WHERE IS THE RAIN-ON-SNOW ZONE
IN THE WEST-CENTRAL
WASHINGTON CASCADES?
Monte Carlo Simulation
of Large Storms
in the Pacific Northwest
Matt Brunengo
Northwest Scientific Association 81st Annual Meeting
March 2009
Does this sound familiar?
After snowfall in December – accumulations of
up to several feet by Christmas at Olympia,
Cowlitz Landing, Portland, and in the
mountains; then —
8 Jan: “ … we would observe that the almost
constant mild, warm rains of the last two
weeks have completely annihilated the snow
on the various prairies, and it is fast
disappearing from the woodland.”
A storm to remember …
15 Jan: “The heavy rains which have followed
in the wake of the unprecedented depth of
snow, have almost flooded the country
between [Olympia] and the Columbia river.
The Cowlitz, we understand, has become as
wild as a cataract—completely forbidding
navigation of its usually indignant waters,
and all the intermediate streams come ‘like
Alpine torrents down’.’’
A storm to remember …
15 Jan: “The Newaukum and Skookum Chuck,
particularly, … are said to be higher than ever
before known in the recollection of the ‘oldest
inhabitant’ … A large portion of the Chickeeles
country has been inundated—all intercourse
with …. Oregon has been at an end for
weeks, and all travel between this place and
Cowlitz Landing has been suspended.”
The Columbian, Olympia, Puget Sound,
Northern O.T., 1853
Introduction

So – what is rain-on-snow?
• snowmelt during rainfall
– commonly big storms
– temps typically warmer than seasonal
• rain + melt  can be major water inputs
• rain doesn’t cause most of the melt
When does ROS occur?

Synoptic scale
• southwesterly frontal–cyclonic storms
– warm air holds more water

Seasonal storm occurrence
• most in fall & winter: rain > melt
• some in spring: melt > rain

Seasonal & interannual climate
• circulation patterns, MJO, ENSO, PDO
Where is ROS most common?

Timing
• warm rain + snowpack
• requires storm arrival in aut–win–spr

Western North America
• intercepts winter storms
• snowpacks on mountains, at least

Most other regions: ROS is uncommon
When / where?

McCabe et al., Bull. AMS, March 2007
•
•
•
•
•
spatial & temporal characteristics of ROS
data from 11 western states
WY 1949 – 2003
4318 NWS sites, detailed analysis of 477
“ROS event”  a day on which precip fell
and snow depth decreased ( SM)
When / where?

McCabe et al., Bull. AMS, March 2007
• spatial:
– ROS occurs almost everywhere in the West
– more common in the PNW
• temporal:
– most common Oct – May (Nov – Mar in PNW)
– possible in any months (esp at higher elevs)
When / where?

McCabe et al., Bull. AMS, March 2007
• correlations of ROS events with
– number of rainy days
– snow on the ground (temperature, elev)
• interannual variability: apparent trends –
– ENSO: more common in north during La Nina
– recent years: declining frequency at lower elevs
How does ROS work?

Sequence: shift from
• T < 0C to build a snowpack
• T > 0C with warm air and rain

Energy supplied for aut–win melt
• long-wave >> short-wave radiation
• some heat supplied by rain
• minor ground heat
How does ROS work?

Energy for melt, cont’d
• sensible and latent heat – major
– latent heat of condensation  600 cal/g
– latent heat of fusion  80 cal/g
• both depend on turbulent transfer – wind

ROS melt function
• snowmelt = f [ temp, precip, wind ]
simplified
Major ROS events in the PNW







Wellington disaster 1910 ?
Vanport flood, Memorial Day 1948
Christmas 1964 
January 1965
1975–76
1977
1980
Major ROS events in the PNW







1983
1989–91
Feb 1996 
1996–97
(November 2006 – not really ROS)
December 2007
January 2009
February 1996
February 1996
Why does ROS matter?


Important water-input process in the PNW
ROS affects people:
• floods
• landslides, avalanches
• roof loads, transportation problems, etc.

Geomorphic work (long-term)

People affect the process:
• effects of land use, climate change
Why does ROS matter?

Issue of hydrologic significance
•
•
•
•
ROS input relative to simple precipitation
magnitude?
frequency?
geography?
Background: previous work


Public & scientific recognition
Field studies
• Pacific NW, Calif, SE Alaska, Alps, …

Use of instrumental records
• weather and snow data

Physical / mathematical modeling
• accumulation, melt, infiltration
Using the record of ROS events

Limited sites and instruments
• National Weather Service
– usually temp, precip, snow depth (wind)
– typically low elevations
• Cooperative Snow Survey (NRCS)
– snow courses to SNOTEL
– depth & SWE, later temperature
– typically higher elevations


Most sites only 60+ years maximum
Record is too short and limited to get a
large & broad sample of events
Probabilistic modeling

Monte Carlo methods
– (capital of Monaco  gambling)
• use of random sampling techniques
(commonly computer simulation)
to obtain approximate solutions
for mathematical & physical problems
M–C modeling, cont’d

Use Monte Carlo simulation to
• sample frequency distributions
– precipitation, snow, temperature, etc.
 initial and hourly conditions during storms
• combine with deterministic models
– snow accumulation and / or melt
– infiltration through snowpack
M–C modeling, cont’d

for ROS simulation
• advantages
– simpler than continuous models
– sample from many large populations
– simulate long “records” of realizations
• limitations
– still tied to the actual record (distributions)
– watch for false assumptions

Running a virtual experiment
The problems

Hydrologic significance
• amount of water delivered (WAR) w/r/to precip?
–
(i.e., relative magnitude)
• frequency of occurrence?

Elevation
• where is the peak ROS zone?

Later
• effects of climate change?
• vegetation and land use: forests vs clearings?
Objectives

Create a computer-based model
• M–C simulation for probabilistic factors
– choose weather & snow conditions
• deterministic parts
– calculate snow accumulation and melt
– track percolation of rain + SM
• keep account of hourly values
• store conditions and output for analysis
 Run
the model – test hypotheses
Study region

Data to feed the model
• area: west-central Washington Cascades
– Tacoma and Seattle watersheds
– data-rich environment
• National Weather Service
– 7 Coop stations: T, P, snow depth
– 1 first-order site – Stampede Pass
Stampede Pass
• NWS airways obs’n site
• 3958 ft / 1206 m
(1065 m eff elev)
• T, P, W, snow, etc.
• hourly or better
• staffed: high-quality observations
• almost continuous since mid-1940s
• snow course & pillow downhill
(3860 ft / 1175 m)
Stampede Pass Precipitation
70
1 hr PD
1 hr PD trend
6 hr PD
6 hr PD trend
12 hr PD
12 hr PD trend
24 hr PD
24 hr PD trend
48 hr PD
48 hr PD trend
LCS PD
LCS PD trend
60
Precipitation ( cm )
50
1 hr AM
1 hr AM trend
6 hr AM
6 hr AM trend
12 hr AM
12 hr AM trend
24 hr AM
24 hr AM trend
48 hr AM
48 hr AM trend
LCS AM
LCS AM trend
LCS
48 hr
40
30
24 hr
20
12 hr
6 hr
10
1 hr
0
0.1
1.0
10.0
100.0
Recurrence ( yr )
1000.0
10000.0
Stampede Pass – Snow
250
Snow-water equivalent ( cm )
200
NWS max
Course max
Snotel max
NWS avg
Course avg
Snotel avg
NWS min
Course min
Snotel min
150
100
50
0
-30
0
30
60
90
120
150
Water year day
180
210
240
270
300

Snow measurement
•
•
•
•
20 courses
8 SNOTEL
many co-located
 10 yr record
Meadows Pass SNOTEL site, Cedar River basin

Cascades terrain
• Swath 1 (south)
• Mitchell & Montgomery,
Quaternary Research,
2005
•
Cedar River area
Modeling of storms / ROS

Extension to long times and large areas
• models + weather records  long series of realizations

Problems
• requirements for input data
• spatial extrapolations difficult

Not a ROS model, exactly

(still can’t answer some questions)
Model architecture


Parameters, random numbers, codes
Probabilistic section: calculate values
• probabilities from random numbers
• inversion  values

Deterministic section
• snow accum / melt section
• infiltration section
WORKBOOK: MONTE CARLO SIMULATION
PARASTART
METERS
RANDOM
NUMBERS
SIM
TEMP
CODES
CODES
TABLES
TEMPLATE
SITE
ELEV
# YRS
PRECOPT
WORKING
EVTs/YRs
NEXT
YR/EVT
SAVE EVENT PAGE
PARAMS + R #s + CODES 
EVENT QUANTITIES 
CALCULATIONS:
Hourly
temp
wind
precip
Snow ?
Liq water ?
± accum
± melt
net ?
precip+SM
perc 
WAR
R #s, codes,
quantities
hourly T, W, P, snow
perc  WAR
END
time
SUMMARY OF RESULTS
FINAL
?
SAVE
WORKBOOK
for all events:
EVENT INFO:
R #s, codes, quantities
REALIZATION AMOUNTS:
precip  liquid  WAR
P L WAR
R I
Table 4 .2 Summary of model parameters, sources, statistical models and associations
Parameter
Source in Record
Series
Model,
Distribution
Correlations &
Functions
Notes
“Storm” Timing
Number of model
events per water year
Event starting date
(water year: 1 Oct ? 1)
Starting time
hourly precip, all stations
LCS PD series
hourly precip, all stations
LCS PD series
truncated normal
( ? 1/yr)
normal (adjusted around
modal dates)
uniform
Event duration
hourly precip, all stations
LCS PD series
log-normal
NWS stations: 7 Coop,
1 airways;
hourly rain gauges
(heated at StpP)
hourly precip, all stations
PD series on long
continuous storms
(also: PD & AM on 1 to 48-hr periods)
1000 LCS events
(SIM codes)
exponential
(by regression)
(alternate: extreme-value
type 1)
4th-order polynomial on
cumulative precip
+ random component
Initial SWE
NRCS: 20 snow courses
8 SNOTEL sites
NWS: StpP station
all avail daily data
(?10 yr record)
mixed log-normal
with P[0]
Initial depth
NRCS: 20 snow courses
NWS: 8 stations
all avail daily data
(?109 yr record)
mixed log-normal
with P[0]
direct calculation
direct calculation from
solid & liquid phases
PD series comprise 1 69–290
events per sta
 snow depth & WE
 central temperature
 radiation melt in
mid-day
 elevation
 total precip
 temperature code
bivariate log-normal with
total precip
Precipitation
Total amount
Hourly distribution
 elevation
 event duration
derive separate elev fcns for
high- and low-precip stations
 total precip
 hourly snow accum
or melt
event SIM codes chosen
randomly
 elevation
 event date
(polynomial fcns)
 change in depth
during accum or melt
 initial SWE
 percolation rate
 SWE/depth
 density &
irreducible saturation
 percolation rate
 density &
grain diameter
 percolation rate
P[0] estimated for depth &
SWE together
 event date
 elevation (lapse of
site from StpP)
 duration
 hourly snowmelt rate
set of 120–133 StpP LCS
events with 1-h to 3-h obs 
all T params
number & duration of frontal
segments bivariate normal
with event duration
Snow
Initial density
Porosity (effective)
Permeability 
hydraulic conductivity
Shimizu empirical eqn
k (cm2)  hyd K (cm/h)
bivariate log-normal with
SWE
see notes
Temperature
Central temperature
Temperature range
StpP station: 1-h to 3-h temp;
with other sta’s 
winter/storm lapse rate
Hourly temperature
133 LCS events
100 T codes
(based on LCS events ;
123 avail)
both normal
central: std dev constant
range: params constant
frontal (T codes) +
diurnal + random
components on range
Wind
Central wind speed
Wind speed std dev
Hourly wind speed
StpP station: 1-h to 3-h wind
122 LCS events
both normal
both with constant params
(no date or elev fcns)
random component on
range
 hourly snowmelt rate
set of 120–133 StpP LCS
events with 1 -h to 3-h obs 
all W params
no good model
Model input, cont’d

Initial snow depth and SWE
• mixed distribution
– probability of no snow P [0]
– log-normal distribution of non–0 values
• trend surfaces for parameters
– functions of date and elevation
Model input, cont’d

Temperature during event
• central value & variation/range
• hourly: combines frontal, diurnal &
random elements

Wind speed
• central value & variation
• hourly: no good model – just random
Deterministic components

Accumulation and melt
• mass and energy balance methods
• melt calculations based on USACE
snowmelt equations

Infiltration: Colbeck and others
• porous-medium flow
• kinematic-wave approximations
Model architecture, cont’d

Single event mode (S–E)
• specify hourly T, W, P values
• single event  hyeto- / hydrographs
• good for calibration & validation
– test model parts
– check against snow / WAR in actual events
– sensitivity analysis
Model architecture, cont’d

Monte Carlo mode (M–C, “full Monte”)
•
•
•
•
specify parameters of fcns & distributions
model generates values
many events  long series of realizations
good for hypothesis testing
– comparison of “long-term” F-M curves
Model products

Output files for model event runs
• comparison with actual events
• could be used as input for other models
 Output
files for long series
• for frequency analysis
• tests of hypotheses
WORKBOOK: MONTE CARLO SIMULATION
PARASTART
METERS
RANDOM
NUMBERS
SIM
TEMP
CODES
CODES
TABLES
TEMPLATE
SITE
ELEV
# YRS
PRECOPT
WORKING
EVTs/YRs
NEXT
YR/EVT
SAVE EVENT PAGE
PARAMS + R #s + CODES 
EVENT QUANTITIES 
CALCULATIONS:
Hourly
temp
wind
precip
Snow ?
Liq water ?
± accum
± melt
net ∆
precip+SM
perc 
WAR
R #s, codes,
quantities
hourly T, W, P, snow
perc  WAR
END
time
SUMMARY OF RESULTS
FINAL
?
SAVE
WORKBOOK
for all events:
EVENT INFO:
R #s, codes, quantities
REALIZATION AMOUNTS:
precip  liquid  WAR
P L WAR
R I
Model operations

Program based on Excel workbooks &
spreadsheets
• total file size ~ 30 Mb


Code in Visual Basic for Applications (VBA)
Monte Carlo mode, 1000 “years”
(about 4400 events)
• run takes approx 1 hour at > 2.8 GHz
Model operations, cont’d

Keep in mind: model built for simplicity
– (originally on 1981 computer)
• based on events (not continuous)
• based on sites (not distributed)
• lots of numbers (realizations) for post-run
analysis

 some things poorly / weirdly / not at all
Results 1: S–E for DEMO 4
DEMO Event #4: SE Model Input / Output
0.5
Precip: Liquid
Precip: Solid
Data from Wetherbee, 1995
Net SWE
0.4
Rain + SM
WAR (liquid to ground)
Water ( cm )
0.3
0.2
0.1
0.0
-0.1
0
12
24
36
48
Run hour
60
72
84
Results 1: S–E for DEMO 4
DEMO Event # 4: Comparison with SE Model
0.5
Data: WAR
SE model: WAR
0.4
Net SWE
Water ( cm )
0.3
0.2
0.1
0.0
-0.1
0
12
24
36
48
Run hour
60
72
84
Results 2: basic M–C
Results 2: basic M–C
Results 2: basic M–C
Results 3: example ROS
Results 3: example ROS
Results 3: example ROS
Event 3230: Summary of Precip / Liquid R+SM / WAR Amounts
25
total precipitation
Precip / liquid R+SM / WAR (cm)
20
liq R + SM to pack
WAR to ground
15
10
5
0
1-hr
6-hr
12-hr
24-hr
Duration of Filter
48-hr
Dur
DurE
Results 4: Precip vs WAR
Results 4: Precip vs WAR
Results 5: Elevation
Table 6.1. Statistics of model realizations for elevation experiment (MC version EXE)
Parameters
500 m
Storm duration : true event value
Mean
46.22
Std dev
20.76
Skew
1.540
Temperature : nominal central value
Range
–8.92 to +20.85
Mean
3.615
Snow: event initial amount
% starting with no snow
84.13
SWE:
maximum
325.2
Mean
1.917
Std dev
9.026
Skew
15.24
Depth:
maximum
562.7
Mean
3.025
Std dev
14.45
Skew
17.09
800 m
Site elevation modeled
1100 m
1250 m
57.77
25.35
1.367
72.33
31.77
1.298
1500 m
Notes
97.31
40.73
1.032
min = 7 (500 m) –14 h (1500
m); max = 252 h at all elevs
 program limit
80.95
34.94
1.224
Results 5: Elevation
–10.78 to +18.98
1.740
–12.67 to +17.10
–0.135
–13.61 to +16.16
–1.073
–15.17 to +14.60
–2.635
38.96
23.67
25.45
22.84
445.0
13.00
23.57
5.29
1113
33.59
47.84
5.12
748.1
20.19
34.09
5.36
Precipitation : event total P
Rank # 1 (# 2)
52.93 (38.90)
76.85 (56.00)
minimum
5.655
6.748
Mean
9.416
12.152
Std dev
3.474
5.166
ExpD regression
intercept: on event / year
5.949 11.154
6.995 14.73 8
log slope
7.985
11.876
ln slope
3.468
5.158
Liquid rain + snowmelt : event total R+SM
Rank # 1 (# 2)
52.93 (46.54)
76.85 (54.25)
Mean
8.186
8.922
Std dev
4.017
6.358
Skew
1.353
1.4015
ExpD regression
intercept: on event / year
4.312 10.129
2.701 12.040
log slope
8.924
14.325
ln slope
3.876
6.221
Water available for runoff : event total WA R
Rank # 1 (# 2)
52.93 (46.48)
76.85 (54.23)
Mean
8.171
8.873
Std dev
4.023
6.363
Skew
1.342
1.400
ExpD regression
intercept: on event / year
4.293 10.116
2.648 11.994
log slope
8.931
14.336
ln slope
3.879
6.226
Volume of WAR vs P: % of events
WAR = 0
0.58
2.81
0 < WAR < P
41.27
57.23
WAR = P
49.63
20.79
1256
1577
4135
93.67
113.40
2.305
15771
336.7
654.26
6.63
48.42
64.17
3.76
1472
62.30
84.52
3.89
116.15
193.91
6.08
100.8 (73.095)
7.9915
14.8835
6.851
112.7 (81.64)
8.273
16.248
7.694
8.045 18.312
15.749
6.840
8.568 20.098
17.686
7.681
100.8 (59.64)
7.618
7.591
1.904
112.7 (61.38)
6.600
7.704
2.199
0.0853 11.395
17.348
7.534
–1.0575 10.439
17.635
7.659
–2.4805 8.558
16.9325
7.354
100.8 (59.62)
7.523
7.587
1.909
112.7 (59.76)
6.466
7.684
2.207
125 (58.76)
4.6425
7.4005
3.054
–0.0062 11.2985
17.341
7.531
–1.170 10.294
17.585
7.637
–2.609 8.278
16.701
7.253
10.01
66.68
12.01
17.52
66.73
9.85
33.81
57.86
6.46
125
(95.89)
8.701
18.5185
9.077
9.451 23.065
20.882
9.069
125
(58.76)
4.872
7.480
2.972
std dev = 3.513, skew =
0.812 for all elevs
min = 0 for SWE and depth
at all elevations;
extreme maxima  lowprobability events in lognormal distributions;
statistics derived from real izations at all dates – many
low values with fewer e xtremes  high variance &
skew
means and maxima of precip
rise with elevation
all skew = 2.30 8 except 2.24
at 1500 m (more events at
program max = 125 cm)
min R+SM = 0 at all
elevations
close to R+SM though
slightly smaller
min R+SM = 0 at all
elevations
counted as WAR = 0 if
< 0.01 cm
Results 5: Elevation
Summary of EXE Simulations: Series Means & Standard Deviations
(High–Precipitation Sites)
20
precip mean
18
precip std dev
liquid R+SM mean
Mean or std dev value ( cm )
16
liquid R+SM std dev
14
WAR mean
12
WAR std dev
10
8
6
4
2
0
0
100
200
300
400
500
600
700
800
Elevation ( m )
900
1000
1100
1200
1300
1400
1500
1600
Results 5: Elevation
Results 5: Elevation
Results 5: Elevation
Results 5: Elevation
Table 7.1. Evaluation of rain-on-snow zone using various parameters from MC – EXE experiments
Parameters
Rain-on-snow zone elevations
maximal (peak ROS)
Results 5: Elevation
lower boundary
upper boundary
Elevation range of EXE experiment runs
High-precipitation sites
200 m
1500 m
Low-precipitation sites
400 m
1250 m
Statistics: event WAR relative to event precipitation
High-precip
~400 m ?
700 – 800 m
~1000 – 1100 m ?
mean P – mean WAR > 10%
peak of mean WAR curve
WAR < lower bdy, COV > 1
Low-precip
~500 m ?
—
~1100 m ?
mean P – mean WAR > 10%
steeper curve > 600 m ?
WAR COV > 1
Exponential regression parameters: event WAR relative to event precipitation
High-precip
intercepton event / yr
~500 m ?
800 – 900 m
~1300 m
intcpt P – intcpt WAR > 10%
peak of WAR intcpt curve
WAR < WAR at lower bdy
log and ln slope
~500 – 700 m ?
~1100 – 1250 m
< ~1500 m ?
increasing WAR > P curve
peak of WAR slope curves
slope curve for WAR < P
Low-precip
intercept on event / yr
—
—
—
steeper curve > 800 m ?
log and ln slope
~500 – 700 m ?
800 – 1100 m
—
increasing WAR > P curve
peak of WAR slope curves
Volume of WAR vs P: %ROS vs rainy vs snowy events
High-precip
~500 – 600 m
700 – 800 – 900 m
1100 m
sharp increase in %ROS;
peak of WAR > P curve
WAR > P and WAR = P
WAR = P below WAR < P
curves fall below WAR = 0
Low-precip
~500 – 600 m
700 – 800 – 900 m
1100 – 1200 m
sharp increase in %ROS;
peak of WAR > P curve
WAR > P and WAR = P
WAR = P below WAR < P
curves fall below WAR = 0
Integral of magnitude x frequency:  M × F dR of event WAR relative to event precipitation
High-precip
> 500 m
800 – 1100 m
—
 WAR >  P
max difference  WAR >  P,
gentle decline of  WAR
curve for > 1250 m
up to peak of  WAR curve
Low-precip
> 400 m
1200 m
800 – 1000 m
 WAR >  P
 WAR falls below  P curve
max difference  WAR >  P,
and peak of  WAR curve
Notes
Ldbg+Pmr3+CdrL+SnqP+StpP
Ldbg+MMtD+Grnw+XcWR
see Fig. 6.2
WAR falls off P thru all elevs
examined  no WAR peak
see Fig. 6.3a
modeled magnitudes for P &
WAR depend on both intercept
& slope in combination
see Fig. 6.3b
small differences in most values
 subtle curves
see Fig. 6.4a
see Fig. 6.4b
see Fig 6.5
integral values in cm / yr × yr 
cm; evaluated range 1–1000 yr
Results 5: Elevation
Summary of Rain-on-Snow Zone Elevations
High– (H) and Low–Precipitation (L) Site Functions
1600
<?
1400
>?
1200
?
parameters using low-precipitation
elevation functions
?
?
?
Elevation ( m )
1000
800
?
?
600
>
?
?
400
?
>
parameters using high-precipitation
elevation functions
200
0
mean (H)
ExpD int (H) ExpD sl (H) %ROS (H)
∫ M × F (H)
mean (L)
Data and parameter types
ExpD sl (L)
%ROS (L)
∫ M × F (L)
EXE Model Realizations: 800 m (High-Precipitation Sites)
Frequency & Magnitude of Precipitation vs WAR Realizations, R # Seeds 1, 2, 3
80
R# 1 precip
70
R# 1 WAR
R# 2 precip
60
Precipitation, WAR ( cm )
R# 2 WAR
R# 3 precip
50
R# 3 WAR
40
30
20
10
0
0.1
1.0
10.0
100.0
Recurrence ( yr )
1000.0
10000.0
Results 6: Climate Change
Conclusions
Monte Carlo simulation is a useful tool
for studying highly variable phenomena
such as storms producing ROS
 The M–C model seems to work

• reproduces the input series
• generates reasonable snow and percolation
responses
• calculates interesting F-M distributions
Conclusions

The model isn’t perfect or universal
• some limitations and oddities
• even though simplified, still requires data

Model suggests a maximum in the
frequency & hydrologic significance of
ROS between 500 – 1200 m in the
study region, peak at ~ 800 m
Conclusions

Future efforts:
• confirm F-M relations for other stations
• confirm F-M relations for representative
elevations
• find easier ways to extrapolate the model
to other regions
• use the model to examine issues of climate
change, forests/clearing, ??
M–C modeling
• In essence:
– distribution function
established from data series
(mean, std dev, etc.)
– random number 
probability of occurrence or
exceedance
– inversion routine returns the
value corresponding to that
probability for that function
• (repeat …..)
Problem statement, cont’d

Hypothesis 1: at individual sites
• low elevation:
– frequency-magnitude for WAR  F-M for gross Pr?
• middle elevation:
– F-M for WAR > F-M for Pr?
• high elevation:
– F-M for WAR < F-M for Pr?
• null hypothesis: no or other differences
Problem statement, cont’d

Hypothesis 2: by elevation
• can we find the peak ROS zone ?
– where F-M for WAR is max > F-M for Pr?
• null hypothesis:
– no / other differences among elevations
Objectives, cont’d

Collect hydrometeorological data
• storm: precip, duration, temp, etc.
• snow: depth, SWE, etc.

Use these data to generate frequency
distributions for the model
Objectives, cont’d

Validate the model
• deterministic parts:
– against actual ROS events
• stochastic parts:
– against input parameters
– against available frequency distributions at
sites (especially Stampede Pass)
Objectives, cont’d

Run the model: virtual experiments
• test hypotheses
– hydrologic significance
– elevation zones
– (later – climate change, forest vegetation)
30
40
25
35
20
30
15
25
10
20
5
15
0
10
-5
5
-10
0
Oct
Nov
Dec
Jan
Feb
Mar
Apr
Month
May
Jun
Jul
Aug
Sep
Precipitation ( cm )
Temperature ( ° C )
Stampede Pass: Average Monthly Climate 1971–2000
Stampede Pass: Storm Start Dates (LCS, PD)
12
100%
11
90%
10
80%
9
70%
60%
7
6
50%
5
40%
4
30%
3
20%
2
10%
1
Water year date (Oct 1 = 1, 1944 – 2004)
260
250
240
230
220
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
-10
-20
-30
-40
-50
-60
-70
0%
-80
0
-90
Frequency
8
0%
Duration ( hr )
264
0
252
10%
240
5
228
20%
216
10
204
30%
192
15
180
40%
168
20
156
50%
144
25
132
60%
120
30
108
70%
96
35
84
80%
72
40
60
90%
48
45
36
100%
24
50
12
Frequency
Stampede Pass: Storm Duration (LCS, PD)
Stampede Pass: Snow Sites
1.1
1
0.9
Probability of no snow ( P [ 0 ] )
0.8
0.7
y = -2.6754E-09x4 + 1.4908E-06x3 - 1.7374E-04x2 - 9.6189E-03x + 9.4844E-01
R2 = 9.7286E-01
0.6
0.5
0.4
NWS d
course d
NWS we
course we
NWS d - we corln
Snotel we
course d - we corln
all - avg
all - wtd avg
all - wtd avg trend
0.3
0.2
0.1
0
-0.1
-30
0
30
60
90
120
150
180
Water year day
210
240
270
300
330
Stampede Pass: Snow Sites
6.0
y = -2.1330E-07x3 - 1.3812E-04x2 + 5.7684E-02x + 7.3366E-01
R2 = 9.6293E-01
5.0
ln of depth and swe ( cm )
4.0
3.0
y = -5.0715E-07x3 + 1.8589E-05x2 + 3.8289E-02x + 3.6546E-03
R2 = 8.8568E-01
2.0
1.0
0.0
y = -4.4121E-08x3 + 8.7803E-07x2 + 3.8581E-03x + 4.7201E-01
R2 = 8.4848E-02
y = -4.3889E-08x3 + 3.2772E-06x2 + 3.4907E-03x + 5.0275E-01
-1.0
R2 = 9.7443E-02
-2.0
-3.0
-30
0
NWS mean d
NWS mean we
snotel mean we
course mean d
course mean we
wtd avg - mean d
wtd avg - mean we
30
60
90
wtd avg, mean d
wtd avg, sd of d
NWS std dev d
NWS std dev we
snotel std dev we
course std dev d
course std dev we
wtd avg - s d of d
wtd avg - s d of we
120
150
180
wtd avg, mean we
wtd avg, sd of we
210
240
270
Water year day
300
330
SWE Trend Surface: 4th date x 1st elev
8.0
7.0
6.0
5.0
4.0
3.0 ln avg SWE
2.0
1.0
0.0
-1.0
-2.0
0
750
Effective elev
(m)
1500
2250
-50
50
15 0
25 0
35 0
W Y date
Stampede Pass: 30 Nov – 3 Dec 1977
14
12
Temperature (°C), wind speed (m/s)
10
8
6
temperature
4
wind speed
2
0
-24
-12
0
12
24
-2
-4
Hour
36
48
60
72
WORKBOOK: MONTE CARLO SIMULATION
PARASTART
METERS
RANDOM
NUMBERS
SIM
TEMP
CODES
CODES
TABLES
TEMPLATE
SITE
ELEV
# YRS
PRECOPT
WORKING
EVTs/YRs
NEXT
YR/EVT
SAVE EVENT PAGE
PARAMS + R #s + CODES 
EVENT QUANTITIES 
CALCULATIONS:
Hourly
temp
wind
precip
Snow ?
Liq water ?
± accum
± melt
net ?
precip+SM
perc 
WAR
R #s, codes,
quantities
hourly T, W, P, snow
perc  WAR
END
time
SUMMARY OF RESULTS
FINAL
?
SAVE
WORKBOOK
for all events:
EVENT INFO:
R #s, codes, quantities
REALIZATION AMOUNTS:
precip  liquid  WAR
P L WAR
R I
Results 1: S–E for DEMO 4
DEMO 4 vs SE Comparison: Cumulative WAR
12
10
SE model cumulative WAR ( cm )
cum comparison
1:1 line
8
6
4
2
0
0
2
4
6
DEMO 4 cumulative WAR ( cm )
8
10
12
Results 3: example ROS
Event 3230: Wind
12
Wind K
10
WindHr
avg wind speed 4.6 m/s, s.d. 2.3
m/s
Wind Speed (m/s)
8
6
4
2
0
-2
-4
0
12
24
36
48
60
Event hour
72
84
96
108
Results 5: Elevation
Results 5: Elevation