. . 

.. .,.t. ···········: ....... ... --~- ... ............. . . . . . . . . . . . . . . 

. . ·· ······· ··:·· ····· .. .. . . . . . . . -~ .. ........ .. : .... . .. . .. .. ~- . 

. . .. . 
Cover: The spatial distribution of correlation coefficients for the North Pacific between sea-surface temperatures and 

reservoir inflow. The front cover shows the correlation pattern from August-14 months prior to the October inflows, 

and the back cover shows the correlation pattern from September-13 months prior to the October inflows. The reds 

represent the highest positive correlation and the blues represent the highest negative correlation. 
United States Department of the Interior 

U.S. GEOLOGICAL SURVEY 

Water Resources Division 

Washington District 

1201 Pacific Avenue, Suite 600 

Tacoma, Washington 98402 

(253) 428-3600 . • FAX (253) 428-3614 

http:/ /wwwdwatcm. wa. water. usgs.gov/ 

November 30,2000 

MEMORANDUM 

To: Publications Coordination, U.S. Geological Survey, Reston, VA /\ k 

From: J.A. Wayenbei-g, Physical Scientist, WRD, Tacoma, WA C.. ~~ . Y 

/ 0· c) Subject: PUBLICATIONS -Recently Released Report 

Enclosed are 2 copies of the following report. 

Enclosures: (2) 

cc: 

Development, Testing, and Assessment of Regression Equations) 

for Experimental Forecasts of Fall-Transition-Season Inflows to 

the Howard A. Hanson Reservoir, Green River, Washington,, by 

J.J. Vaccaro, WRIR 00-4153. 

Natural Resources Library, Washington, D.C. (3 copies) 

Library of Congress, Washington, D.C. (2 copies) 

USGS Library, Reston, VA (2 copies) 

USGS Library, Menlo Park, CA (2 copies) 

USGS Library, Spokane, WA (2 copies) 

USGS Library, Lakewood, CO (2 copies) 

Regional Hydrologist; Menlo Park, CA (1 copy) 

Public Inquiries Office, Spokane, WA (1 copy) 

Development, Testing, and Assessment of 

Regression Equations for Experimental 

Forecasts of Fall-Transition-Season Inflows 

to the Howard A. Hanson Reservoir, 

Green River, Washington 

By J.J.Vaccaro 

U.S. GEOLOGICAL SURVEY 

Water-Resources Investigations Report 00-4153 

Prepared in cooperation with the 

TACOMA PUBLIC UTILITIES, 

SEATTLE PUBLIC UTILITIES, and 

U.S. ARMY CORPS OF ENGINEERS 

Tacoma, Washington 

2000 
U.S. DEPARTMENT OF THE INTERIOR 

BRUCE BABBITT, Secretary 

U.S. GEOLOGICAL SURVEY 

Charles G. Groat, Director 

Any use of trade, product, or firm names in this publication is for descriptive purposes only and does not imply 

endorsement by the U.S. Government. 

For additional information write to: 

District Chief 

U.S. Geological Survey 

1201 Pacific Avenue - Suite 600 

Tacoma, Washington 98402 

Copies of this report can be purchased 

from: 

U.S. Geological Survey 

Information Services 

Building 810 

Box 25286, Federal Center 

Denver, CO 80225-0286 
CONTENTS 

Abstract ................................................................................................................................................................................. . 

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

Purpose and scope . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3 

Physical and hydrometeorological setting................................................................................................................... 4 

Development of the forecasting equations............................................................................................................................. 5 

Selection of explanatory variables............................................................................................................................... 5 

Development of final explanatory variables . . .... .. . ...... .. . . ...... .. . .. . .. . . . . .. .. . .. . . . .. ...... .. .. ........ .. ................ .. . .. ...... .. . . . . . . . . . . . . . . 6 

Function of explanatory variables in equations . .. . .......... ... . . ... . . . . .. . . . . .. ........ .. ...... .. .. . . .. . . . . . . . ........... .. . . .. . . .. ... . ... . . . . . . . . . . . .. . 7 

The forecasting equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 

Results from calibration and testing of equations . . ..... .. ... .... .. . . .. . ....... .. .. . . . . .. ........ .. ...... .... .. .. . . . . .. ................ .. . . . ...................... 10 

Assessment of performance and reliability of equations....................................................................................................... 17 

Potential use and problems with equations............................................................................................................................ 24 

Summary ................................................................................................................................................................................ 24 

References . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 

Appendix 1. Descriptive statistics for monthly, annual, and seasonal (September-October) inflows of the 

Green River to the Howard A. Hanson Reservoir, Washington............................................................................... 27 

Appendix 2. Other techniques considered for developing equations to forecast inflows for the 

fall-transition season................................................................................................................................................ 30 

FIGURES 

1. Map showing location of the Howard A. Hanson Reservoir, in the Green River Basin, and 

Chester Morse Lake, in the Cedar River Basin, Washington...................................................................................... 2 

2-6. Graphs showing observed and predicted values for the: 

2. September-October inflow season, equation 1 .................................................................................................. 12 

3. September-October inflow season, equation 2 .................................................................................................. 13 

4. October inflow season, equation 3 ..................................................................................................................... 14 

5. October inflow season, equation 4 ..................................................................................................................... 15 

6. November inflow season, equation 5 ................................................................................................................. 16 

7. Graph showing observed November inflows and residual values calculated from equation 5 ................................... 17 

8-10. Graphs showing standardized values of observed and predicted inflows for the: 

8. September-October inflow season, equations 1 and 2....................................................................................... 19 

9. October inflow season, equations 3 and 4 .......................................................................................................... 20 

10. November inflow season, equation 5 ................................................................................................................. 22 

11. Graph showing standardized values of observed and predicted inflows for the October and November 

inflow seasons . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . .. . . . . . . . .. . . . . . . . .. . . . . . .. . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . 23 

TABLES 

1. Atmospheric-circulation, sea-surface temperature, hydrometeorological, and sunspot variables 

used in regression equations for estimating September-October, October, and November 

seasonal inflow . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 

2. Results from the initial calibration and testing of five multiple linear regression equations and 

the final calibration of the equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 

VERTICAL DATUM 

Sea Level: In this report "sea level" refers to the National Geodetic Vertical Datum of 1929 (NGVD of 1929)--a geodetic 

datum derived from a general adjustment of the first-order level nets of both the United States and Canada, formerly called 

Sea Level Datum of 1929. 

Ill 

Development, Testing, and Assessment of 

Regression Equations for Experimental Forecasts of 

Fall-Transition-Season Inflows to the Howard A. 

Hanson Reservoir, Green River, Washington 

By J.J. Vaccaro 

ABSTRACT 

A method for forecasting inflows to reservoirs at 

long-lead times (14 months to 2 months prior) for the 

September-November fall-transition season was 

formulated, analyzed, tested, and applied to the 

Howard A. Hanson Reservoir on the Green River in 

western Washington. The method uses multiple linear 

regression to estimate the monthly mean streamflow 

for the combined months of September and October (a 

low-flow period), October (initial fall transition), and 

November (onset of the fall precipitation season). The 

predictors in the equations are monthly values and 

3-month averages of the monthly values of 

hemispheric-to-regional variables calculated using 

atmospheric and sea-surface-temperature data, 

hydrometeorological data, and sunspot numbers. 

Five equations were calibrated and tested, two 

for September-October inflows (14- and 13-month 

lead times), two for October inflows (11- and 2-month 

lead times), and one for November inflows (14-month 

lead time). 

The equations were initially calibrated using 

27 years of data from the 1952-96 period and were 

tested using the remaining 18 years of record. The 

calibrated equations were significant at greater than a 

95-percent level. The results of the testing indicated 

that the equation for November had the largest rsquared value (0.80) and the equations for the other 

inflow periods had values that ranged from 0.43 to 

0.57. For all equations, the standard error for the inflow 

estimates of the testing period was less than the 

standard deviation of the observed values. This initial 

testing suggested the potential future performance of 

the equations. 

The final equations were calibrated using data 

for all years from the 1952-96 period. The SeptemberOctober, October, and November equations had rsquared values of 0.71 and 0.70, 0.74 and 0.80, and 

0.82, respectively. These values were smaller than 

those from the initial 27 -year calibration period and 

larger that those from the 18-year testing period. The 

r-squared values, residual values, and standard errors 

all indicate that the equations, on the average, will 

perform reasonably well. 

The equations provide a tool for the management 

of reservoir outflows, especially to help guide existing 

management decisions such as releasing water in the 

spring for flushing salmon smolt and, concurrently, 

storing water for fall releases for steelhead migration 

and spawning. The ability to estimate the inflows for 

the fall-transition season, well before the beginning of 

the fall runoff, may decrease uncertainty in the task of 

regulation decisions. 

INTRODUCTION 

The ability to forecast seasonal inflows to 

reservoirs at long-lead times provides an important tool 

for the complex task of regulating reservoir outflows. 

In cooperation with the Tacoma Public Utilities, the 

Seattle Public Utilities, and the U.S. Army Corps of 

Engineers, the U.S. Geological Survey developed a 

method for' calculating forecasts of inflow of the Green 

River to the Howard A. Hanson Reservoir (Hanson 

Reservoir) for the September-November fall transition 

season. Hanson Reservoir is located on the west slope 

of the Cascade Range in Washington State (fig. 1). 
KING 

COUNTY 

Basin boundary 

County boundary 

Cedar Lake weather 

station 

(,chester 

\ Lake 

CANADA 

UNITED STATES 

Figure 

location 

I 

I 

_) 

) 

I 

r- KITTITAS 

COUNTY 

./"l .. "' 1.. . ' ~ ''l .. /'•""' ).· ~··--J·., --nLiwn.rlJ A. Hanson '"'""''· \.: · Reservoir · 

r-.. 1 ·"' 

'\ Green River C:.... • • )_ 

··'--··.I"""··"'· .. , 

PIERCE 

C01JNTY 

"..../ ·. <' /:--- """""\ , ' -- . -- - ··~··-··) .. ....__ .. ) ---::--... , .. ,.r 

\ 

''----'II 

\t..__"'"'_--,, 

I 

I YAKIMA 

I COUNTY 

/ 

0 5 10 15 20 MILES 

0 5 10 15 20 KILOMETERS 

Figure 1. Location of the Howard A. Hanson Reservoir, in the Green River Basin, and Chester Morse Lake, in the 

Cedar River Basin, Washington. 

2 
The fall transition period was chosen because it is a -

critical period for water supply and fisheries and there 

are currently no forecasts of streamflow at long-lead 

times for the September-November period. The 

forecasts for the Green River also are applicable, in 

terms of either percentiles or standardized values, to 

the inflows of the Cedar River to Chester Morse Lake 

(fig. 1), which is operated by the Seattle Public 

Utilities. The Green and Cedar Rivers share a common 

basin boundary and drain the west slopes of the 

Cascade Range, and their highly variable inflows 

during the fall-transition season are strongly correlated. 

The correlation coefficients for the Septemberthrough-November flows between the Green and Cedar 

Rivers are all greater than 0.90, reflecting the fact that 

the shape of the outflow hydrographs for these two 

basins are very similar during this season. 

Because each year about 26,000 acre-feet of 

storage is captured in the Hanson Reservoir for 

augmentation of summer-fall streamflow, the major 

issue in this study was to develop a means to forecast 

inflows, at long-lead times, that would indicate the 

potential for a wet or a dry fall-transition seasonespecially for October. The forecasts would help to 

facilitate the management of outflows, in order to meet 

the conflicting demands of releasing water in the spring 

for flushing salmon smolt and at the same time storing 

water for release during the fall-transition season for 

steelhead, coho, and chinook migration and spawning, 

as well as for meeting the seasonal water demand. 

The method chosen to forecast inflows was to , 

formulate multiple linear regression equations to 

estimate the mean monthly flow for the combined 

months of September and October (a low-flow period), 

for October (initial fall transition), and for November 

(onset of the fall precipitation season). This technique 

is the simplest of the currently used techniques, and it 

identifies explanatory variables that are the most 

related to the inflows. Initially, other, more complex 

techniques were also explored, and they are briefly 

described in Appendix 2 at the end of the report. The 

predictors, or explanatory variables, in the equations 

are monthly values, and 3-month averages of monthly 

values, of hemispheric-to-regional variables calculated 

using atmospheric and sea-surface temperature data, 

hydrometeorological data, and sunspot data. This 

approach is based on the well-documented strong link 

in the Pacific Northwest between natural climateforcing mechanisms (on a hemispheric-to-regional 

scale) and such hydrometeorological aspects as 

precipitation and streamflow. Thus, the possibility of 

using equations to forecast streamflow was evident 

from the persistence and strength of these climate 

variables and their strong link to temperature, 

precipitation, snowpack, and streamflow in the Pacific 

Northwest. The prediction results, although not 

excellent, indicate that this is a promising method that 

should be pursued further. 

Purpose and Scope 

The purpose of this report is to ( 1) describe the 

selection of explanatory variables for, and the 

development of five regression equations that forecast 

inflows to the Hanson Reservoir; (2) document the 

cal~bration and testing of the equations with historical 

inflow data for the Hanson Reservoir; (3) assess the 

performance and reliability of the equations for 

predicting inflow for the fall transition season and for 

predicting a wet or a dry year; and ( 4) assess the 

performance and reliability of the equations, using the 

most current data, as a measure of future performance. 

The explanatory variables used to formulate the 

equations were selected from four general categories of 

climate data: sea-surface temperature, regional-climate 

and atmospheric-circulation measures, North Pacific 

atmospheric-circulation measures, and hemispheric 

measures, including sunspots. 

The inflow data used to calibrate and test the 

equations were for the 45-year period 1952-96 from the 

historical record, which includes a major regional 

hydrometeorological regime shift in 1967. A "partial 

period" of 27 years from the 1952-96 data was used to 

calibrate the initial equations, and the remaining 

18 years of record were used to test them, including 

water years 1992-96 to reflect current conditions. The 

final equations were calibrated for the complete 1952-

96 record and were tested using inflow data for water 

year 1997. 

The seasonal inflows addressed in this report are 

the sum of the monthly mean flows for the SeptemberOctober period, the monthly mean October flows, and 

the monthly mean November flows. Descriptive 

information for seasonal, monthly, and annual inflows 

for the Hanson Reservoir is given in Appendix 1; the 

monthly inflows were provided by Thomas Murphy 

(written commun., U.S. Army Corps of Engineers, 

1994-97). 

3 
The inflows exhibit very low autocorrelation, with a 

lag-1 coefficient not exceeding 0.06. The September 

flow volumes exhibit little variability-the range 

between the 1st- and 70th-percentile flows is nearly 

equivalent to the standard deviation and is only about 

one-half of the mean. 

Five regression equations were developed for the 

seasonal inflows. The explanatory variables for the 

equations use atmospheric-circulation data, sea-surface 

temperatures, hydrometeorological data, and, for two 

equations, sunspot numbers. A preliminary 

exploratory study indicated that monthly values and 

3-month seasonal averages (or sums) of the variables 

were most appropriate for analysis. These two 

temporal domains are consistent with the seasonal 

progression of most atmospheric circulation patterns 

and with the transport of sea-surface temperatures 

(which also change seasonally) along oceanic currents. 

Of the five equations, two are for the SeptemberOctober inflows, two are for the October inflows, and 

one is for the November inflows. The lead times for the 

September-October equations are virtually the same, 

and the lead times for the October equations are 

different. Because the October flows are generally 

about double the September flows, the equations for the 

September-October inflows typically capture more of 

the inherent variability of the October flows, which 

have a standard deviation that is about one and one-half 

times larger than the mean September flow. In 

addition, for the complete period of record used in this 

study (1915-97), the September and October monthly 

flows are in phase up to 1967, and thereafter they are 

not in phase. (Being in phase means that the years of 

occurrence of higher and lower flows are generally the 

same.) This change in 1967 is one of the regional 

hydrometeorological regime shifts described by 

Vaccaro (1996). 

The lead time of an equation used to predict 

reservoir inflow is the shortest time span between the 

beginning of the inflow season and the time represented 

by any of several explanatory variables of the equation. 

For example, one of the equations used to predict 

October inflows uses values of explanatory variables 

that precede the October inflows by anywhere from 9 to 

33 months. In this case, the lead time of the equation is 

9 months, because it is the shortest lead time between 

October and any of the explanatory variables. 

4 

Physical and Hydrometeorological Setting 

The Hanson Reservoir has a storage capacity of 

105,463 acre-feet and Chester Morse Lake about 

38,137 acre-feet. Mean annual inflow to the Hanson 

Reservoir is about 1,000 cubic feet per second (ft3s-1) 

and to Chester Morse Lake about 400 ft 3s-1, 

representing runoff from 220 square miles (mi2) and 

79 mi2, respectively. Streamflow in both basins is 

supported by both rainfall runoff and snowmelt runoff. 

The outflow point of the Hanson Reservoir is about 

500 ft feet lower in altitude than that of Chester Morse 

Lake, so not only is the Hanson Reservoir drainage area 

larger, but more of it lies in the transient-snow zone at 

lower altitude and with less relief. 

Mean annual precipitation in the Cedar and 

Green River Basins ranges from about 60 to more than 

135 inches. April I snow-water equivalent (1969-95) 

in the Green River Basin is minimal at snow-course 

sites below about 2,100 feet (ft) in altitude, 20 inches 

for sites at 3,500 ft, and 34 inches at the highest site at 

4,700 ft. The snow-water equivalent at the highaltitude site has historically ranged from 6 to 66 inches. 

Generally, snow-water equivalent from March to 

April decreases for areas below about 2,800 ft and 

increases at higher altitudes. The Green River's mean 

monthly flows for November-May range from about 

1,200 ft3s-1 in November to 1,600 ft3s-1 in December 

(Appendix 1). August has the lowest mean monthly 

flows (200 ft 3s-1 ), followed by September (220 ft3s-1). 

These months of low flows reflect the overall droughtlike character of the summer season in western 

Washington, which receives about 60 percent of its 

total water-year precipitation during the months 

November through March. 

For the Green River, the October inflows have 

the largest coefficient of variation, followed by the 

November inflows. A large coefficient of variation for 

October ·inflows reflects the fact that the onset of the 

fall-precipitation season may or may not occur during 

this month. Thus, October inflows appear to cons~st of 

two very distinct populations, even more so than the 

other monthly inflow values. For the Cedar River, 

September inflows have the largest coefficient of 

variation, reflecting the limited ability of the smaller 

Cedar River Basin to store water, especially base flows, 

in comparison with the larger Green River Basin. 

October and November have the next largest coefficient 

of variation in the Cedar River. 
DEVELOPMENT OF THE 

FORECASTING EQUATIONS 

Numerous investigators have documented the 

strong linkage, at various time scales, between 

hemispheric-to-regional climate forcing and such 

aspects as precipitation and streamflow in the Pacific 

Northwest. See, for example, Yarnal and Diaz (1986), 

Speers and Mass (1986), Cayan and Peterson (1989), 

Redmond and Koch ( 1991 ), Ebbesmeyer and others 

(1989), and Cayan and Webb (1992). The persistence, 

strength, and interconnections of quasi-stationary 

circulation patterns and sea-surface temperature 

anomalies has also been well documented. See, for 

example, Bjerknes (1969), Horel and Wallace (1981), 

Namias (1981), Rogers (1981), Barnston and Livezey 

(1987), and Namias and others (1988). The persistence 

and strength of these climate variables and, their strong 

linkages to streamflow suggested the possibility of 

formulating atmospheric-circulation and sea-surface 

temperature variables that are strongly linked to 

temperature, precipitation, and streamflow in the 

Pacific Northwest. These climate-related variables 

resulted from analysis of regime shifts in precipitation, 

temperature, snowpack, and streamflow, and their 

linkage to climate information (Vaccaro, 1996). The 

variables are monthly time-series and include such 

previously well-documented and well-known variables 

as the Southern Oscillation Index (SOl), which is 

obtained from the Climate Analysis Center (CAC), 

National Oceanic and Atmospheric Administration 

(NOAA). The variables are spatially invariant because 

they are based on data from fixed points in space. 

A screening process was used to obtain a small 

set of variables from the potentially large number of 

variables possible. The explanatory variables were 

used to develop five regression equations: two for the 

period September-October (at 14- and 13-month lead 

times), two for October (at 9- and 2-month lead times), 

and one for November (at a 14-month lead time). 

Selection of Explanatory Variables 

Generally, the explanatory variables that were 

chosen had not only a statistical link, but also a 

climatological basis for a physical link between the 

variables and Pacific Northwest hydrometeorological 

variables (Vaccaro, 1996). Some of the variables 

considered are not necessarily independent because 

they are only calculated differently using the same or 

nearly the same data points. For example, values from 

two grid points of a data set may be subtracted to obtain 

one variable and added to obtain another, resulting in, 

say, a pressure-difference variable and a pressureheight variable, which may or may not be correlated. 

Some of the variables formulated using 700-millibar 

(mb) height values (units of tens of feet minus 700) are 

also represented as standardized series. 

Most explanatory variables were hemisphericto-regional variables that are measures of the climate 

system. Although they also provide estimating ability, 

few site-specific variables, such as precipitation, 

snowpack, or temperature, were used because the 

intent was to make the results from the regression 

equations (in terms of percentiles or standardized units) 

more regional in nature. There is a significant relation 

between the inflows to Hanson Reservoir and 

streamflow at 22 sites in western Washington. For 

example, there is a correlation coefficient of 0.93 

between the inflows to Hanson Reservoir and the timeseries of the first principal component of the 

streamflow for the 22 western Washington sites. There 

is a correlation coefficient of 0.96 between the inflows 

and the average of the standardized values of 

streamflow for 11 of the sites that had first-principalcomponent values that were similar to those of the 

Green River. This suggests that, by using mainly largescale variables and few basin-specific data as 

explanatory variables, the equations developed for the 

Green River may also be applicable for some other 

streams in western Washington. For example, a 

forecasted 20th-percentile inflow value for Hanson 

Reservoir may indicate that some other nearby streams 

might also have a value in the 20th-percentile range. 

For three of the five equations, however, sitespecific precipitation and temperature data from the 

Seattle Public Utilities's Cedar Lake weather station 

(altitude of 1,460 ft at latitude 47.4167°N and 

longitude 121.7333°W) and sunspot numbers were 

used in order to improve the robustness of the 

equations. The site-specific weather data are used 

because ( 1) the weather data are highly correlated to 

the fall-transition period inflows for both the Green and 

Cedar Rivers, and (2) the site is considered a key 

indicator of meteorological conditions for the west 

slope of the Cascade Range in Washington (Rasmussen 

and Tangborn, 1976; Ebbesmeyer and others, 1989; 

Vaccaro, 1996). Mean water-year precipitation at 

Cedar Lake is 102 inches and has ranged from 67 to 

135 inches. Mean monthly precipitation ranges from 

5 
14.5 inches in January to 2.2 inches in July. The 

sunspot numbers are correlated to seasonal inflows, but 

they are poorly correlated with most of the climaterelated variables; therefore, they provide additional 

forecasting ability that is relatively independent of the 

climate variables. 

The atmospheric information used in this study 

consists of the monthly SOl (CAC, NOAA) and 

diamond-grid, monthly mean 700-mb heights for the 

Northern Hemisphere (data from 1947-95 from D. R. 

Cayan, written commun., U.S. Geological Survey, 

1995; data from 1996 to present from CAC, 

NOAA-extracted from the monthly mean values 

contained in the climate data assimilation system 

(CDAS) data set). The sea-surface temperature data set 

used in the preliminary stages of this study consisted of 

5° -gridded monthly means for the latitude range 20°-

60°N and the longitude range 11 0°W-130°E (D. R. 

Cayan, written commun., U.S. Geological Survey, 

1995). The final sea-surface temperature data set 

consisted of 4 o -gridded monthly means for the same 

area. This data set was obtained by averaging the 1 ogridded monthly means of the optimally interpolated 

sea-surface temperature data for the period 1982 to 

present, obtained from the National Center for 

Environmental Prediction (NCEP), NOAA. For the 

period 1950-81, reconstructed sea-surface 

temperatures on a 2° -grid (Smith and others, 1996) 

were averaged to the 4°-grid. The sunspot number 

time-series was obtained from the National 

Geophysical Data Center (NGDC) operated by NOAA. 

The SOl, CDAS, and sea-surface temperature data can 

be obtained from the CAC/NCEP world wide web site, 

and the sunspot data can be obtained from the NGDC 

site. 

The water years of record for the SOl were 

1933-96, for the CDAS were 1950-96, and for the seasurface temperature were 1947-96; the sea-surface 

temperature data set started in January 1950, and the 

700-mb data set started in December 1946. Thus, most 

of the atmospheric and sea-surface temperature data 

used to formulate the variables have a common period 

of water years 1950-96 (47 years). The actual record 

length used in the analyses ranged from 44 to 48 years 

because of the long-lead aspect of the equations. The 

sunspottime-series began in 1750. 

6 

Development of Final Explanatory 

Variables 

Each explanatory variable in an equation is a 

time-series measurement for some specific prior time: 

for example, a December value of a 4 o -grid sea-surface 

temperature would be used in an equation to predict the 

following October inflow. Because of the potentially 

large number of possible explanatory variables, a 

screening process was used to obtain a small set of 

variables to be used in a stepwise regression analysis. 

As a result of this process, a lead time was selected for 

each equation and the understanding of the relation 

between inflows and the climate system was improved. 

Values of monthly explanatory variables and 

their 3-month averages initially were screened using a 

correlation analysis to determine which variables 

correlated most strongly to each of the seasonal inflow 

periods. The correlation coefficients were calculated by 

lagging the variables 1 month at a time, starting in 

October of the fall-transition season and ending 3 years 

prior to the October. For example, for the 3-month 

averages, calculation of a correlation coefficient might 

be based on the 45-year period of record for, .say, a 

March-April-May average of some variable and the 

following October inflow; the next correlation 

coefficient would be for the February-March-April 

average of the variable and the October inflow. 

Each monthly time-series therefore has 72 potential 

values (36 months plus the 3-month averages for the 

36 months) for use in regression analysis. This -

screening analysis was repeated with cubed values of 

the 3-month and monthly variables because cubing 

emphasizes the extremes of some variables. 

Several thousand correlation coefficients were 

examined. It was determined that, for each of the three 

seasonal inflows, a large number of variables were 

significantly correlated at various time lags. To further 

screen the significantly correlated variables, they were 

stepwise regressed against the corresponding inflows. 

The regression was done for several partial periods 

consisting of 26 years of inflows and also for the 

1952-96 period. The 1952-96 period was used 

because there have been distinct temporal changes, 

including shifts and trends, in the persistence, strength, 

existence, and centers of major circulation features that 

are not fully captured by partial periods or by spatially 

invariant variables (Vaccaro, 1996). 

The results of both the compilation of the 

significantly correlated variables and the regression 

analysis were used to select five sets of 20 to 50 
variables, one set for developing each of the five 

regression equations. This selection was oriented to 

finding the fewest number of variables common 

between the sets, and was also aimed at obtaining sets 

of variables that were somewhat consistent with the 

seasonal progression of hemispheric-North Pacific 

circulation features (Barnston and Livezey, 1987) and 

North Pacific features (Vaccaro, 1996). Both the 

September-October and the October equations share 

common variables because most of the climate 

signature is represented in the highly variable October 

inflows. 

A preliminary exploratory analysis determined 

that five to nine variables is a reasonable number for an 

equation. Using fewer than five variables allows one to 

two variables to be too influential. Using more than 

about nine variables leads to stability problems that are 

probably due to (1) the effects of large changes in 

variables that are not as strongly correlated as the first 

one to nine variables, (2) increasing cross correlation 

between variables as the number increases, and (3) 

increasing noise with increasing variables (because the 

less-dominant modes of hemispheric-to-regional 

circulation features contain more noise). 

Function of Explanatory Variables in 

Equations 

Nineteen variables were selected to be used in 

the formulation of the five regression equations and 

they are listed in table 1. The variables can be grouped 

into four general categories: sea-surface temperatures 

(sst), regional climate and atmospheric-circulation 

measures, North Pacific atmospheric-circulation 

measures, and hemispheric measures. 

The sst variables (variables 1-4 in table 1) 

essentially account for the persistence of Pacific Ocean 

temperatures in the upper part of the water column in 

certain regions of the North Pacific. This persistence is 

associated with the development of anomalous sst 

patterns and affects atmospheric-circulation features. 

Sstl accounts for sst persistence during the summer for 

a region located offshore of the Pacific Northwest that 

borders the eastern edge of a cold-water pool 

associated with a major circulation feature called the 

Aleutian Low. Sst2 and sst3 account for changes and 

persistence in sst's from summer into winter near the 

influential California Current. Based on the 

preliminary exploratory analysis, sst4 is an indicator of 

west-east sst gradients and the movement of water 

across the northern part of the Pacific. 

All but one of the regional measures (variables 5-

1 0) indicate the importance of persistence during the 

May through July period. The lead time for the 

variables also corresponds to the dominant frequencies 

contained in the reservoir inflows for the fall-transition 

season as determined by harmonic analysis. Variables 

5 to 8 essentially measure the tropospheric atmospheric 

conditions centered over northwestern Washington and 

southwestern British Columbia, Canada, and are 

influential for the following reasons (Vaccaro, 1996): 

( 1) the variables are indicative of the strength and 

location of an influential high-pressure cell (HIGHPRESSURE) that is one of the centers of a dominant 

hemispheric circulation pattern; (2) the height field 

over northwestern Washington determines or affects 

the magnitude of the upper-level winds, the location of 

storm tracks, and the strength of the zonal (latitudinal) 

or meridional (longitudinal) atmospheric flow. These 

three aspects are accounted for by the HEIGHT, 

HEIGHT-STAND, and NSGEO variables. Variables 9 

and 10, the data for the Cedar Lake weather station, 

capture the persistence in wet/dry and warm/cold 

patterns for part of the west slope of the Cascade 

Range. The climate signal in the data for this site is 

very strong at scales from hemispheric to North Pacific 

to regional. For example, the winter precipitation at 

this site is highly correlated to the SOl, and is strongly 

linked to the existence, location, strength, and 

persistence of the Aleutian Low and an associated coldwater pool. At the regional scale, air temperatures for 

the Cedar Lake site are strongly related to geostrophic 

flow over northwestern Washington, and the variable 

captures major climate-regime changes. 

The North Pacific atmospheric-circulation 

variables (variables 11-16) measure the strength of 

circulation patterns that influence streamflow in 

western Washington during the fall-transition season. 

The variables WPOl, WP04, and NPAC contain 

features (represented as 700-mb grid-point data) of 

circulation patterns that are described by Barnston and 

Livezey (1987). The EASTPAC circulation pattern 

was clearly defined as an influential dipole in the 

preliminary exploratory analysis completed in this 

study. 

7 
8 

Table 1. Atmospheric-circulation, sea-surface temperature, hydrometeorological, and sunspot variables used in 

regression equations for estimating September-October, October, and November seasonal inflow 

Variable 

name 

1 sst1 

2 sst2 

3 sst3 

4 sst4 

S HEIGHT-STAND 

6 HIGH-PRESSURE 

7 HEIGHT 

8NSGEO 

9 Cedar L. Precip. 

10 Cedar L. Temp. 

11 EASTPAC 

12 WP04 

13 WP01 

14 LOW-PRESSURE 

1S HT-PNA-STAND 

16NPAC 

17 SOl 

18 SUNSPOT 

19SUNSPOT2 

Month Typel 

June M 

Dec M 

Aug 3M 

June 3M 

June 3M3 

July M 

July 3M 

May M3 

Nov M 

May M 

Mar M 

Mar M 

Aug 3M 

June 3M 

Jan 3M3 

May 3M 

Feb M 

Oct M3 

June M3 

Lead 

time2 

(months) Descriptive information 

Sea-surface temperature variable3 

17 4°-grid cell, average for 48°N,l38°W 

11 4°-grid cell, average for 36°N,126°W 

1S 4°-grid cell, average for 36°N,126°W 

s 4°-grid cell, average for 44°N,162°E 

Regional climate and atmospheric-circulation variables4 

17 Measures geopotential height field, centered at S0°N,12S0 W 

28 Measures strength of a high-pressure cell, centered at SS0

N, 11S0W 

28 Measures geopotential height field, centered at S0°N, 12S0

W 

30 Measures strength of northerly flow between S0°N, 120°W -130°W, a 

measure of the 700-mb geostrophic flow from the north 

36 Measures precipitation at Cedar Lake weather station at 1 ,460 feet 

30 Measures monthly mean temperature at Cedar Lake weather station 

North Pacific atmospheric-circulation variables4 

20 Measures strength of north-south dipole, 2S0 N,l3S0

W and 60°N,1S0°W 

20 Measures strength of north-south dipole, 2S0 N,16S0

W and SS0 N,16S0W 

27 Measures strength of northeast -southwest dipole, 2S0 N, 11 0°W and 

SS0N,l7S0E 

29 Measures strength of Aleutian Low centered at 4S0N, 16S0W 

22 Measures geopotential height field (variable 7), centered at S0°N,l2S0W 

multiplied by strength of pattern with centers at SS0N,l1S0 W;4S0 N,16S0W; 

and 30°N,8S0W 

30 Measures strength of a pattern with centers near 40°N, 170°W; 70°N, 180°W; 

and 40°N,l30°E 

Hemispheric variables4 

33 Measures east-west sea-level pressure gradient in the tropics 

2S Measures sunspot numbers 

17 Measures sunspot numbers 

1M, monthly variable; 3M, 3-month mean centered at indicated month; M3 or 3M3, value of variable is cubed. 

2Lead time of listed month prior to the November seasonal inflow; for example, 1 month would be October. 

3Jn units of degrees Fahrenheit. 

4Except for the SOl (variable 17), all atmospheric-circulation variables are calculated from 700-millibar diamond-gridded data; the SOl 

is calculated using sea-level pressure at Tahiti and Darwin, Australia; STAND, standardized variable: monthly value of variable is standardized by subtracting the mean monthly value for the complete period of record and then dividing by the standard deviation; a dipole consists of 

700-mb grid points, generally representing the height difference between a high and a low pressure; a pattern or height field consists of more 

than two grid cells used to calculate the variable. 
This north-south dipole measures the relative strength 

of a low-pressure region near the typical location of the 

Aleutian Low (LOW-PRESSURE) and a high-pressure 

region centered well offshore of central Mexico that is 

consistent with the generation of storms and storm 

tracks. The WP04 variable is similar to the EASTPAC 

except that the high-pressure region is much more to 

the west. Both WP04 and EASTPAC, which have the 

same lead time, account for persistence in the 

atmospheric signal over broad areas just after the 

winter season. The apparent signal of these two 

patterns, as seen in the inflow data, changes from the 

EASTPAC for the September-October inflows to the 

WP04 for the October inflows, suggesting the 

importance of a westward translation of a highpressure region during this lead time. The NPAC 

variable measures a pattern that covers a reasonably 

large area of the northern part of the western North 

Pacific. The strength of the NPAC is also an indicator 

of the future strength of both the EASTPAC and the 

WP04, and the lead times of the three suggest not a 

seasonal, but rather an annual progression of influential 

atmospheric-circulation patterns. 

The last three measures (variables 17-19) are 

classified as hemispheric. The strong influence of the 

Southern Oscillation/El Nifio phenomenon on global 

climate and hydrology is well documented. In 

addition, its influence on the hydrology of the Pacific 

Northwest also is well documented (Cayan and 

Peterson, 1989; Redmond and Koch, 1991; Cayan and 

Webb, 1992; Vaccaro, 1996). At first, the SOl was 

excluded from the equations because its large-scale 

influence on the climate system might interfere with 

the signal from other important variables. However, 

the February SOl was significantly related to both 

the September-October and October inflows at the 

33-month lead time. As a result, the SOl was included. 

Moreover, because of the overall weak precursor 

climate signal contained in these first 2 months of the 

fall-transition season, some kind of explanatory 

variable was needed that accounted for variance in the 

inflow data. Sunspot numbers are known to be 

correlated with many geophysical phenomena, but the 

cause-effect relations are unknown and vigorously 

debated. The cube of the sunspot number provides 

estimating ability and is not significantly correlated to 

most other variables. Thus, sunspot variables 18 and 

19 are included in two equations, resulting in more 

reliable and independent equations. 

The Forecasting Equations 

The five multiple linear regression equations 

developed to forecast inflow to the Hanson Reservoir 

are presented below for each inflow period and each 

equation lead time. The equations were developed 

using the inflows for the complete 1952-96 period. 

Equations 1 and 2 are for September-October 

inflows and they use explanatory variables with values 

from about 3 years to 13 months prior to September. 

Equations 1 and 2 have lead times of 13 and 14 months, 

respectively. Equations 1 and 2 share five variables, 

and equation 2 has two additional variables, HIGHPRESSURE and sstl. Sst1 essentially builds on the 

concurrent strong relation between a central North 

Pacific cold-water pool and the average geopotential 

height-field over northwestern Washington (HEIGHTSTAND), and the former apparently accounts for 

persistence in the high-pressure cell. Both equations 

calculate nearly the same inflow values. 

Equations 3 and 4 are for October inflows and 

they use explanatory variables with values from about 

3 years to 2 months prior to October. Equations 3 

and 4 have lead times of 9 months and 2 months, 

respectively. The lead times of the variables in 

equation 4 (see table 1) indicate that they capture a 

distinct temporal progression of climate influences on 

subsequent October flows. 

Equation 5 is for November inflows and it uses 

explanatory variables with values from 30 to 14 months 

prior to November. Equation 5 has a lead time of 

14 months. All the variables but one are for the period 

May-August, and four of the seven variables are for the 

period of May-June. The clustering of explanatory 

variables during these times indicates that the summer 

transition is an important period for defining 

persistence in the system, which is obviously 

complexly related to the November inflows. 

The five regression equations are as follows. 

Regression equations for September-October inflows 

• With a 13-month lead time: 

SEASONAL INFLOW= -22.97 x Cedar L. Precip 

-70.72 x SOl- 8.767 

x NPAC + 5.734 

x EASTPAC + 139.04 

x HEIGHT-STAND+ 2065.3 (1) 

9 
• With a 14-month lead time: 

SEASONAL INFLOW= -21.65 x Cedar L. Precip 

- 64.897 x SOl - 9.085 

x NPAC -7.591 

x HIGH-PRESSURE 

+ 4. 795 x EASTPAC 

+ 119.499 x HEIGHT-STAND 

+ 31.85 X SS 1 + 2965.3 

Regression equations for October inflows. 

• With a 9-month lead time: 

SEASONAL INFLOW= -47.38 x SOl -13.32 

x NPAC- 14.767 

x HIGH-PRESSURE 

-13.165 x HEIGHT 

+ 3.771 X WP04 

+ 130.013 x ss2 + 2607.2 

• With a 2-month lead time: 

SEASONAL INFLOW= -50.58 x SOl- 10.778 

xNPAC-20.732 

x HIGH-PRESSURE 

- 1.893 X 10-5 

x SUNSPOT + 3.246 

X WP04 + 77.47 X ss2 

+ 122.17 x sst4 - 3056.6 

Regression equation for November inflows 

• With a 14-month lead time: 

SEASONALINFLOW= -281.16xNSGEO 

10 

+ 71.81 x Cedar L. Temp 

-29.46 x LOW-PRESSURE 

+ 29.242 X WP01 - 428.32 

X HT-PNA-STAND + 7.088 

x 10-5 x SUNSPOT2 

+ 130.75 x sst3 + 608.4 

(2) 

(3) 

(4) 

(5) 

RESULTS FROM CALIBRATION 

AND TESTING OF EQUATIONS 

The initial equations were calibrated using 

stepwise regression based on 27 years of record from 

the 1952-96 period. To obtain the partial period, the 

lower, middle, and upper thirds of the years were 

ranked by September-October inflows, and nearly 

equal numbers of years were randomly selected 

from each third. The equations were tested using 

the remaining 18 years, which is a conservative 

60/40-percent split of the record. Water years 1992-96 

were included only in the 18-year test period in order to 

assess the equations under more current conditions. 

This split-sample procedure allowed for an assessment 

of potential performance of the equations under both 

past and future conditions. The final equations were 

calibrated with the complete 45-year record and were 

tested for water year 1997. Results from the calibration 

and testing are shown in table 2. 

The testing results from the split-sample 

calibrated equations (table 2) indicate that the 

equations provide reasonable and somewhat . 

independent estimates of the seasonal inflows and 

suggest how the final equations may perform in the 

forecast mode. The r-squared, standard error, and 

standard deviation values reflect the estimating ability 

of the equations. About 80 percent of all the residual 

values (observed minus predicted) were less than th~ 

standard error determined during calibration. That is, 

more than 68 percent (the expected percentage from 

regression analysis) of the predicted values from the 

testing period were less than a standard error from the 

observed values. The ranges and averages of the 

residuals of the five equations for the testing period 

were as follows. 

Inflow Period 

September-October 

(equations 1 and 2) 

October 

(equations 3 and 4) 

November (equation 5) 

Residuals 

(ft3s-1) 

Range Average 

-345 - 557 78 

-375 - 458 5 

-986 - 590 -156 
.. .. 

Table 2. Results from the initial calibration and testing of five multiple linear regression equations and the final calibration of the equations 

[Equations developed using data from 1952-96. Equations initially were based on a sample size of 27 years for calibration and 18 years of testing. Final equations were calibrated 

using the complete 45 years; --, not applicable; all values rounded] 

Inflow, in cubic feet per second, except for r-squared, dimensionless 

Standard deviation 

Mean Standard error 

calibration Testing Calibration Testing r-square of estimate 

-- --

Calcu- Calcu- Calcu- Calcu- Cali- CaliEquation1 Observed lated Observed lated Observed lated Observed lated bration Testing bration Testing 

SeQtember-October inflow eguations 

Eguation 1 

Initial 688 684 765 663 453 421 392 246 0.86 0.44 205 303 

Final 719 719 -- -- 427 357 -- -- 0.71 -- 251 

Eguation 2 

Initial 688 688 765 658 453 420 392 246 0.87 0.43 194 305 

Final 719 719 -- -- 427 359 -- -- 0.70 -- 248 

October inflow eguations 

Eguation 3 

Initial 448 448 514 516 364 351 313 347 0.84 0.57 167 210 

Final 474 475 -- -- 343 295 -- -- 0.74 -- 188 

Eguation 4 

Initial 448 448 514 512 364 334 313 286 0.92 0.48 113 232 

Final 474 475 -- -- 343 307 -- -- 0.80 -- 166 

November inflow eguation 

Eguation 5 

Initial 1,229 1,208 1,429 1,585 851 769 960 675 0.81 0.80 419 440 

Final 1,309 1,309 -- -- 891 806 -- -- 0.82 -- 415 

1 Equations given in text in the section "Forecasting Equations." 
The average residuals and the testing results in 

table 2 show that the September-October equations (1 

and 2) have a bias toward estimating smaller-thanaverage values, the October equations (3 and 4) have 

little bias, and the November equation (5) has a bias 

toward predicting larger-than-observed values. Some 

of the bias in the split-sample testing is retained in the 

final calibrated equations, discussed later in the section 

Assessment of Performance and Reliability of 

Equations. The equations generally performed well 

under the more recent 1992-96 conditions. The testing 

results indicate that the performance of the final 

equations when applied in the forecasting mode should 

be reasonable. 

The observed and predicted values from the final 

five equations are shown on figures 2-6 and the 

calibrated statistics are presented in table 2. The rsquared value for the final September-October -

equations 1 and 2 was about 0.70 with a standard error 

of about 250 ft3s-1, about 40 percent smaller than the 

standard deviation of the observed values and 

65 percent smaller than the mean (table 2). Although 

there is a slight difference in estimating ability between 

equations 1 and 2 (table 2), it is not easily discerned 

when comparing the difference in the scatter about the 

diagonal between figures 2 and 3. The additional two 

variables used in equation 2 should help to produce 

somewhat different estimates from those of equation 1, 

but the overall results will be similar. 

~ 2600 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

12 

0 

(.) 

~ 2400 

a: 

w 

a.. 2200 .__ 

w 

w 

u.. 2000 

(.) 

CI.i a 1800 

z 

g 1600 

_J 

u.. 

~ 1400 

w 

a: tn 1200 

a: 

w 

aJ 1000 § 

0 800 

a: 

w 

aJ 

:2 

w 

.__ 

a.. 

w 

en 

0 

w 

t3 

0 

600 

400 

200 

+ 

+ 

+ 

+ 

Equation 1 

w 

a: 

a.. 

0 ~~~~-L~~~~~~~~~~~~~~~~~~~~~~~~ 

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 

OBSERVED SEPTEMBER- OCTOBER STREAMFLOW, IN CUBIC FEET PER SECOND 

Figure 2. Observed and predicted values for the September-October inflow season, equation 1. 

The central diagonal line represents a theoretical line of equal observed and predicted values, 

and the overall scatter about the line reflects the standard deviation of the residuals. Values on 

the axes are the full range of observed values for the inflow period. 
In practical terms, for potential future use of the 

equations the forecasted inflows from equations 1 and 

2 may best be averaged. 

The final October equations (3 and 4) had rsquared values of 0.74 and 0.80 and standard errors of 

188 ft3s-1 and 166 ft3s-l, respectively (table 2). 

Comparing figures 4 and 5 illustrates the slightly 

il)lproved fit of equation 4 versus equation 3. The 

figures also illustrate that the equations produce 

estimates that generally differ, and yet retain about the 

same distribution. 

0 

z 

0 

(_) 

w 

(J) 

a: 

w 

a.. 

1-

w 

w 

u.. 

(_) 

ill 

:::> 

(_) 

~ 

g 

....J 

u.. 

~ 

<1:: 

w 

a: 

1-

(J) 

a: 

w 

en 

§ 

0 

a: 

w 

en 

~ 

w 

1-

a.. 

w 

(J) 

0 

w 

I-

(_) 

0 

2600 

2400 

2200 

2000 

1800 

1600 

1400 

1200 

1000 

800 

600 

400 

200 

In comparison to the other equations, equation 5 

better represents the observed values (fig. 6), with an rsquared value of0.82 and a standard error of 415 ft3s-l 

(table 2). Most of this improvement can be accounted 

for by the fact that the November inflows generally 

contain a much stronger climate signal than either the 

September or October inflows, and this signal is 

apparently captured by the variables in equation 5. In 

western Washington, the third largest precipitation of 

all the months happens in November, and thus 

November is strongly linked to the winter atmosphericcirculation regime. 

+ 

+ 

Equation 2 

w o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

g: 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 

OBSERVED SEPTEMBER- OCTOBER STREAMFLOW, IN CUBIC FEET PER SECOND 

Figure 3. Observed and predicted values for the September-October inflow season, equation 2. 

The central diagonal line represents a theoretical line of equal observed and predicted values, 

and the overall scatter about the line reflects the standard deviation of the residuals. Values on 

the axes are the full range of observed values for the inflow period. 

13 
The persistence in the atmospheric-oceanic system 

(which the equations essentially attempt to account for) 

is reflected in the November inflows. The standard 

error, which reflects the standard deviation of the 

residuals, is considerably less than the standard 

deviation of the observed values, as demonstrated on 

figure 7. This figure also illustrates the bias to 

predicting larger-than-observed values (negative 

residuals) 

A good estimate of the error in values calculated 

by each equation is the standard error of estimate of the 

1800 

0 

z 1650 0 

() 

w 

(/) 1500 

a: 

w 

a. 1350 Iw 

w 

u... 

() 1200 

a5 

::> 

() 1050 ~ 

~ 900 0 

_J 

u... 

~ 750 + <l: 

w 

a: 

I-

(/) 600 

a: 

w 

Ill 450 ~ + () 

0 300 + 0 

w 

I-

() 150 0 

w 

a: 

a. 0 

final calibrated equations (table 2). The potential error 

sin predicted values for equations 1-5 are, respectively, 

251 ft3s-1, 248 ft3s-1, 188 ft3s-I, 166 ft3s-l, and 

415 ft3s-l. The smaller an estimated inflow value is, the 

larger its potential percent error would be. The 

equations will yield good results on average, but not 

necessarily in every individual year. The reason is that 

the equations are statistical; that is, forecasts for some 

years may be out on the distribution tail strictly by 

chance or because of changes in the climate regime. 

+ 

Eq~ation 3 

-15o~~~~~~~~J_~~~~~~~~~~~~~~~~~~~ 

14 

-150 0 150 300 450 600 750 900 1050 1200 1350 1500 1650 1800 

OBSERVED OCTOBER STREAMFLOW, IN CUBIC FEET PER SECOND 

Figure 4. Observed and predicted values for the October inflow season, equation 3. The central 

diagonal line represents a theoretical line of equal observed and predicted values, and the 

overall scatter about the line reflects the standard deviation of the residuals. Values on 

the axes are the full range of observed values for the inflow period. 
1800 

0 

z 1650 0 

(.) 

w 

(/) 1500 

a: 

w 

a.. 

1- 1350 

w 

w 

LL 

(.) 1200 

i]j 

:::> 

(.) 1050 

z 

;:~ 

900 0 

.....J 

LL 

~ 750 <t: 

w 

a: 

1-

(/) 600 

a: 

w 

m 450 ~ (.) 

0 300 0 

w 

1-

(.) 150 0 

w 

a: 

a.. Equation 4 0 

-15o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

-150 0 150 300 450 600 750 900 1050 1200 1350 1500 1650 1800 

OBSERVED OCTOBER STREAMFLOW, IN CUBIC FEET PER SECOND 

Figure 5. Observed and predicted values for ,the October inflow season, equation 4. The central 

diagonal line represents a theoretical line of equal observed and predicted values, and the 

overall scatter about the line reflects the standard deviation of the residuals. Values on 

the axes are the full range of observed values for the inflow period. 

15 
16 

4500 

0 

z 

0 4000 (.) 

w 

en 

a: 

w 

a.. 3500 

1--

w 

w 

LL 

(.) 3000 OJ 

:::> 

(.) 

z 

$:~ 2500 q.. 

0 

.....J 

LL 

::E 

<( 

w 2000 

a: 

1--

en 

a: 

w 1500 Ill 

::E + 

w 

> 0 

z 1000 

0 

w 

1--

(.) 

0 500 w 

a: Equation 5 a.. 

0 ~~~~~~J_~~-L~~~~~~~~~~~~~~~~~~~ 

0 500 1 000 1500 2000 2500 3000 3500 4000 4500 

OBSERVED NOVEMBER STREAMFLOW, IN CUBIC FEET PER SECOND 

Figure 6. Observed and predicted values for the November inflow season, equation 5. The 

central diagonal line represents a theoretical line of equal observed and predicted values, and 

the overall scatter about the line reflects the standard deviation of the residuals. Values on 

the axes are the full range of observed values for the inflow period. 
4000 

0 

z 

0 3600 Observed 

() 

w 

C/) 3200 Residual, equation 5 

a: 

w 

c.. 2800 

Iw 

w 

u. 2400 

() 

co 2000 :::> 

() 

z 1600 

~ 0 1200 .....J 

u. 

~ 

<( 800 

w 

a: 

IC/) 400 I 

a: 

w 

CD 0 ~ 

w 

> -400 0 

z 

I 

1\ I 

I I I I 

I II I \I I I I I I I I I I I \I I I \ " I 

I I \I \I I I \ I I I I 

II I I I I I I I " I 1 I I IJ IJ 

' \I 1 I IJ II I 

-800 

1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 

WATER YEAR 

Figure 7. Observed November inflows and residual values calculated from equation 5. 

ASSESSMENT OF PERFORMANCE 

AND RELIABILITY OF EQUATIONS 

The performance and reliability of the final 

equations were assessed by ( 1) applying the equations 

to forecast inflow for the 1997 water year and then 

comparing forecasted with observed values for the 

three inflow periods for calendar year 1996 (September 

of water year 1996 and October-November of water 

year 1997-the fall-transition season inflows for water 

year 1997), and (2) analyzing the results for high 

(generally the upper 70th-percentile values) and low 

(generally the lower 30th-percentile values) inflow 

periods. The latter was done because the primary 

interest is in providing long-lead estimates for potential 

for wet or dry fall-transition seasons. In addition, the 

ability of the equations to capture the concurrency of 

high or low inflows for October and November was 

also assessed. 

The observed inflows, forecasted inflows, and 

percentiles for the fall-transition period for water year 

1997 are: 

ForeObserved! casted 

inflow Percen- inflow PercenInflow period (ft3s-l) tile (ft3s-l) tile2 

September-October 

Equation 1 1,623 96th 1,020 83rd 

Equation 2 1,623 96th 959 79th 

October 

Equation 3 1,464 98th 848 85th 

Equation 4 1,464 98th 774 83rd 

November 

Equation 5 2,771 91st 1,512 69th 

lThomas Murphy, written commun., U.S. Army Corps of 

Engineers, 1997. 

2Based on the observed historical record. 

17 
The observed inflows for the fall-transition 

period for water year 1997 were some of the largest on 

record, and the forecasted inflows were also well-above 

average. Although the monthly mean flow for 

September was a 30th-percentile value (based on the 

1952-97 period of record), both the observed and 

forecasted inflows for the September-October period 

were large. The large values demonstrate both the 

differences between the monthly mean September and 

October inflows and the fact that equations 1 and 2 

typically capture the variability in October inflows. In 

terms of standardized series (standardized series have a 

mean of zero and a standard deviation of 1 ), the 

observed flows ranged from 1.63 for November to 2.66 

for October. The forecasted inflows ranged from 0.26 

for November to 1.16 for October, and except for 

November, all values were more than one-half of a 

standard deviation above the mean. Overall, this first 

Observed low flowsl 

The number of times that the 

calibrated discharge is in these 

Equations ranges of standardized units 

used to 

compute Less -0.5 0 Greater 

calibrated than to to than 

discharge -0.5 0 0.5 0.5 

application of the equations to future conditions 

produced reasonable estimates. 

The reliability of the equations for forecasting 

low- and high-flow periods was assessed using 

standardized series for the observed and calibrated 

values. Using standardized data allows for the 

comparison of two series whose distributions or values 

may not be the same. For the inflow periods during the 

years that the observed values were more than half a 

standard deviation from the mean (values greater than 

0.5 or less than -0.5), the observed and calibrated 

values (figures 8-10) are compared graphically for 

each inflow period. Values less than -0.5 represent low 

flows and values greater than 0.5 represent high flows. 

The calibrated values for the corresponding observed 

low- and high-flow inflows are shown in the table 

below by inflow period and by the number of 

occurrences in each standardized discharge range. 

Observed high flowsl 

The number of times that the 

calibrated discharge is in these Percentage of 

ranges of standardized units calibrated low 

and high flows 

Less -0.5 0 Greater corresponding 

than to to than to observed low 

-0.5 0 0.5 0.5 and high flows 

September-October inflow period; 14low flows and 9 high flows observed from 1953-96 

1 

2 

10 

10 

3 

2 

1 

2 

0 

0 

0 

0 

0 

0 

2 

2 

7 

7 

74 

74 

October inflow period; 19low flows and 13 high flows observed from 1952-96 

3 

4 

13 

13 

3 

5 

3 

1 

0 

0 

1 

0 

1 

2 

0 

0 

11 

11 

75 

75 

November inflow period; 13low flows and 10 high flows observed from 1952-96 

5 7 6 0 0 0 0 3 7 61 

1 Low flow is defined as a discharge of less than -0.5 standardized units; high flow is defined as a discharge of greater than 0.5 

standardized units. 

If the match between observed and calibrated 

values was extremely good-that is, the number of 

calibrated low- and high-flow values would be equal to 

the respective number of observed values-then all 

values would plot in either the lower left (low-flow 

values) or the upper right (high-flow values) quadrant 

on figures 8-10. For equations 1 and 2 (September18 

October inflows with 23 observed values-14low flow, 

9 high flow), 74 percent of the calibrated values plot in 

the ~orrect quadrant (fig. 8, above table). For equations 

3 and 4 (October inflows with 32 values-19 low flow, 

13 high flow), 75 percent of the calibrated values plot 

in the correct quadrant (fig. 9). 
OBSERVED DISCHARGE, IN STANDARDIZED UNITS 

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 

- - - - -I- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

4.8 

4.4 

4.0 

3.6 

3.2 en 

1-

z 

2.8 ::::> 

2.4 

2.0 

1.6 

0 

w 

N 

0 

a: 

(§ 

z 

~ en 

z 

u.i 

1.2 ~ 

<( 

I 

() 

en 0.8 

0.4 

0.0 

-0.4 

-0.8 

-1.2 

-1.6 

0 

0 

w 

t;( 

a: 

Ill 

:::i 

<( 

() 

----~----~--------------------------------------------~-2.0 

EXPLANATION 

Area of both observed and calibrated high-flow values 

Area of both observed and calibrated low-flow values 

----- Theoretical line of equal observed and calibrated values 

--- Line of half a standard deviation from the mean 

~ Equation 1 values 

0 Equation 2 values 

Figure 8. Standardized values of observed and calibrated inflows for the September-October inflow season, 

equations 1 and 2. Calibrated values are only shown for observed high flows (observed values> 0.5) and 

observed low flows (observed values< -0.5). 

19 
20 

OBSERVED DISCHARGE, IN STANDARDIZED UNITS 

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 

7( 

4.8 

4.4 

4.0 

3.6 

3.2 U) 

~ 

z 2.8 ::::> 

2.4 

2.0 

1.6 

0 

w 

N 

iS 

a: 

(3 

z 

~ U) 

z 

u.i 

1.2 (!J 

a: 

<( 

0.8 

0.4 

0.0 

I 

() 

U) 

iS 

0 

w 

~ a: 

CD 

::J 

_ 0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -0.4 

<( 

() 

7( 

-0.8 

-1.2 

-1.6 

----~--------------------------------------------------~-2.0 

EXPLANATION 

Area of both observed and calibrated high-flow values 

Area of both observed and calibrated low-flow values 

----- Theoretical line of equal observed and calibrated values 

- - - Line of half a standard deviation from the mean 

7( Equation 3 values 

0 Equation 4 values 

Figure 9. Standardized values of observed and calibrated inflows for the October inflow season, equations 

3 and 4. Calibrated values are only shown for observed high flows (observed values> 0.5) and observed 

low flows (observed values< -0.5). 
For October, fully 68 percent of the inflows were more 

than half a standard deviation from the mean, further 

showing that October inflows clearly represent two 

distinct populations. For equation 5 (November 

inflows with 23 observed values-13low flow, 10 high 

flow), 61 percent of the calibrated values plot correctly 

(fig. 10). Thus, with a reasonable probability, the five 

equations are able to capture the occurrence of lowand high-flow discharge values. All but 7 of the 79 

calibrated low-flow values were less than the mean and 

only 4 of the 54 possible high-flow values were less 

than the mean. This suggests that the equations also 

provide reliable estimates of below- or above-average 

inflows for the fall-transition season. 

The closeness of values to the diagonal on 

figures 8-10 demonstrates the match of the calibrated 

values to observed values for extreme years, and 

grouping of values above or below the diagonal 

indicates the potential bias. A bias toward overprediction-predicting larger-than-observed values for 

high-flow years or smaller-than-observed for low-flow 

years-would be better than a bias toward underprediction. Over-prediction identifies occurrences of 

inflows more than half a standard deviation from the 

mean. In under-prediction, calculated values are not 

only closer to the mean but they may also be above the 

mean during a low-flow year and below the mean 

during a high-flow year. 

Equations 1 and 2 have a bias to under-predict 

the extremes (fig. 8). The preponderance of the 

calibrated values lie above the diagonal for low flows 

(in three cases greater than the mean) and below the 

diagonal for high flows (no occurrences of less than the 

mean). Although eight values from equations 3 and 4 

were of the opposite sign of the observed values 

(fig. 9), these equations have less overall bias than 

equations 1 and 2. There also is a tendency for the 

standardized values for low- and high-flow years from 

both equations 1 and 2 to be more similar to those from 

equation 4 than from equation 3. 

The calibrated values for November (fig. 10) 

show little bias and a good fit for the high-flow years. 

For low-flow years, there is a reasonable fit but a bias 

for calculating larger-than-observed values (underprediction). This bias also occurred for the low-flow, 

split-sample testing years. All equations generally 

capture the occurrences of high flows better than low 

flows, especially equations 3 and 4 (fig. 10, above 

table). 

For each inflow period there were more 

occurrences of low-flow years than high-flow years, 

indicating the increased probability for the falltransition season to be dry. Indeed, for October, there 

were 6 more low-flow years than high-flow years, 

representing about 13 percent of the 45-year period of 

record used in the analysis. However, the range of the 

low-flow values is much smaller than the range of high-

. flow values (figs. 8-10). For example, for the three 

inflow periods, the range between the upper 30thpercentile values varies from four to nine times the 

range between the lower 30th-percentile values. These 

differences, which are also exhibited by the Cedar 

River inflows, identify important aspects of the 

hydrology of the two basins and their relation to 

climate forcing-the basins can only get so dry, but 

they can get very wet. The equations are better able to 

capture the wet extremes because of the much larger 

range in the wetter years. 

The climate regime differs greatly between lowand high-flow years because low and high flows 

generally are derived from two separate populations, 

each with specific characteristics. If one considers only 

the question of whether the fall-precipitation season 

will start in October or November, then these 

differences are quite important. For example, for the 

45 years of record used in this analysis, 13 of the 

standardized October inflows had values greater than 

0.5 and one October had a value of 0.44. In practical 

terms, about 30 percent of the time the precipitation 

season began in October and the amount of 

precipitation was enough to sustain higher flows. The 

remaining 70 percent of the time, October inflows were 

smaller than the mean (about 70 percent of which could 

be considered low-flow years). The October equations 

captured the occurrence of the onset of the fallprecipitatio~ season about 85 percent of the time. The 

climate regime that drives the wet Octobers is 

apparently accounted for in equations 3 and 4. There 

were only six occurrences of concurrent high flows 

during both October and November, but all of the 

November standardized values during high-flow 

Octobers were greater than -0.5. Given that in the 

historical record no very-low-flow November years 

occurred during high-flow Octobers, the October 

equations should provide reasonable estimates if the 

fall precipitation season will not only start in October, 

but may continue through November. 

21 
22 

OBSERVED DISCHARGE, IN STANDARDIZED UNITS 

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 

4.8 

4.4 

4.0 

3.6 

3.2 en 

1-

z 

2.8 ::.> 

2.4 

2.0 

1.6 

0 

w 

N 

Ci 

a: 

(§ 

z 

~ en 

z 

w 

1.2 CJ 

a: 

<( 

0.8 

0.4 

-0.4 

-0.8 

-1.2 

-1 .6 

I 

() 

CJ) 

Ci 

0 

w 

t;: 

a: 

co 

:J 

<( 

() 

-----L----~------------------------------------------~-2.0 

EXPLANATION 

Area of both observed and calibrated high-flow values 

Area of both observed and calibrated low-flow values 

----- Theoretical line of equal observed and calibrated values 

- - - Line of half a standard deviation from the mean 

~ Equation 5 values 

Figure 10. Standardized values of observed and calibrated inflows for the November inflow season, 

equation 5. Calibrated values are only shown for observed high flows (observed values> 0.5) and 

observed low flows (observed values< -0.5). 
One reason that there were only six concurrent 

occurrences of high flows during both October and 

November is that for several of the high-flow Octobers 

with average November inflows, the precipitation 

during November was above normal but the air 

temperatures were such that much of the precipitation 

fell as snow. 

Given the two distinct climatic regimes, another 

question of interest is whether the equations capture the 

concurrency of a low-flow October and a belowaverage (but not necessarily low-flow) November. For 

the 19low-flow Octobers, 15 Novembers met this 

criterion; the equations calculated 14 of these 15 

3.8 

concurrent low-flow periods correctly. In addition, of 

the 28 years that November standardized inflows were 

less than 0.0, 13 were less than -0.5 and all but 2 of 

these years had concurrent low-flow Octobers; that is, 

low-flow Novembers are generally preceded by lowflow Octobers. The results of equations 3-5, taken 

together, should provide useful information on whether 

October and November will be wet or dry; see, for 

example, figure 11. The information shown on 

figure 11 also suggests that a standardized value 

provides reliable information and should be considered 

in conjunction with the actual forecast value. 

October - Observed I 

~ 

II 

II 

11 

II 

II 

3.4 

3.0 

(/) 

1-

z 2.6 

::J 

0 2.2 

w 

N 

Ci 1.8 

a: 

<( 1.4 0 

z 

~ 1.0 

(/) 

z 0.6 

u.i 

(!J 0.2 a: 

<( 

[5 -o.2 

(/) 

Ci -0.6 

-1.0 

-1.4 

-1.8 

November - Observed 

October - Predicted, Equation 4 

November - Predicted, Equation 5 

\I 

·.y 

I 

I 

1951 1956 1961 1966 1971 1976 

WATER YEAR 

1981 1986 1991 1996 

Figure 11. Standardized values of observed and predicted inflows for the October and 

November inflow seasons. The predicted inflows are from equations 4 and 5. 

23 
POTENTIAL USE AND PROBLEMS 

WITH EQUATIONS 

The equations may be used for reservoir 

management in several ways. One way is to calculate 

the inflows and then account for the values when 

making reservoir release decisions throughout the year. 

Another way that they may be used is, if equations 

1 and 2 (13- and 14-month lead times) calculate a 

20th-percentile value for the following SeptemberOctober inflow, then the observed inflows from January 

through October for the years that produced, say, 1stthrough 40th-percentile inflows could be used in a 

management model to assess potential management 

strategies. This approach could also be used with the 

results of equation 3 at 9 months lead time and equation 

4 at 11 months lead time. Management strategies 

might include hedging releases at certain times in order 

to augment flows for the September-October period. 

The accuracy of the equations is influenced by 

the strength and position of atmospheric- and oceaniccirculation patterns. Indeed, at times particular 

circulation patterns may not even exist. This inherent 

problem leads not only to a loss of accuracy, but also to 

the fact that the equations essentially act as detectors of 

climate-regime shifts. The equations may no longer be 

valid at some future period because of an overall 

realignment of atmospheric circulation and sea-surface 

temperature anomaly patterns. Much of this problem is 

due to the highly nonlinear and dynamic nature of the 

ocean-atmosphere system and because, 

climatologically, the period of record used in this study 

is short. This problem can be partly overcome by using 

variables that are more dynamically based; for 

example, using a pressure-gradient variable rather than 

the currently used pressure-difference variable. 

Developing more dynamically based variables for the 

equations may increase their overall accuracy. 

SUMMARY 

The U.S. Geological Survey, in cooperation with 

the Tacoma Public Utilities, Seattle Public Utilities, 

and the U. S. Army Corps of Engineers, developed a 

method for estimating, at long-lead times, the falltransition seasonal inflow to the Howard A. Hanson 

Reservoir. The method uses multiple linear regression 

equations to estimate inflow values. The equations 

were developed for three inflow periods: September24 

October, October, and November. Two regression 

equations were developed for each of the first two 

inflow periods and one equation for the November 

inflows. The equations for the September-October 

inflows are at 14- and 13-month lead times, the October 

equations are at 9- and 2-month lead times, and the 

November equation is at a 14-month lead time. The 

equations use monthly and 3-month averages of 

variables that are calculated mainly from atmospheric 

and sst data. 

The equations were initially calibrated using 

27 years of data from the 1952-96 period of record. 

Inflows for years 1992-96 were saved for testing the 

equations in order to assess the estimating ability of the 

equations for the most recent period of record used in 

this study. Testing the equations using 18 years of data 

resulted in r-squared values that ranged from 0.43 to 

0.80. This initial calibration and testing procedure 

suggested how the equations may perform under future 

conditions. The final five equations were calibrated 

using the complete 1952-96 period of record. The final 

calibrated equations had r-squared values that ranged 

from 0.70 to 0.82. The equation for the November 

inflows has the largest r-squared value, and the 

equations for the September-October inflows have the 

smallest r-squared values. These results are consistent 

with the fact that the climate signature becomes 

stronger as the fall-transition season progresses. 

The results and assessment of the final equations 

showed that the equations generally are able to capture 

the occurrence of low-flow and, especially, high-flow 

years. The October equations can provide a reasonable 

estimate of whether the onset of the fall-precipitation 

season will begin in October. In addition, the 

assessment indicates that standardized values should 

be considered in conjunction with the actual forecast 

values, especially for the November forecasts, due to 

the error and bias in the estimates. 

The ability to predict reservoir inflows with long 

lead times can be used to help manage releases from the 

reservoir throughout the water year. The final 

equations, with results reformulated in terms of 

percentiles or standardized units, are also applicable to 

Chester Morse Lake on the nearby Cedar River. 

Similarly, the equations may also be applicable to other 

western Washington streams. 
REFERENCES 

Barnett, T.P., and Preisendorfer, R.W., 1978, Multifield 

analog prediction of short-term climate fluctuations 

using a climate state vector: Journal of Atmospheric 

Science, v. 35, p. 1771-1787. 

Barnston, A.G., and Livezey, R.E., 1987, Classification, 

seasonality and persistence of low-frequency 

atmospheric circulation patterns: Monthly Weather 

Review, v. 115, p. 1083-1126. 

Bjerknes, J.H., 1969, Atmospheric teleconnections from the 

equatorial Pacific: Monthly Weather Review, v. 97, 

p. 162-172. 

Cayan, D.R., and Peterson, D.H., 1989, The influence of 

North Pacific atmospheric circulation on streamflow in 

the west, in Peterson, D.H., ed., Aspects of climate 

variability in the Pacific and Western Americas: 

Washington, D.C., American Geophysical Union, 

Geophysical Monograph 55, p. 375-397. 

Cayan, D.R., and Webb, R.H., 1992, El Nino/Southern 

Oscillation and streamflow in the western United 

States, in Diaz, H.F. and Markgraf, V.M., eds, El Nino, 

Historical and paleoclimatic aspects of the Southern 

Oscillation: Cambridge, England, Cambridge 

University Press, p. 29-68. 

Christensen, R.A., Eilbert, R.F., Lindgren, O.H., and Rans, 

L.L., 1981, Successful hydrologic forecasting for 

California using an information theoretic model: 

Journal of Applied Meteorology, v. 20, p. 706-713. 

Christensen, R.A., and Eilbert, R.F., 1985, Seasonal 

precipitation forecasting with a 6-7 month lead time in 

the Pacific Northwest using an information theoretic 

model: Monthly Weather Review, v. 113, p. 502-518. 

Ebbesmeyer, C.C., Coomes, C.A., Cannon, G.A., and 

Bretschneider, D.E., 1989, Linkage of ocean and fjord 

dynamiCs at decadal period, in Peterson, D.H., ed., 

Aspects of climate variability in the Pacific and Western 

Americas: Washington, D.C., American Geophysical 

Union, Geophysical Monograph 55, p. 399-417. 

Gladwell, J.S., 1970, Runoff generation in western 

Washington as a function of precipitation and 

watershed characteristics: Pullman, Wash., 

Washington State University, College of Engineering, 

Research Division, R. L. Albrook Hydraulic 

Laboratory, Bulletin 319, 341 p. 

Horel, J.D., and Wallace, J.M., 1981, Planetary-scale 

atmospheric phenomena associated with the Southern 

Oscillation: Monthly Weather Review, v. 109, p. 813-

829. 

Namias, J., 1981, Teleconnections of 700mb height 

anomalies for the northern hemisphere: La Jolla, Calif., 

Scripps Institution of Oceanography, CALCOFI 

Atlas 29, 265 p. 

Namias, J., Yuan, X., and Cayan, D.R., 1988, Persistence of 

North Pacific sea surface temperature and atmospheric 

flow patterns: Journal of Climate., v. 1, p. 682-703. 

Rasmussen, L.A., and Tangborn, W.V., 1976, Hydrology of 

the North Cascades region, Washington 1. Runoff, 

precipitation, and storage characteristics: Water 

Resources Research, v. 12, no. 2, p. 187-202. 

Redmond, K.T., and Koch, R.W., 1991, Surface climate and 

streamflow variability in the Western United States and 

their relationship to large-scale circulation indices: 

Water Resources Research, v. 27, no. 9, p. 2381-2399. 

Rogers, J.C., 1981, The North Pacific Oscillation: Journal of 

Climatology, v. 1, p. 39-57. 

Smith, T.M., Reynolds, R.W., Livezey, R.E., and Stokes, 

D.C., 1996, Reconstruction of historical sea surface 

temperatures using empirical orthogonal functions: · 

Journal of Climate, v. 9, p. 1403-1420. 

Speers, Pamela, and Mass, C.F., 1986, Diagnosis and 

prediction of precipitation in regions of complex 

terrain: Washington State Department of 

Transportation, WA-RD 91.1, 166 p. 

Torranin, Padoong, 1972, Application of canonical 

correlation in hydrology: Fort Collins, Colo., Colorado 

State University, Hydrology Papers, No. 58, 30 p. 

Vaccaro, J.J., 1996, Changes in the hydrometeorological 

regime in the Pacific Northwest, in Isaacs, C. M. and 

Tharp, V. L, eds., Proceedings of the twelfth annual 

Pacific climate (PACLIM) workshop: Technical Report 

46 of the Interagency Ecological Program for the 

Sacramento-San Joaquin Estuary, Apri11996, p. 143. 

Yarnal, B., and Diaz, H.F., 1986, Relationships between 

extremes of the Southern Oscillation and the winter 

climate of the Anglo-American Pacific coast: Journal 

of Climatology, v. 6, p. 197-219. 

25 
APPENDIXES 
1\) 

...... 

Appendix 1. Descriptive statistics for monthly, annual, and seasonal (September-October) inflows of the Green River to the Howard A. Hanson Reservoir, 

Washington 

[Monthly unregulated flows provided by the U.S. Army Corps of Engineers. Abbreviations used are: Station ID,U.S. Geological Survey identification number for 

reservoir stage gage; MEDN, median value; MAXM, maximum value; MINM, minimum value; SDEV, standard deviation; CV AR, coefficient of variation; Sept.-Oct., 

sum of monthly mean values for September and October] 

Howard A. Hanson Reservoir Green River, Storage Ca12acity of 105,463 acre-feet 

Station Drainage 

identifi- area Altitude Location of dam 

cation (square Number years gage 

number miles) Start year End year of record (feet) Latitude 0

N Longitude ow 

12105800 220 1915 1996 82 1,141 47.2778 121.7856 

Inflow, in cubic feet 12er second 

Oct. Nov. Dec. Jan. Feb. March Apr. May June July Aug. Sept. Annual S~pt.-Oct. 

MEAN 500.46 1,236.07 1,615.28 1,495.36 1,342.89 1,169.55 1,504.23 1,478.25 955.98 381.25 201.02 220.75 1,008.42 723.80 

MEDN 392.00 943.50 1,432.00 1,409.00 1,197.50 1,055.00 1,501.00 1,466.00 812.50 308.00 190.50 170.50 997.25 618.00 

MAXM 1,818.50 3,974.50 5,293.00 4,394.00 3,445.92 3,439.00 2,490.50 2,842.00 2,802.00 1,076.00 389.50 788.50 1,567.37 2,558.50 

MINM 90.50 108.50 202.50 208.00 265.00 491.50 600.00 434.00 186.00 110.00 116.00 97.00 488.08 207.00 

SDEV 368.79 857.77 948.56 826.28 689.92 496.33 413.03 564.92 574.96 199.06 60.22 124.17 231.41 440.48 

SKEW 1.56 1.19 1.51 0.65 0.91 2.02 -0.08 0.39 1.20 1.43 1.08 2.45 0.15 1.77 

CVAR 0.74 0.69 0.59 0.55 0.51 0.42 0.27 0.38 0.60 0.52 0.30 0.56 0.23 0.61 
1'1.) 

co 

Appendix 1. Descriptive statistics for monthly, annual, and seasonal (September-October) inflows of the Green River to the Howard A. Hanson Reservoir, 

Washington-Continued 

Quartile 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

65 

Inflow, in cubic feet per second 

Oct. Nov. Dec. Jan. Feb. March Apr. May June July 

119.77 120.00 424.00 383.00 426.00 542.00 754.50 517.00 252.00 164.00 

155.50 334.69 601.50 481.00 540.00 677.50 878.50 745.00 301.00 197.00 

190.48 421.00 749.50 555.00 636.00 738.00 1,004.50 873.72 429.00 214.20 

207.50 520.00 807.50 633.50 690.00 797.50 1,069.00 920.00 444.50 225.50 

232.00 606.00 879.50 707.50 779.00 833.00 1,179.00 1,031.00 513.50 237.00 

252.00 691.00 975.00 907.50 879.00 877.50 1,254.00 1,050.00 538.02 257.00 

258.50 796.50 1,120.50 1,006.00 967.00 926.50 1,335.00 1,199.00 686.50 263.00 

280.50 882.00 1,314.00 1,093.00 1,050.82 946.50 1,382.50 1,250.50 730.00 276.50 

318.00 917.00 1,390.50 1,286.00 1,131.50 983.07 1,483.50 1,341.50 758.00 284.50 

392.00 943.50 1,432.00 1,409.00 1,197.50 1,055.00 1,501.00 1,466.00 812.50 308.00 

412.00 1,002.00 1,524.00 1,477.00 1,271.50 1,081.50 1,555.00 1,509.00 861.00 329.50 

461.00 1,116.00 1,641.00 1,563.50 1,342.50 1,166.00 1,607.00 1,599.50 915.50 350.00 

502.00 1,225.00 1,687.00 1,683.00 1,422.50 1,193.50 1,680.00 1,613.00 968.00 390.00 

70 538.00 1,447.50 1,799.00 1,830.00 1,517.50 1,257.00 1,713.00 1,655.00 1,057.50 416.00 

75 648.50 1,617.00 1,958.50 2,030.00 1,618.50 1,380.00 1,795.00 1,788.00 1,131.00 445.00 

80 730.50 1,763.50 2,154.50 2,242.00 1,734.00 1,462.50 1,861.00 1,878.00 1,418.50 474.50 

85 783.00 1,988.50 2,162.00 2,482.00 2,021.50 1,561.00 1,908.00 2,169.50 1,464.50 579.00 

90 916.00 2,462.00 2,481.00 2,613.00 2,292.00 1,610.00 1,939.00 2,233.00 1,641.50 686.00 

95 1,247.00 2,786.00 3,231.50 2,696.00 2,537.00 1,732.50 2,072.50 2,416.00 2,092.00 754.50 

Aug. 

124.00 

132.00 

142.00 

150.00 

154.50 

159.50 

161.50 

172.50 

181.00 

190.50 

193.50 

202.00 

209.00 

211.00 

221.00 

231.00 

269.00 

280.00 

299.50 

Sept. Annual Sept.-Oct. 

115.00 637.04 

120.00 688.29 

131.00 744.25 

139.00 785.96 

147.50 834.08 

153.50 848.25 

158.50 885.08 

161.50 911.83 

165.00 933.54 

170.50 997.25 

175.00 1023.00 

199.50 1052.79 

206.00 1061.79 

234.50 1135.33 

238.00 1188.71 

246.00 1214.08 

335.00 1244.75 

344.00 1346.00 

424.00 1375.83 

257.00 

288.00 

328.50 

348.76 

400.50 

425.00 

455.50 

571.00 

591.50 

618.00 

644.00 

664.00 

731.00 

788.00 

834.00 

941.00 

1037.00 

1169.50 

1513.00 
1\) 

<0 

Appendix 1. Descriptive statistics for monthly, annual, and seasonal (September-October) inflows of the Green River to the Howard A. Hanson Reservoir, 

Washington-Continued 

Lag 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

Oct. 

0.04 

-0.06 

-0.12 

-0.04 

-0.01 

0.03 

0.07 

0.13 

0.07 

-0.07 

-0.16 

0.03 

0.20 

0.19 

-0.13 

-0.11 

-0.08 

-0.07 

-0.16 

-0.06 

0.16 

0.01 

-0.05 

-0.17 

-0.02 

0.17 

0.07 

-0.05 

0.01 

Nov. 

0.06 

-0.08 

-0.14 

-0.02 

0.20 

-0.12 

-0.21 

0.05 

0.01 

-0.05 

-0.13 

-0.09 

0.35 

-0.07 

0.02 

-0.21 

-0.06 

0.15 

-0.08 

-0.24 

-0.05 

0.06 

0.03 

-0.12 

-0.20 

0.22 

0.20 

0.25 

-0.17 

Dec. 

0.03 

0.04 

0.00 

-0.11 

-0.08 

-0.09 

-0.08 

0.04 

0.01 

-0.18 

-0.14 

-0.15 

0.04 

-0.02 

0.05 

0.25 

0.00 

-0.17 

-0.03 

0.03 

0.03 

0.24 

-0.07 

0.13 

0.02 

-0.07 

-0.20 

-0.09 

0.14 

Autocorrelation coefficients, number years = 82; number of lags = 29 

Jan. 

0.19 

0.03 

-0.13 

-0.26 

-0.04 

0.04 

0.06 

0.18 

-0.02 

-0.03 

-0.10 

-0.08 

-0.06 

0.09 

0.15 

0.00 

-0.01 

-0.15 

-0.19 

-0.03 

-0.14 

-0.11 

-0.14 

-0.27 

0.02 

-0.11 

0.00 

0.01 

0.03 

Feb. 

-0.03 

-0.13 

-0.06 

0.12 

-0.11 

-0.07 

0.15 

-0.05 

0.05 

0.21 

0.04 

-0.21 

-0.10 

0.18 

-0.10 

-0.02 

-0.08 

0.01 

0.20 

0.01 

0.29 

-0.25 

0.12 

-0.15 

-0.10 

-0.16 

0.09 

0.04 

0.05 

March 

-0.08 

0.06 

-0.04 

0.08 

-0.11 

-0.16 

0.11 

-0.14 

-0.21 

-0.17 

0.10 

-0.01 

-0.17 

0.11 

0.07 

0.18 

-0.13 

0.31 

-0.08 

-0.12 

-0.02 

0.20 

-0.03 

-0.06 

0.05 

-0.01 

-0.16 

-0.24 

0.16 

Apr. 

0.06 

0.01 

-0.01 

-0.10 

0.04 

0.01 

0.02 

-0.02 

-0.07 

-0.07 

-0.01 

-0.07 

0.13 

-0.05 

-0.09 

-0.09 

-0.03 

-0.02 

0.03 

0.12 

-0.25 

-0.12 

-0.05 

0.06 

-0.01 

-0.07 

-0.07 

-0.01 

0.02 

May 

0.22 

0.00 

0.15 

0.17 

0.15 

-0.01 

-0.04 

0.00 

-0.16 

0.09 

0.04 

0.01 

-0.01 

-0.04 

0.08 

-0.01 

-0.14 

-0.01 

0.04 

-0.05 

0.14 

-0.04 

-0.09 

0.04 

0.02 

0.16 

0.06 

-0.12 

-0.09 

June 

0.10 

-0.03 

-0.07 

0.11 

0.11 

0.07 

-0.10 

-0.04 

-0.04 

0.18 

-0.04 

-0.08 

-0.08 

0.00 

-0.04 

0.11 

-0.01 

-0.04 

0.07 

-0.05 

0.07 

0.01 

-0.14 

-0.12 

-0.05 

0.11 

-0.03 

-0.07 

0.05 

July 

0.24 

0.05 

-0.01 

0.08 

0.08 

-0.07 

-0.03 

0.02 

0.05 

0.05 

-0.09 

-0.04 

-0.14 

-0.12 

-0.17 

0.09 

0.13 

-0.04 

0.10 

-0.06 

0.00 

0.07 

-0.24 

-0.23 

-0.13 

-0.09 

-0.09 

-0.17 

0.04 

Aug. 

0.13 

0.05 

-0.09 

0.25 

0.10 

0.02 

0.11 

0.22 

0.10 

-0.08 

-0.03 

0.11 

0.03 

-0.01 

-0.16 

0.09 

-0.07 

0.00 

-0.07 

0.00 

0.07 

-0.02 

-0.23 

-0.31 

0.05 

-0.02 

0.01 

-0.12 

0.03 

Sept. 

0.01 

-0.11 

-0.09 

0.26 

0.13 

0.04 

-0.07 

0.00 

0.33 

0.00 

-0.09 

-0.02 

0.21 

0.15 

-0.05 

-0.15 

-0.04 

0.11 

0.00 

-0.10 

-0.08 

-0.05 

0.05 

0.08 

-0.19 

-0.14 

0.11 

-0.06 

-0.21 

Annual 

0.13 

0.07 

-0.02 

0.05 

-0.03 

-0.13 

-0.01 

-0.05 

-0.07 

-0.04 

0.00 

0.00 

0.03 

0.12 

0.27 

0.01 

0.07 

-0.09 

-0.18 

-0.26 

-0.09 

0.12 

-0.17 

-0.14 

0.03 

0.17 

-0.05 

-0.06 

-0.04 

Sept.-Oct. 

0.01 

-0.10 

-0.13 

0.01 

-0.04 

0.02 

-0.03 

0.06 

0.12 

-0.07 

-0.17 

0.07 

0.25 

0.14 

-0.17 

-0.13 

-0.01 

-0.02 

-0.15 

-0.16 

0.12 

0.05 

-0.02 

0.17 

-0.03 

0.17 

0.13 

-0.11 

-0.06 
Appendix 2. Other Techniques Considered For Developing Equations To Forecast 

Inflows For The Fall-Transition Season 

Initially, several techniques were explored for developing equations for forecasting inflows, from which the 

regression-equation method was chosen. One technique 

was to develop variables using principal-component analysis (PCA), a tool that has been used by many investigators in 

a large range of atmospheric and hydrologic studies. One of 

the early uses of PCA in western Washington was reported 

by Gladwell (1970), who applied PCA to climatological and 

physiographic variables to form explanatory variables for 

regression analysis in order to determine which ones were 

the most influential for explaining streamflow variability in 

western Washington. For the PCA used in the early stages 

of this study, the first three to four principal component (PC) 

time-series of about 25 variables were calculated using a 

common period of record (1947-94) for all variables. The 

PC time-series were then used as the explanatory variables 

in regression analysis. Conceptually, in using such a regression technique, current values of the time-series would be 

updated on the basis of the current values of the variables 

and the historical values of the loadings for each variable. 

This method assumes that the loadings would not need to be 

updated using PCA because they should be relatively stable 

due to the length ( 48 years) of the time-series used to 

develop them. Although initial results from this technique 

proved promising, it was not pursued because the additional 

data and work needed to calculate the many variables used 

in the PCA, and the formulation of the PC series, would 

tend to make a monthly, operational technique more complex. 

Another technique was tested that also used PCA. In 

this technique, prior to calculating the variables for the 

atmospheric-circulation features, a time-series representing 

the first PC of the North Pacific sea-surface temperatures 

was used as a weighting factor for multiplying the 700-mb 

data. This time-series and the streamflow series in western 

Washington are coherent on the decadal scale. This type of 

weighting was further tested by standardizing each of the 

5°-gridded sea-surface temperature time-series, and then 

multiplying each of the monthly 700-mb gridded timeseries by the corresponding month of the nearest gridded 

30 

*U.S. GOVERNMENT PRINTING OFFICE: 2000- 589-121 / 03022 Region No. 10 

value of the sea-surface temperature data. For both cases, 

the resulting 700-mb grid was reduced to the spatial extent 

of the sea-surface temperature data; thus, variables based on 

the 700-mb atmospheric-circulation features that extended 

beyond the spatial domain of the sea-surface temperature 

data set were not calculated. The weighted 700-mb data 

were then used to calculate variables for multiple linear 

regression equations. Results from this testing were not as 

good as other techniques. Use of only major hemispheric 

variables was also explored. For this case, only variables 

such as the SOl were analyzed in conjunction with the seasonal inflows. It was found that, for the fall-transition season, the signal of large-scale variables was both highly 

variable and not very strong. 

Another technique explored was that of Barnett and 

Preisendorfer ( 1978), which they called multivariate analog 

prediction. This technique identifies regions where data, 

such as pressure heights and sea-surface temperatures, are 

highly related to the dependent variable. Data from these 

regions are then used to formulate independent series for 

analysis. The entropy minimax nonparametric-technique 

(Christensen and others, 1981; Christensen and Eilbert, 

1985), canonical correlation analysis (CCA), and non-linear 

multiple regression were also briefly explored. These three 

techniques were felt to be either too complex for operational 

use or there was not enough time to fully explore them. Previously, Torranin (1972) tested CCA for "long-range prediction" using gridded sea-surface temperatures as independent 

variables and coastal precipitation in the Pacific Northwest 

(which is highly correlated to the streamflow in western 

Washington and precipitation at Cedar Lake) as a dependent 

variable; he found the results to be unreliable. However, 

the use of CCA by National Oceanic and Atmospheric 

Association's Climate Analysis Center for producing longlead forecasts 'over much of the continental United States 

suggests that the spatial scale of the explanatory data used 

in this study may be such that CCA may be worth pursuing 

in conjunction with the already completed work.