. 

Records of instantaneous daily SWE depth started within a year 

of the sensor's construction (table 1) and concluded with the 

1986 water year. Some of the records have days, weeks, or entire 

years missing because of equipment malfunctions. Each record 

was screened for anomalies by automatic search routines and by 

eye. Daily values that were negative, extremely large, or showed 

large deviations from prior and post values were coded as 

missing values. The resultant data file was termed the "complete" or "all-data" file. Simple linear regression was performed 

on the daily SWE depths of each pair of sensors (Minitab 1985). 

The high- and low-elevation sensors were the dependent and 

independent variables, respectively, and the number of matching dates (sample size) was recorded, as well as the SE and the 

R2 (table 2). 

Record Reduction 

The typical snow sensor record has frequent non-storm 

periods during which the SWEs remain largely unchanged. 

4 

Because there are so many unchanging values, the regression 

may weight those values more than the changing values. To 

overcome this problem, a reduction process was used to delete 

consecutive points with identical values. This data reduction 

also would reduce the serial correlation that is typical of time 

series data; this type of day-to-day "persistence" violates the 

independence assumption typically made with regression. As a 

result, the sample sizes decreased by a mean value of 18 percent 

with a range between 9 and 46 percent. Regression was then 

performed with the "reduced data set" file (table 2). The procedure reduced the R2

 insignificantly and increased the standard 

errors between 0.9 and 2.3 cm. 

Record Smoothing 

A three-value moving average algorithm was applied to the 

all-data file to reduce the magnitude of the chatter (table 3). This 

type of smoothing is usually chosen for its simplicity. A longer 

moving average was not used because of the desire to retain the 

timing of small (5-10 cm) changes associated with storms of a 

day's duration or less. The smoothed daily SWE for day "i", k'i, 

was the average of ki-1, ki, and ki+1 (Bloomfield 1976). The 

USDA Forest Service Res. Paper PSW-RP-211. 1992 
Table %Regression results of daily snow water equivalent (SWE) values for the surrogate snow 

sensor pairs for the complete and reduced data sets 

Sensor pair and 

data set 

Blue CanyonÃ‘Sno Lab 

- all data 

- reduced data 

- reduced accum. 

- reduced ablation 

Robbs Saddle-Schneiders 

- all data 

- reduced data 

- reduced accum. 

- reduced ablation 

Bloods Creek-Mud Lake 

- all data 

- reduced data 

- reduced accum. 

- reduced ablation 

Poison Ridge-Green Mountain 

- all data 

- reduced data 

- reduced accum. 

- reduced ablation 

Graveyard-Green Mountain 

- all data 

- reduced data 

- reduced accum. 

- reduced ablation 

Poison Ridge-Kaiser Pass 

- all data 

- reduced data 

- reduced accum. 

- reduced ablation 

Green Mountain-Mammoth Pass 

- all data 

- reduced data 

- reduced accum. 

- reduced ablation 

West Woodchuck-State Lakes 

- all data 

- reduced data 

- reduced accum. 

- reduced ablation 

Giant Forest-Upper Tyndall 

- all data 

- reduced data 

- reduced accum. 

- reduced ablation 

Quaking Aspen-Pascoes 

- all data 

- reduced data 

- reduced accum. 

- reduced ablation 

Sample 1 Standard I Percent of I Coefficient of 

size 

624 

620 

519 

103 

2146 

1959 

1285 

579 

1133 

1031 

628 

372 

508 

47 1 

26 1 

212 

334 

284 

158 

127 

635 

608 

324 

275 

972 

838 

534 

306 

3216 

1734 

995 

682 

2857 

1499 

922 

536 

772 

565 

343 

195 

1 error 

cm 

23.2 

23.3 

21.8 

16.3 

34.4 

35.3 

19.1 

46.0 

25.0 

25.5 

10.4 

22.3 

43.8 

44.9 

15.8 

58.7 

43.4 

44.3 

16.4 

54.6 

36.1 

36.0 

13.0 

39.6 

40.0 

41.1 

28.5 

35.3 

11.1 

13.3 

5.8 

18.2 

28.5 

31.3 

22.2 

33.7 

13.6 

15.0 

6.3 

20.0 

1 April 1 SWE1 1 determination (R2) 

USDA Forest Service Res. Paper PSW-RP-211. 1992. 
Table 3–Regression results of daily snow water equivalent (SWE) values for the surrogate snow 

sensor pairs for the smoothed and reduced data 

Sensor pair and 

data set 

Sample 

size 

Standard 

error 

Percent 

April 

of 

I SWE' 

Coefficient of 

determination (R2

) 

cm pct 

Blue Canyon-Snow Lab 

-all smooth 623 23.2 27 50 

-reduced smooth 620 23.1 27 50 

-red. smooth accum. 518 21.7 25 52 

-red. smooth ablation 104 16.2 19 62 

Robbs Saddle-Schneiders 

-all smooth 2145 34.3 39 58 

-reduced smooth 1960 35.3 40 57 

-red. smooth accum. 1379 18.6 21 82 

-red. smooth ablation 590 45.8 52 40 

Bloods Creek-Mud Lake 

-all smooth 1131 24.9 22 83 

-reduced smooth 1040 25.4 22 82 

-red. smooth accum. 665 10.1 9 97 

-red. smooth ablation 381 22.3 20 81 

Poison Ridge-Green Mountain 

-all smooth 507 43.7 56 51 

-reduced smooth 484 44.4 57 49 

-red. smooth accum. 267 15.7 20 88 

-red. smooth ablation 220 58.0 74 40 

Graveyard-Green Mountain 

-all smooth 333 43.4 55 54 

-reduced smooth 297 43.8 56 50 

-red. smooth accum. 165 16.1 21 88 

-red. smooth ablation 133 54.1 69 38 

Poison Ridge-Kaiser Pass 

-all smooth 634 36.1 45 60 

-reduced smooth 617 36.0 45 60 

-red. smooth accum. 342 12.6 16 94 

-red. smooth ablation 277 39.2 49 57 

Green Mountain-Mammoth Pass 

-all smooth 971 40.0 37 41 

-reduced smooth 873 40.7 38 38 

-red. smooth accum. 568 28.0 26 64 

-red. smooth ablation 309 34.7 32 27 

West Woodchuck-State Lakes 

-all smooth 3215 11.0 15 91 

-reduced smooth 2314 12.6 17 89 

-red. smooth accum. 1546 5.2 7 97 

-red. smooth ablation 778 18.5 25 81 

Giant Forest-Upper Tyndall 

-all smooth 2855 28.5 41 46 

-reduced smooth 2125 30.0 43 42 

-red. smooth accum. 1466 20.9 30 59 

-red. smooth ablation 672 34.5 49 35 

Quaking Aspen-Pascoes 

-all smooth 770 13.6 21 91 

-reduced smooth 618 14.5 23 90 

-red. smooth accum. 416 5.8 9 98 

-red. smooth ablation 206 19.6 31 89 

1

Percentage is the SE (equation 3) divided by the long-term mean of the April 1 SWE of the 

high-elevation sensor, multiplied by 100. 

6 USDA Forest Service Res. Paper PSW-RP-211. 1992. 
subsequent calculation for k i+1 used ki, not k’i, in the calculation. 

When missing values were encountered as the ki-1, or ki+1, value, 

the ki value was retained. When ki was missing, the missing 

value code was retained in the smoothed file. 

Reduction was also applied to the smoothed files, but smoothing decreased the efficiency of the reduction process by spreading out chatter to surrounding values. Reduction of the smoothed 

files reduced the size of the all-data file by an average of 12 

percent and ranged from less than 1 to 28 percent. Smoothing 

increased the standard errors from 0 to 1.6 cm, and by 0.7 cm on 

the average. The greatest increases in standard error were 

matched with the largest decrease in sample size. Although the 

sums of squared differences decreased after smoothing and 

reduction, the sample size decreased more. Larger standard 

errors reflect the position of the sample size as a divisor in the 

calculation of SE (equation 3), and indicate that regression 

yielded a poorer fit after reduction and smoothing. 

Record Division-Accumulation and Ablation 

When SWEs for high- and low-elevation sensors are plotted, 

an hysteresis curve results: the accumulation curve has a different path than the ablation curve (fig. 3). The maximum SWE at 

the Quaking Aspen and Pascoes sensors occurred close to April 

1 in 1981, which supports a common assumption made by snow 

hydrologists in California. There was a rapid loss of SWE at the 

lower elevation site in April and early May. The curve becomes 

essentially vertical when the lower site reaches 0 cm SWE and 

the high-elevation site is still snow covered. 

The ablation curve differs from the accumulation curve 

because of an elevation-influenced melt process. Although 

higher elevations receive slightly more intense solar radiation, 

the air temperatures are low enough to retard melting. This 

process results in the snow at lower sites melting out before that 

at the higher sites, even in cases in which the two sites get nearly 

an equivalent amount of seasonal precipitation (fig. 3). 

Figure 3-Hysteresis curve for snow water equivalent (SWE) at Quaking Aspen vs. Pascoes for water year 1981. 

USDA Forest Service Res. Paper PSW-RP-211. 1992. 7 
Although each year has a different time of peak SWE, April 

1 was used as the division date for all years and sites. To 

incorporate the hysteresis effect, the multi-year records for each 

site were split into accumulation and ablation files, and regression equations were fit to the split files. The split typically 

yielded a noticeable improvement in the SE and R2

 for the 

accumulation season (tables 2 and 3). Except for one case, there 

was a corresponding minor worsening in the SE and R2

 for the 

ablation period. Before splitting the files, only three of the 10 

sensor pairs had R2 in excess of 80 percent (correlation of 0.89). 

After splitting, seven of the ten pairs had an accumulation R2

 in 

excess of 80 percent. Two of the three remaining sensor pairs 

could be corrected to achieve a better-than-85-percent R2

 by 

eliminating questionable data and very low snowfall years with 

little or no snow at the lower sensor. 

Meltout at Low-Elevation Sensors 

A regression equation for prediction of high-elevation SWE 

is useful only when there is snow at the low-elevation site. When 

the low-elevation SWE is zero, the predicted high-elevation 

SWE is the y-intercept of a linear regression equation. In that the 

y-intercept is a constant, it is poorly suited to predict a varying 

high-elevation SWE as snow disappears at the low-elevation 

site. The time of meltout at low-elevation sensors varies from 

early May to early June, and the SWE at Kaiser Pass varied from 

30 to nearly 160 cm over a 5-year period (fig. 4). The rate at 

which snow at Kaiser Pass melted was relatively linear, and the 

slope was relatively consistent from year to year. A plot of a 5-

year period revealed that the high-elevation SWE at the time of 

meltout of the snow at the low-elevation sensors might be 

refined by adjusting it on the basis of classification of the water 

year SWE into low, normal, or high years (fig. 4). A constantvalue decay rate might then be applied to predict meltout at highelevation sensors. 

A multiple regression equation was developed with the alldata file using the SWE value of the high-elevation sensor when 

the low-elevation sensor melted out as the dependent variable. 

Independent variables included the SWE of the low-elevation 

Figure 4-Ablation at Kaiser Pass after meltout at Poison Ridge for water years 1982-1986. 

8 USDA Forest Service Res. Paper PSW-RP-211. 1992. 
sensor on April 1, the number of days from April 1 to meltout 

(lag) at the low-elevation site, and dummy variables representing the water year magnitude: 

SWEHigh = A +B(SWELow, April1) + C (MeltoutlagLow) 

+D(High/Med. Peak SWE)+ E(LowPeak SWE)+error 

(4) 

where high and low refer to the high- and low-elevation sensors, 

respectively, and A through E are least-squares coefficients. A 

low-SWE class had an April I SWE of less than 80 percent of the 

long-term April 1 mean SWE. A medium-SWE class had an 

April 1 SWE of 80-120 percent of the long-term April 1 mean 

SWE, and a high-SWE class had an April 1 SWE greater 

than 120 percent of the long-term April 1 mean SWE. Three pairs 

of sensors were used to evaluate this technique: Robbs Saddle -

Schneiders, West Woodchuck - State Lakes, and Giant Forest -

Upper Tyndall. Of the three equations, the best had an adjusted 

R2

 of 36 percent, and an SE of 23.9 cm, indicating a weak 

relationship between high-elevation SWE and the independent 

variables. 

Another potential way of incorporating water year magnitude would be to include a second variable in the regression 

equation that would represent the year effect. Although this 

might have improved the least squares fit in the historical data 

set, no information would exist for this variable when the 

equation was being used in real time as a year progressed and 

water supply forecasts were being made. 

Multiple Regression 

Although it is overly simplistic to assume that the highelevation SWE is solely a function of low-elevation SWE, 

reliable information exists only for the low-elevation SWE. 

High-elevation SWE also varies in response to the magnitude of 

the water year, the number of events with rain-snow elevations 

between the two sensors, and other less quantifiable factors. 

Two cases were evaluated by multiple regression to determine 

whether a pair of low-elevation sensors would produce a better 

predictive relationship than a single sensor. The "all-data" files 

for Green Mountain and Poison Ridge were used to predict SWE 

at Kaiser Pass, and Robbs Saddle and Mud Lake were used to 

predict SWE at Schneiders. In both cases, the multiple regression had higher R2

 and lower SE than the simple regression. 

However, analysis of the sums of squares showed that most of 

the improvement was due to selection of a low-elevation site that 

was a better predictor of the high-elevation site than the sensor 

that was originally selected. 

USDA Forest Service Res. Paper PSW-RP-211.1992. 

Discussion 

Although the theme of this paper is prediction of SWE for a 

high-elevation sensor that has been "removed," no prediction 

can completely compensate for the discontinuation of a sensor's 

data. When a sensor is discontinued, one sample of a time series 

of interest is gone. The variation and potential extremes are 

thereafter lost from the record. This loss may or may not be 

important to the process of predicting the seasonal water supply 

or flood probability. 

Current and Future Uses 

The primary current use of the snow sensors is as an input to 

updates of the monthly water supply forecasts for California's 

water users. DWR's Cooperative Snow Surveys pool the data 

from both the sensors (for the updates) and the snow courses for 

the monthly forecasts) to obtain basin indices. The indices are 

then used in a multiple regression model that was derived with 

the full complement of courses. The regression model also 

contains variables for expected future seasonal precipitation, the 

prior season's water year magnitude, and other terms. If a basin 

snow index is based on ten or more courses or sensors, dropping 

a single sensor would have a small but relatively unimportant 

effect on the forecast of the AJRO. If a whole group of highelevation sensors was excluded, however, a rather serious bias 

could result. Because higher elevations typically receive more 

snowfall, the bias would likely result in an underprediction of the 

AJRO. The bias could be compensated for by recalculating the 

forecasting equations, but the loss in information from the highelevation areas of the basin would weaken the predictive 

relationship. 

California's water management agencies and irrigation districts base their water allocation and purchasing plans on the 

water supply forecasts. The economic consequences of the 

forecasts are extensive: irrigated acreages, crop selections, and 

million-dollar purchases of supplementary water are based on 

the water supply forecasts. Pacific Gas & Electric and Southern 

California Edison use the data to optimize hydroelectric production and meet water quality and flow requirements. The sites for 

the recently-installed sensors were chosen because of the 

perceived need to have data for areas that are currently 

underrepresented by sensors. The trend of discontinuing measurements at selected snow courses will leave some basins with 

few sample sites. Sensors, once installed, provide a cost-effective and safe alternative to monthly visits to snow courses by 

snow surveyors. 

Although the current forecasting system pools the data from 

many sensors to yield a basin index, it is likely that future 

techniques will use more modern, distributed models. The 

current lumped-basin regression technique was developed before current computer and telemetry capabilities. As demands 

9 
for more frequent and more accurate forecasts increase, DWR 

will likely shift to more sophisticated tools. These improved 

models will require better data than simple monthly snow course 

basin indices. The daily temperature and SWE values from each 

of the sensors in a basin will be crucial components for the new 

models. The elimination of a group of sensors at the top of a 

number of California's basins may limit the implementation of 

the next generation of prediction techniques. 

Regression Problems 

Regression is the obvious tool for predicting a high-elevation 

SWE from a low-elevation sensor's SWE, but some of the 

technique's rules are violated in this situation. An assumption of 

regression is that the independent variable (low-elevation SWE) 

has no or very low error compared to the dependent variable. In 

this case, both sensors' SWEs have approximately the same 

error magnitude. This violation of theory may result in a weaker 

relationship and larger error bands (standard error) around the 

predicted values. 

Another theoretical problem with the independent variable is 

termed independence or serial correlation. A wide range of the 

independent variable's values should be possible in successive 

samples. In this application, however, an SWE from today is 

probably not more than a few centimeters different than one 

from yesterday or tomorrow. In spite of the violation of regression theory due to serial correlation, the regression coefficients 

are almost unbiased. The SE, however, is almost certain to be 

underestimated, so actual SEs may be even larger than those 

listed in this study. 

Correlation versus Regression 

The Environmental Assessment signed by the Forest Service 

defined 0.95 as a successful correlation between two sensors. 

This is equivalent to an R2 of 90 percent, but neither statistic is 

the best measure of how good the relationship will be for water 

supply forecasting. The SE is a more appropriate measure of 

accuracy in terms of forecasting in that it provides a method for 

estimation of how the predicted SWE may vary from the actual 

SWE. If the SE is relatively large compared to the long-term 

April 1 SWE, the uncertainty of the prediction may be too great 

for it to be useful. In general, SE decreases as R2

 increases, so the 

two measures of the relationship are inversely related (tables 2 

and 3). For these data, an SE of 17-21 cm generally corresponds 

to an R2 of between 80 and 90 percent, and is generally between 

19 and 25 percent of the April 1 SWE for a site. 

The typical SE in this study is much greater than the 2-3 

percent transducer error. The SE is also frequently greater than 

the difference between the high- and low-elevation sensors (fig. 

3). For Quaking Aspen and Pascoes, the SE for the accumulation 

period is 6.3 cm, and the SE for the ablation period is 20.0 cm 

(table 2). In 1981, the accumulation portion of the curve is less 

than 5 cm from the 1:1 line (fig. 3). For the ablation portion of 

the curve, the difference between the two sensors is less than 15 

10 

cm in all cases. The regression results are based on 5 years of 

data, however, and other years show a more complicated pattern 

(fig. 5). The example from water year 1984 is illustrative of why 

the SE is large for the regression equation: the 30 cm depth at 

Quaking Aspen has four different depths at Pascoes at various 

times, ranging from a SWE of 25 to 50 cm. During January, the 

SWE at Quaking Aspen declined by more than 12 cm while the 

Pascoes SWE remained fairly constant. This period produced an 

"offset," which was due to the reasonably common midwinter 

dry spell, during which warmer temperatures at the lower site 

allowed melting. Little melt occurred at the higher site. This 

pattern was repeated near the peak SWE, and there are indications that an event with a rain-snow line between the two sensors 

may have occurred. 

Confidence Intervals 

SEs are also used in the calculation of confidence intervals. 

A 95 percent confidence interval around a new predicted highelevation SWE near the mean of the low-elevation sensor is the 

SWE plus or minus approximately twice the value of the SE. For 

example, the Poison Ridge - Kaiser Pass pair has an SE of 36.1 

cm and a mean low-elevation SWE of 22.4 cm. If a confidence 

interval for that mean value were predicted for a following year, 

the high-elevation SWE would be 72.9 cm ± 72.2 cm. This wide 

interval results when the SE is over 45 percent of the long-term 

April 1 SWE, illustrating the seriousness of large SEs. The 

interval would become even wider as SWE values diverged from 

the mean SWE. A range of that width is no better than a random 

selection of a value limited only by the April 1 mean SWE, 

indicating that the prediction equation was essentially useless. 

In contrast to the near "worst case" example shown above, 

the West Woodchuck - State Lakes pair has an SE of 11.1 cm and 

a mean low-elevation SWE of 39.9 cm. The SE is 15 percent of 

the high-elevation sensor April l mean SWE, and the 95 percent 

confidence interval for a value near the mean is 36.8 cm ± 22.2 

cm. By splitting the data into accumulation and ablation files or 

by better pairings, SEs might be reduced to 10 percent of the 

April l mean SWE. This could result in confidence intervals that 

were plus and minus 25 percent of the predicted value. 

Record Reduction and Smoothing 

The reduction algorithm had between a one and 46 percent 

effect on sensor record size and a negligible effect on SE and R2

. 

If a sensor produced a record with little chatter, then a larger 

number of points were dropped (e.g., West Woodchuck - State 

Lakes, table 2). Although this procedure was, in part, designed 

to reduce serial correlation problems, its application did not 

result in marked changes in SE or R2

. A more dramatic effect 

might have occurred if the criteria had been set differently. 

Instead of dropping a value if it was identical to a prior value, a 

wider filter such as "SWE plus-or-minus 0.5 cm" might have 

been used. A band filter might have better accomplished the goal 

of removing points of unchanging SWE during non-storm, nonUSDA Forest Service Res. Paper PSW-RP-211.1992. 
Figure 5-Hysteresis curve for snow water equivalents (SWE) at Quaking Aspen and Pascoes snow sensors for water year 1984. 

melt periods such as occurred during much of December 1984 

(fig. 1). 

The smoothing process was devised to treat the minor 

deviations that result from transducer chatter and other errors of 

electronic or temperature-related origin. These errors typically 

cause small spikes in one record without accompanying spikes 

in the paired sensor record. More complex smoothing or wider 

moving averages would delay sensor response to real storm 

events. The smoothing algorithm reduced the magnitude of the 

spikes, but it transformed a single spike into a three-value blip. 

Smoothing had a negligible effect on the SE and R2 (tables 2 and 

3). 

Smoothing reduced the efficiency of the reduction process. 

For the West Woodchuck - State Lakes pair, the reduced file had 

1734 points as compared to 2314 points for the smoothed and 

reduced file. The smoothing process spread the spike to neighboring points, which then escaped deletion by the reduction 

algorithm. The optimal combination would probably be a band 

filter reduction scheme followed by a smoothing algorithm. 

USDA Forest Service Res. Paper PSW-RP-211.1992. 

Accumulation and Ablation Division 

In almost all cases, splitting the record at April 1 improved 

the regression fit for the accumulation phase of the reduced 

record (tables 2 and 3). The ablation phase then showed a poorer 

linear regression fit, typified by larger SEs and lower R2

. Six of 

the ten pairs had an R2 of 88 percent or higher for the accumulation period, and the mean SE as a percent of the April 1 SWE 

was 14 percent. 

It is not surprising that a linear relationship works less well 

during ablation than during accumulation in light of figures 3 

and 5. A severe problem with fitting a linear equation is caused 

by the string of declining SWEs at the high-elevation sensor 

while the low-elevation sensor has already reached an SWE of 

zero. An approach to the problem would have been to subdivide 

the ablation phase into a before-and-after meltout of snow at the 

low-elevation sensor. A linear equation might have been used 

for the pre-meltout portion, and a temperature index melt equation might have been applied to the high-elevation sensor's SWE 

11 
for the post-meltout period. The prediction of high-elevation 

SWE proved problematic, however, in that the prediction attempt described (equation 4) performed poorly at the three sites 

that were evaluated. The time of meltout is affected by factors 

other than water year class and time of meltout at the lowelevation sensor. For example, water years 1982 and 1986 had 

similar peak SWEs at both sites, but Poison Ridge melted out 

more than two weeks later in 1986 than in 1982 (fig. 4). Inclusion 

of historical data such as average daily temperatures might 

improve the predictive relationship, but these data were not used 

because of their unavailability during real-time prediction. 

Multiple Regression 

For the two cases examined, the addition of a second lowelevation sensor produced a better relationship than when only 

the original low-elevation sensor was used to predict the highelevation SWE. In the Kaiser Pass case, Poison Ridge is clearly 

a poor predictor in that the SE is 36.1 cm (table 4). The inclusion 

of Green Mountain halved the sample size, decreased the SE by 

7.3 cm, and increased the R2

 from 60 to 74 percent. The results 

were nearly as good, however, with Green Mountain as the only 

independent variable. The improvements were even more dramatic for the Schneiders cases in that the SE dropped from 34.4 

cm (paired with Robbs Saddle) to 12.3 cm (paired with Robbs 

Saddle and Mud Lake). Robbs Saddle was paired with Schneiders 

because of its westerly position in the American River basin, but 

the sensors are elevationally and horizontally distant (865 m and 

34 km), resulting in a weak relationship. Mud Lake is 255 m 

lower than Schneiders and 12.3 km away, so even though it is in 

the Mokelumne rather than the Cosumnes basin, it is a better 

predictor of Schneiders than Robbs Saddle. The SE and R2

 for 

the Mud Lake - Schneiders pairing are 12.6 cm and 94 percent, 

as opposed to 34.4 cm and 58 percent for the Robbs Saddle -

Schneiders pairing (table 4). 

The inclusion of a second sensor adds another type of 

theoretical regression violation called multicollinearity. Independent variables should be measures of different processes that 

Table 4-Comparison of regression results from simple and multiple 

regressions for two high-elevation sensors 

Sensor pairs Sample 

size 

Standard 

error 

 Coefficient of 

determination (R2

) 

cm pct

Kaiser from Poison Ridge 635 36.1 60 

Kaiser from Green Mountain 966 29.5 

Kaiser from Poison Ridge 

and Green Mountain 382 8 74 

Schneiders from Robbs Saddle 2146 34.4 58 

Schneiders from Mud Lake 2997 12.6 94 

Schneiders from Robbs Saddle 2105 12.3 95 

74 

28.

are related to the variable of interest. The SWEs at two sensors 

are likely to vary in a parallel manner, and much of the information in one record is contained in the other. If the addition of a 

second sensor improves the prediction capability, such as with 

both Kaiser and Schneiders, it may mean that the initial pairing 

should be dropped in favor of the new pairing. 

Rain-Snow Line 

Storm-to-storm variation of the rain-snow elevation is a 

characteristic of the Sierra Nevada. Much of a season's snow is 

deposited on both the upper and lower sensors by large, frontal 

storms that move west-to-east across the Sierra Nevada. Each 

storm has a different elevational boundary between rain and 

snow, and occasionally the lower-elevation sensor may receive 

rain while the higher-elevation sensor receives snow. This 

phenomenon complicates the prediction of SWE at high-elevation sensors, but no easy solutions are available. The obvious 

recourse is to pick as an independent variable a sensor that is 

close to the sensor of interest in elevation and geographic 

proximity. However, this solution is unrealistic because all of 

the California snow zone has only 110 sensors, so finding 

reliable, nearby sensors is very difficult. Indeed, because of the 

high cost of installation and operation, most sensors are separated by several hundred meters elevation and 20-30 km. At 

these distances and elevational differences, it is very likely that 

the rain-snow line will fall between any sensor pair for several 

storms during a season. Five of the ten pairs analyzed here are 

485 m or more apart in elevation, and these five have R2

 values 

of 60 percent or less. The three pairs with an R2 in excess of 0.80 

averaged 425 m in elevational difference. A maximum difference of 500 m might be a good general cutoff value for sensor 

pairs. 

It might be expected that nearby sensors would be likely to 

receive concurrent precipitation and of a similar type. However, 

these ten pairs did not show a good relationship between distance and R2

 or SE (tables 1 and 2). The ten pairs analyzed herein 

ranged from 8 to 34 km apart, and averaged 24.3 km in distance. 

Some distant pairs (West Woodchuck and State Lakes, 29 km) 

had a good predictive relationship, while adjacent pairs (Graveyard and Green Mountain, 8 km) had a poor predictive relationship. 

The distance relationship is obviously confounded by 

elevational and siting influences. Sensors that are 20 km apart 

but at a similar elevation (Bloods Creek and Mud Lake) may be 

more likely to form a good pair than sensors that are 10 km apart 

but at very different elevations. Sensors that are low enough to 

fall in the transient snow zone (Blue Canyon) are also unlikely 

to provide a good match to a higher sensor in the continuous 

snow zone (Central Sierra Snow Laboratory) in spite of a 

moderate elevational difference and reasonable proximity. No 

clear "cutoff" distance can be recommended on the basis of this 

analysis. 

12 USDA Forest Service Res. Paper PSW-RP-211. 1992. 
Conclusions 

The goal of this analysis was to assess the feasibility of 

predicting the SWE at high-elevation snow sensors from lowelevation sensors using least squares regression. The need for 

the prediction arises from the USDA Forest Service policy that 

calls for removal of sensors sited in wilderness after a correlation 

period. The initial criterion for an acceptable correlation was a 

correlation coefficient of 0.95. Because of the short record of 

most wilderness sensors, this analysis selected ten pairs of 

sensors with longer records as surrogates for the wilderness 

pairs. 

Of the ten pairs, only two pairs met the initial criterion. Six 

of the ten had a correlation coefficient of less than 0.75. Record 

smoothing and deletion of unchanging values were done but 

with no improvement in results. 

Improved results were obtained by dividing the records for 

the ten pairs into accumulation and ablation periods. April 1 was 

used as the division date. Six pairs then yielded correlation 

coefficients of 0.94 for the accumulation period, but only one 

pair reached that level in the ablation phase. Attempts to improve 

the SE and R2

 of the ablation relationship by including a 

classification of water year magnitude and time of meltout at low 

sites failed. 

Although the original criterion used correlation as a measure 

of acceptability, the standard error of estimate of the regression 

equation is a better measure. The lowest SEs of the regressions 

for the sensor pairs ranged from 15 to 25 percent of the highelevation mean April 1 SWE. Other sensor pairs yielded SEs of 

30 to 55 percent of the mean April 1 SWE. This error is 10-20 

times larger than the sensor transducer error, and could have an 

adverse effect when used as an input to the water supply forecast 

models. 

The selection of the ten sensor pairs was based on minimizeing horizontal and elevational difference. The pool of potential 

sensors was additionally constrained by eliminating sensors that 

had frequent technical problems or short records. Colocation 

within a drainage basin was also a goal. The analysis suggested 

that elevational difference is an important attribute for pairing, 

but distance and basin were less important. Future analyses 

could define pairs by calculating correlation coefficients for all 

sensors within 500 m elevation and a large distance such as 75 

km. The 500-m difference in elevation may minimize the cooccurrence of rain at the low sensor and snow at the high sensor. 

Adding a second low-elevation SWE and performing mul ­

tiple regression improved the prediction capability in both cases 

examined. In one case the SE decreased by 20 percent but was 

still 25 percent of the site's April l mean SWE. The second case 

provided dramatic improvement in SE, but the attempt revealed 

that the initial pairing was actually a poor one. In both cases 

simple regression with the new sensor yielded results almost as 

good as with the multiple equation. This result underscored the 

difficulty in a priori selection of pairs. Greatly improved results 

might have been obtained for each of the ten high-elevation 

sensors considered in this analysis if between five and 15 lowelevation sensors had been screened by a correlation matrix 

process. 

Although improved sensor selection and alternative prediction methods may yield better pairings and lower SEs, even the 

best SEs obtained in this analysis were 10 percent of the highelevation sensor's long-term April 1 mean SWE. Confidence 

intervals (95 percent) around predictions based on these "best" 

equations would be at least ±20 percent of the April 1 mean 

SWE. If this prediction error is acceptable to water supply 

forecasters, then data collection through 2000 or 2005 in the 

wilderness areas and determination of the best wildernessnonwilderness pairs may produce predictive relationships that 

are useful after the wilderness sensors are removed. 

References 

Bloomfield, Peter. 1976. Fourier analysis of time series: an introduction. 

New York: Wiley & Sons; 258 p. 

Brandow, Clay A.; Azuma, David L. 1988. Toward correlating California's 

wilderness snow sensors. In: Proceedings of the Western Snow Conference; volume 56; 1988 April 19-21; Kalispell, MT. Fort Collins: Colorado 

State University; 136-147. 

Haan, Charles T. 1977. Statistical methods in hydrology. Ames: Iowa State 

University Press; 378 p. 

Hart, David M. Letter to Bruce McGurk. 11 October 1989. Located at California 

Department of Water Resources, Cooperative Snow Surveys Program, 

Sacramento. 

Kattelmann, R.C.; McGurk, B.J.; Berg, N.H.; Bergman, J.A.; Baldwin, J.A.; 

Hannaford, M.A. 1983. The isotope profiling snow gage: twenty years of 

experience. Proceedings of the Western Snow Conference; volume 51;

1983 April 19-21; Vancouver, WA. Fort Collins: Colorado State University; 

1-8. 

McGurk, B.J. 1986. Precipitation and snow water equivalent sensors: an 

evaluation. Proceedings of the Western Snow Conference; volume 54; 1986 

April 15-17; Phoenix, AZ. Fort Collins: Colorado State University; 71-80. 

Minitab, 1985. Minitab reference manual. State College, PA: Minitab, Inc.; 

227 p. 

Suits, R.D. 1985. California snow sensor performance. Davis: University of 

California Unpublished Master's Thesis; 142 p. 

USDA Forest Service Res. Paper PSW-RP-211. 1992. 13 
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