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Supplement
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Temperature Modeling
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The historical and future seasonal distribution of stream temperature  on day-of-year x is
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expressed in terms of a seasonal cycle, f A ( x) , driven by the seasonal air temperature and a
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deviation, f D ( x) , representing the cooling effects of snowmelt runoff (Beer and Anderson
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2011) (Fig. 1):
 ( x)  f A ( x)  f D ( x)
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(1)
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where f A ( x) is characterized by the annual average water temperature, A, the amplitude of the
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seasonal variation from heat exchange with the atmosphere B, and a seasonal phase offset C:
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 2
f A ( x)  A  B sin 
 x  C  
 365

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and f D ( x ) is characterized by the maximum springtime depression of water temperature, D, the
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day-of-year beginning the snowmelt season (Xbegin) and the end of the season (Xend ):
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D 
  x  X begin
1 
    for X begin  x  X end
 1  sin  2 

f D ( x)   2 
.
  X end  X begin 4   


otherwise
0
(2)
(3)
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An additional constraint is that snowmelt contribution is zero after the maximum temperature,
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thus f D ( x)  0 for x > Xmax. The parameters were fit to each year’s stream temperature data over
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the water year (October 1 – September 30) with a Gauss-Newton algorithm that minimizes the
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sum of squares.
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For each stream i, a baseline set of site-specific parameters, Pi   Ai , Bi , Ci , Di , X begin ,i , X end ,i  , was
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estimated by fitting the model to data aggregated across all available years of stream
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temperatures. Parameters A and D, which exhibit yearly variations, were also needed, so a set of
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parameters was fit to water-year, y, giving site- and year-specific parameters
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Pi , y   Ai , y , Bi , y , Ci , y , Di , y , X begin ,i , y , X end ,i , y  . Parameters A and D were then related to environmental
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trends, and hence projected forward to mid-21st century. We assumed that the effects of climate
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change on other parameters were small.
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Data
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Stream temperature data for CA, OR, WA, and ID were obtained from the United States
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Geological Survey (http://waterdata.usgs.gov/nwis/), California Department of Water Resources,
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Data Exchange Center (http://cdec.water.ca.gov/), and the Washington State Department of
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Ecology (http://www.ecy.wa.gov/science/data.html). Daily water temperature values were
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distinguished by water year (October – September) for fitting the temperature model. Several
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sites in Washington had only monthly averages.
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Fitting the data was an iterative process. For example, first the temperature model was fit for a
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water year within a site (henceforth a “fit-year”). It was then inspected visually to insure that the
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model was fitting as intended. Any conspicuously anomalous input data was excluded and if
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model parameters were outside of existing constraints then new fitting constraints were adopted
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and the entire procedure reapplied to all the data. By the end of the fitting process, all parameters
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met basic constraints (e.g. 0 <  < 20, Xbegin before Xmax and both between the coolest and
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warmest days, Xmax  Xbegin at least 50 days, etc.). All fits used here and many others are shown
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in supplementary material (available 9 January 2013 at
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http://www.cbr.washington.edu/data/Streams/data.html). An individual site had to have ≥10
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years-worth of annual parameters so that the air and snow trends could be used to generate site-
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specific baseline conditions to the year 2010. Missing data were tolerated as long as the six
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parameters could be obtained.
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Historic annual mean air temperatures (AIR) for the EPA basins were obtained from Westmap
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(http://www.cefa.dri.edu/Westmap/). Historical SWE measures for each basin in early April were
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compiled from Natural Resources Conservation Service SNOTEL data
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(http://www.nrcs.usda.gov/) and California Department of Water Resources Snow Course data
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(http://cdec.water.ca.gov/ ).
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For incomplete series, SWE’s measured over the available years were normalized to find the in-
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year basin anomaly, then re-computed as a basin-wide measure (Clark et al., 2001; Mote et al.,
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2005). Normalized SWE is i , y  SWEi , y  SWE i  SWE ,i . A missing normalized value at site i in
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year w was defined i , w 
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then SWEi , w  i , w i  i and a basin-wide average SWE for each year y was:
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SWE y 
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Trend prediction
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For historical trend estimates, available Ai , y data was related to the mean annual air temperature
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AIRj,y across 22 EPA basins j as Ai , y  ai  bi AIR j , y   i , y . Then, air temperature trends were
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determined as AIR j , y  c j  d j y , where y is year. The resulting projected parameter for any year


1
  j ,w where m is the number of basin sites. The missing value was
m j i
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 SWEi , y where m is the number of basin sites i.
m i
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is then Aˆi , y ( x )  ai  bi ci  bi di y and a mid-century estimate was obtained with y = 2050 (Table
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S1).
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Variability in Aˆi , y is assumed to remain the same as in the historic record, so up to k future
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conditions at site i were generated using: Aˆi ,k  Aˆi   i , yk with a one-to-one mapping of the (y)
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set of years to the k new values.
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Di,y was developed in an analogous manner, beginning with basin-wide, annual estimates of the
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Snow-Water Equivalent (SWE) for April 1 within basin j. To avoid negative values, basin trends
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were derived from the log-relationship: log( SWE y )  log( )   log( y ) (Table S1) and the across-
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year trends in Di,y , were defined Dˆ i , y   SWE j , y SWE j ,2010  Dˆ i ,2010 . In addition, Dˆ i ,2050  0 for
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basins with insignificant snow (South Mohave, North Mohave, South Coast, North Lahontan,
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Central Coast, San Francisco, and Oregon Basins) .
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