cap20 hidro.docx

8/10/2019 cap20 hidro.docx

1/145

Section 5: EnvironmentalModels


2/145


3/145

71)7

Chapter 20

STOCHASTIC EVENT FLOOD MODEL(SEFM)

M.G. Schaefer, Ph.D. P.E and B.L. Barker, P.E.

MGS Engineering Consultants, Inc., 7326 Boston Harbor Road NE, Olympia, WA 98506

ABSTRACT

The Stochastic Event Flood Model (SEFM) was developed for analysis ofextreme floods resulting from 72-hour general storms and to providemagnitude-frequency estimates for flood peak discharge, runoff volume andmximum reservoir level for use in hydrologic rislc assessments at dams. Itcan also be used to assess the variability of floods produced by design stormssuch as Probable Mximum Precipitation. The model was developed

specifcally for application in mountainous areas of the western United Stateswhere snowmelt runoff is commonly a contributor to flooding. This chapterprovides a description of the basic concepts employed in developing theComputer model and identifies the various hydrometeorological componentsthat are modeled in the Computer simulations. Results from some recentapplications of the model are also presented. The model is in the early stagesof implementation and changes are being made as more is learned about the

probabilistic characteristics of the hydrometeorological processes. It isanticipated that the model will continu to evolve as improvements are madeto the model.

20.1. OVERVIEW

The basic concept of the stochastic event-based rainfall-runoff model41is toemploy a deterministic flood computation model and to treat the

input parameters as variables instead of fxed vales. Monte Cario samplingprocedures are used to allow the hydrometeorological input parameters to varyin accordance with that observed in nature while preserving the naturaldependencies that exist between some climatic and hydrologic parameters.

Multi-thousand Computer simulations are conducted where each


4/145

20 I M .G. Schaefer, B.L . Barker

71N

simulation contains a set of input parameters that were selected based on thehistorical record and collectively preserves the dependencies betweenparameters. The simulated floods constitute elements of an annual maximaflood series that can be analyzed by standard flood- frequeney methods. Theresultant flood magnitude-frequeney estimates reflect the likelihood ofoccurrence of the various combinations of hydrometeorological factors thataffect flood magnitude. The use of the stochastic approach allows thedevelopment of separate magnitude- frequeney curves for flood peak discharge(Fig. 20.1a), flood runoff volume (Fig. 20.1b), and mximum reservoir level(Fig. 20.1c). Frequency information about mximum reservoir levels is

particularly important for use in hydrologic risk assessments because itaccounts for all the pertinent hydrologic factors - flood peak discharge, runoffvolume, hydrograph shape, initial reservoir level, and reservoir operations. Allof the flood characteristics above, and the hydrologic risk can be evaluated ona monthly, seasonal, or annual basis.

Fig. 20.1a. Magnitude-Frequency Curve for Peak Discharge.


5/145

Stochastic Event Flood Model (SEFM) / 20

7(1'

20.2. CAPABILITIES OF THE STOCHASTIC MODEL

The stochastic event-based flood model41has the capability to simlate a widerange of hydrometeorological and watershed conditions. Computer simulationsare conducted for 72-hour duration general storms based on end-of-monthhydrometeorological conditions. Runoff is computcd on a distributed basis for

polygons of land called Hydrologic Runo IT Uuils (lIRUs) Ihal have eommonmean annual

Fig. 20.1b. Magnitude-Frequency Curve for Runoff Volume.

Fig. 20.1c. Magnitude-Frequency Curve for Mximum Reservoir Level.


6/145

20 / M. G. SchaeJ' er, B .L. Barker

710

precipitation, elevation, and soil characteristics. Hydrometeorologicalparameters that vary seasonally with mean annual precipitation, elevation orsoil type, such as antecedent precipitation, antecedent snowpack, and soilmoisture, are also allowed to vary spatially within the watershed viaaccounting through the HRUs.

The SEFM model can be run in a completely stochastic mode where allhydrometeorological parameters are allowed to vary. It can be run in acompletely deterministic mode with all parameters fixed, or it can be run in amixed mode with some parameters treated as variables and other parametersfixed.

In most cases, the flood response of a given watershed and reservoir issensitive to only a few of the hydrometeorological parameters. Recognizingthis situation, the data entry interface allows the user to specify how each ofthe hydrometeorological parameters is to be treated- variable or fixed valu. This approach allows the user to provide a greaterlevel of detail in the simulation of those hydrologic processes that have thegreatest influence on the watershed/project under study.

Simulations are conducted based on end-of-month conditions for thevarious hydrometeorological input parameters. A monthly time increment waschosen for several reasons. It provides reasonable efficiency in data analysisbecause many hydrometeorological variables are reported on end-of-monthintervals. Use of a monthly time increment results in dates of storm/flood

occurrence that are on-average 7-days, and at-most 15-days, different fromthose obtained if the storm/flood date could occur on any day of the year.Thus, twelve time increments are deemed sufficient for sub-division of theclimatic year (October 1 to September 30) to depict the natural seasonalvariability in hydrometeorological inputs such as soil moisture, snowpack,reservoir level, etc.

20.2.1. Simulation Capabilities of the SEFM model

The simulation capabilities of the model include:

Variable - Month of occurrence of extreme stormVariable - Precipitation in climatic year prior to extreme stormVariable - Snowpack snow-water equivalentVariable - Temperatures for mclling of snowpackVariable - Soil moisture conditions at onset of the stormVariable - Surface infltration rate as function of soil moisturecondition Ability to - Simlate surface runoff and interflow


7/145

Stochastic Event Flood Modcl (SEFM) / 20

711

runoff Ability to - Compute runoff on a distributed basis usingconditions within each HRU Ability to - Check for frozenground conditions and set surface infltration rates accordinglyAbility to - Generate 72-hour general storms with

characteristics observed in historical storms Variable- Temporal distribution of 72-hour general storms Variable -Storm size and spatial distribution of 72-hour general stormsVariable - Storm centering within the watershed Variable -Streamflow prior to extreme storm Variable - Reservo ir levelprior to extreme storm Variable - Time lag and peak discharge of

surface runoff unit hydrographs

20.3. APPLICABILITY OF THE STOCHASTIC MODEL

The stochastic event flood model is currently configured for simulation of 72-hour general storms. There is no computational limit to the size of thewatershed to which it can be applied. However, implicit in the development ofthe model is the condition that some hydrometeorological parameters arehighly correlated spatially. For example, soil moisture accounting is conductedto determine soil moisture conditions at the onset of the extreme storm. Inconducting soil moisture accounting, multi-month periods of precipitaron and

snowpack are taken to be highly correlated throughout the watershed.Specifically, the exceedance probability of multi-month precipitation at agiven location is assumed to not vary significantly from that at other locations.Thus, the exceedance probability can be adequately representcd as one areallyaveraged valu. In a similar manner, the exceedance probability of multi-month snowpack can also be adequately represented as one areally averagedvalu.

As the watershed size increases, the requirement for high spatialcorrelalion of inulti-month precipitation and snowpack becomes more (linicul!lo salisfy. l itis considentlion suggesls that the stochastic model is applicable towatersheds up to a nominal size of about 500 mi2. For larger watersheds, thespatial variability of some hydrometeorological parameters may warrant that

site-specific modules be developed to address the site-specific spatialcharacteristics of the watershed under study.

20.4. DISTRIBUTED RAINFALL-RUNOFF MODELING

A key element in the stochastic approach is the selection of realistic initial


8/145

20 / M.G. Schaefer, B.L . Barker

712

conditions in the watershed at the onset of the extreme storm. This requiresthat a distributed approach be used in modeling the rainfall-runoff process sothat the spatial variability of soil moisture, soil moisture storagecharacteristics, soil infiltration rate, snowpack, and frozen ground conditionscan be properly accounted for in computing runoff.

20.4.1. Hydrologic Runoff Units

To accommodate the distributed approach, the watershed is divided intonumerous sub-areas. These sub-areas are comprised of irregularly shaped landareas having common mean annual precipitation, elevation, and soilinfiltration characteristics and are termed Hydrologic Runoff Units (HRUs).Runoff is computed separately for each HRU and then combined to obtain theresponse of each sub-basin.

20.4.2. Mean Annual Precipitation Zones

Mean Annual Precipitation (MAP) often vares widely across mountainouswatersheds in the arid, semi-arid, and sub-humid western US6,23. This spatialvariability requires that a watershed be sub-divided into zones of similar meanannual precipitation to facilitate the allocation of antecedent precipitation,allocation of winter snowpacks, and computation of soil moisture budgets.

Sufficient zones should be employed to adequately describe the variability ofmonthly antecedent precipitation, snowpack, and soil moisture that occurs dueto differences in the magnitude of annual precipitation. Figure 20.2 depicts anexample delineation of mean annual precipitation zones for the 55 mi2Keeehelus watershed29in Washington State.


9/145

Stoehastic Event Flood Model (SEFM) / 20

Mean Annual Precipitation (n)

Less Than 80 80 to 90 ; j 90 to 1CO

n ioo to nog Greater Than 110

Keechelus Watershed Mean

Annual Precipitation

Keechelus bake

Scale:

1 Mile

Fig. 20.2. Mean Annual Precipitation Zones for Keechelus Watershed,Washington.

20.4.3. Elevation Zones

Elevation information is needed to account fortemperatura changes that occur with elevationduring extreme storms. This temperature

information is required forsnowmelt computationsand for checking for frozen groundconditions. Selection of upper and lowerboundsfor theelevation zones should be based on ahypsometric curve for the watershed toensure proper apportioning of areas. Figure 20.3 depicts an exampledelineation of elevation zones for the Keechelus watershed.


10/145

20/ M .G. Schaefer, t.L . Bar ker

Fig. 20.3. Elevation Zones for Keechelus Watershed, Washington.

20.4.4o Soils ZonesSoil zones are used to delineate contiguous areas with similar soilcharacteristics. Each soil zone represents a unique combination of mximumand mnimum surface infiltration rate, deep percolation rate, and soil moisturestorage capacity. These vales are subsequently refined through calibration ofthe hydrologic model using observed climate and streamflow data. Figure 20.4

depicts an example delineation of soil zones for the Keechelus watershed.

Keechelus WatershedElevation Zones

Elevation (Feet):

LessT han

3000 3000 to 4000

|g 4000 to 5000 l G reater T

ha n 5000

Scale: 1:200,000 1 t/liie


11/145


20.5. HYDROMETEOROLOGICAL COMPONENTS

A number of hydrometeorological inputs are required in employing thestochastic approach. These same inputs are needed in a standard deterministicanalysis. However, in the stochastic approach, the inputs are treated asvariables and allowed to vary in a manner consistent with the historical record.Further, dependencies between hydrometeorological inputs are preserved.

The various hydrometeorological inputs are listed in Table 20.1 and arebriefly described in the following sections. Table 20.1 also identifies thedependencies that are preserved between the hydrometeorological components.Table 20.1. Independent and Dependent Hydrometeorological

Components.

Fig. 20.4. Soil Zones for Keechelus Watershed, Washington.


12/145

20 IM.G. Schaefer, B.L . Barker

20.5.1. Probabilistic Inputs for Initial Watershed Conditions

Date of Occurrence of Extreme Storms -is the end-of-month of occurrence ofthe extreme storm. It is based on the seasonality of extreme storms as depictedby the monthly distribution of the historical occurrences of extreme72-hourgeneral storms. Figure 20.5 depicts an example of the seasonality of 72-hourextreme storms on the west face of the Sierra Mountains in centralCalifornia31. Numeric storm dates are based on a system where September 1,December 1, and February 1 are 9.0, 12.0, and 14.0 respectively.

No HydrometeorologicalComponent

Dependency Vares By Zone

1 Month of Storm Occurrence Independent2 Antecedent Precipitation Dependent upon: 1 Mean Annual

PrecipitationAntecedent Temperatura Dependent upon: 1 Elevation

4 Antecedent Snowpack Dependent upon: 1 and2

Mean AnnualPrecipitation

5 October 1s Soil Moisture Independent Mean AnnualPrecipitation, Soils

6 Initial Streamflow Dependent upon: 1 and2

7 Initial Reservoir Level Dependent upon: 1 and2

8 Precipitation Magnitude- Frequency Independent

9 Precipitation TemporalCharacteristics

Independent

10 Storm Centering Independent11 Precipitation Spatial Characteristics Independent

12

Temperature During Storm Dependent upon: 1 and8

Elevation


13/145

Stochastie Event Flood Model (SEFM) / 20

717

Antecedent Precipitation - is the total precipitation from the start of theclimatic year (October lst) until the given end-of-month for locations within aspecified zone of mean annual precipitation. It is used in computing soilmoisture budgets and as an explanatory variable in correlation relationshipswith other hydrometeorological parameters. Figure 20.6 depicts antecedentprecipitation data27fitted by the three-

Fig. 20.5. Seasonality of 72-Hour Extreme Storms for West Face of SierraMountains iu Central, California.

Barnes Oregon

NON-EXCEEDANCE PROBABILITY

Flg. 20.6. Fxainple of Antecedent Precipitation for October lst to theEiul-of-Fcbruary, llames Oregon.


14/145

20 / M. G. Schaefer, B.L . Barker

parameter Gamma distribution for a zone of mean annual precipitation withinthe Crooked River watershed in Central Oregon.

Soil Moisture at Start of Climatic Year- is the soil moisture for the start of theclimatic year (October lst) for a specified HRU. It is used for computing soilmoisture budgets.

Snowpack -is the snow-water equivalent for a specified zone of mean annualprecipitation for a given end-of-month. It is correlated with antecedent

precipitation and the frequency of snow-free versus snow- on-groundconditions is preserved. Figure 20.7 depicts the spatial distribution ofsnowpack snow-water equivalent for typical end-of- February conditions27 inthe Crooked River watershed in Central Oregon.

Antecedent Temperature - is the average temperature in the two-week periodprior to the selected date of occurrence of the extreme storm. It is used forchecking for frozen ground conditions in each HRU.

Initial Streamflow -is the streamflow at the onset of the storm for the specifiedend-of-month. It may be treated as an independent variable or correlated withantecedent precipitation.

Reservoir Level- is the initial reservoir level at the onset of the storm for thespecified end-of-month. It may be treated as an independent variable orcorrelated with antecedent precipitation. Figure 20.8 depicts the variability ofend-of-month reservoir levels for Keechelus Reservoir29 for the period from1924-1998.


15/145


7IV

Snowpack Distribution

Fig. 20.7. Example Distribution of End-of-February Snowpack for Crooked RiverWatershed, Central Oregon.

Fig. 20.8. Simple Box-Plot for the Variability of End-of-lYIonth ReservolrFevels, Kcechclus Reservoir, Washington.


16/145


720

20.5.2. Probabilistic Inputs Related to the Occurrence of theExtreme Storm

Stochastic simulation of the temporal and spatial distribution of extremestorms is the most complex component of the Stochastic Event Flood Model.The following general descriptions of the stochastic storm elements provide anoverview of the stochastic storm generation process.

Precipitation Magnitude-Frequency- General storms of 72-hour duration areassembled utilizing the 24-hour basin-average 10 mi2 precipitation. Theprecipitation magnitude-frequeney curve for the 24- hour duration is based on

regional analyses12

24,30

33

34

of 24-hour precipitation annual maxima at gageswithin the watershed and in climatologically similar areas. Figure 20.9 depietsa precipitation magnitude-frequeney curve developed from regional analyses3 3conducted for Washington State for application at the Keechelus watershed.

Precipitation Temporal Characteristics - Probabilistic information about thetemporal characteristics of historical 72-hour general storms is used forassembly of hyetographs. This includes probabilistic information25,32 on:

depth-duration rclationships; clapsed time to Ihe high intensity segment of thestorm; sec|ucncing ol'hourly precipitation incremcnts; and macro stormpatterns. lixtrcmo storms seloetod lor

Fig. 20.9. Basin-Average 24-Hour 10 mi2Precipitation Magnitude- FrequencyCurve, for Keechelus Watershed


17/145


0.80

_ 0.70 c=- 0.60 ZO 0.50 t-

< 0.40

9: 0.30O^ 0.20Q.

0.10

0.00

0 6 12 18 24 30 36 42 48 54 60 66 72

TIME (Hours)

analysis are obtained from gages within the watershed and in climatologicallysimilar areas. Figures 20.10a,b depict two temporal distributions generated bySEFM for the Keechelus watershed. The temporal characteristics weredeveloped from analyses2932 of 25 storms in the Cascade Mountains ofWashington.

Precipitation Spatial Characteristics -Probabilistic analyses are conducted ofdepth-area-duration data developed from historical storms. This information isapplied in a probabilistic manner to allow for variable storm areal coverage andto describe the spatial distribution of precipitation over the watershed. Figure

20.11 depicts a family of depth-

0.80

0.70 c^ 0.60 z

24-Hour= 8.0 inches72 Hour = 10.7 inch es

nj i I I | InmFl in

Rll

11

ILD

0 6 12 18 24 30 36 42 48 54 60 66 72

TIME (Hours)


18/145


722

Area-duration curves for the 24-hour duration for Keechelus watershed29developed from analysis of storms in the Cascade Mountains of Washington.

Storm Centering - Storms are allowed to center over different parts of thewatershed and one storm center is allowed per sub-basin. The user specifiesthe spatial allocation of precipitation to the other sub-basins surrounding thesub-basin with the storm center.

20.5.3. Inputs Related to Rainfall-Runoff ModelingRainfall-runoff computations are accomplished in two stages. First, surfacerunoff is computed based on a surface infiltration rate using a decay function 10where the surface infltration rate is dependent on the magnitude of soilmoisture. Next, interflow runoff1is computed based on a deep percolation rate.

1< 0.40

: 0.30U

0.200L0.10

0.00

I ij'. 20.H)n,l>. Ilxiiiiiple I'oniponil Dlslrilmlions lor Keechelus Watershed,

Fig. 20.11. Probabilistic Depth-Area-Duration Curves for Cascade Mountain

Areas in Washington.


19/145


723

Separate rainfall-runoff computations are conducted for each HRU to reflectthe site-specific climatic and soil conditions. The runoff from each HRU isaggregated to the sub-basin level and surface and interflow unit hydrographsare used to compute the surface and interflow flood hydrographs. Inputs forrainfall-runoff modeling are described below. A schematic of this procedure isshown in Fig. 20.12. Table 20.2 lists an example of the soil characteristics for

WIINIIIII((IOII.


20/145


21/145


Rain + Snowmelt

Surface Runoff

(does not contribute to flood)

1U

lis

Keechelus watershed in Washington. The geographic layout of these soils isshown in Fig. 20.4.

Mximum Surface Infiltration Rate -is the mximum rate at which the soil canaccept water at the soil surface for a specified soils zone. This occurs when thesoil is at the wilting point having been desiccated by evapotranspiration.

Soil Moisture

Storage (Rool Zone)

Gravitational or

Intermed ateVadose Zone

Fig. 20.12. Schematic of Soil Moisture and Runoff Processes.

Mnimum Surface Infiltration Rate- is the limiting rate at which the soil canaccept water at the soil surface for a specified soils zone. This occurs when the

soil is fully wetted and soil moisture is at field capacity.Deep Percolation Rate - is the limiting rate that a soil layer, hardpan withinthe soil column, or underlying bedrock can accept water that has infiltrated thesurface of the soil for a specified soils zone. Water that passes through thislimiting soil layer, hardpan, or bedrock contributes to groundwater and doesnot return to the stream during the time interval for modeling of the extremeflood.

Soil Moisture Storage Capacity -is the moisture holding capacity of the soilcolumn to the depth that can be affected by evapotranspiration.

Evapotranspiration - is the average monthly potential evapotranspiration

amount for a specified zone of mean annual prccipitation.


22/145

20 /M.G. Schaefer, B.L . Barker

724

Rank Ali Events n Descending Order of Magnitude andDevelop Magnitude-Frequency CurvesI

Fig. 20.13. Flow Chart for Stochastic Simulation Procedure.

20.6.1. Comparison with Traditional Flood FrequencyAnalysis

A comparison with traditional flood-frequency analysis can be used to obUiin ;iperspcctive on the approach used with the stochastic model. The primary ibcus iuIraclilional lloocl Ircquency analysis is flood peak (INchiirgi. Iu eonihicting mal -si I c Ircquency analysis for peak

Temperatures during Extreme Storms used for Snowmelt Computations- is the temporal sequence of temperatures during the occurrence of the 72-hour general storm for a specified elevation zone that is used for computingsnowmelt runoff

Surface Runoff Unit Hydrographs- are used to convert the computed surfacerunoff volume from each sub-basin into a flood hydrograph. Surface runoffunit hydrographs can have variable lag time and peak discharge to account forthe variability observed in nature.

Interflow Runoff Unit Hydrographs - are used to convert the computedinterflow runoff volume from each sub-basin into a flood hydrograph.Interflow runoff unit hydrographs have fixed lag time and peak dischargebased on calibration to observed floods.

Reservoir Routing and Dam Operations - reservoir operations are simulatedconsistent with standard operating procedures for the project under study. TheComputer program is currently set up for the HEC-1 model36, which uses afixed reservoir elevation-discharge rating curve. Project specific modules canbe developed to simlate more complicated operational procedures.

20.6. SIMULATION PROCEDUREOne of the key features of the stochastic modcl is the use of Monte Cariosimulationmethods (Jain15,Salas el al26) forselocling 1 he

Table 20.2. Soil Characteristics, Keechelus Watershed.Soil

ZoneSoil Moisture

Storage Capacity(in)

Surface Infiltration Rates (in/hr) Deep PercolationRate (in/hr)Mximum Mnimum

l 4.00 2.00 0.60 0.06

2 6.00 2.00 0.60 0.063 6.00 2.00 0.60 0.084 4.00 10.00 closed sub-

basin10.00 closed sub-

basin

0.06

5(reservoir)

0.00 0.0 0.00 0.00


23/145

Stoehastic Event Flood Model (SEFM) / 20

72 I

magnitude and combination of hydrometeorological input parameters forcomputation of floods. While the individual elements of the model can becomplex, the basic concepts used in the simulation are straightforward. Aflowchart for the stochastic simulation procedure is depicted in Fig. 20.13 andthe basic concepts of the simulation procedure are described below.


24/145


726

discharge, the basics steps are to: collect an annual maxima series for theperiod of record; view the magnitude-frequency characteristics of the data byconstructing a probability-plot using a standard plotting position formula toestimate annual exceedance probabilities; and fit a probability distribution tothe annual maxima data in attempting to capture the statistical informationcontained in the dataset. Flood peak discharge magnitude-frequency estimatesare then made using the distribution parameters for the fitted probabilitymodel.

If an extremely long period of flood record were available (multi- thousandyears of flood peak discharge annual maxima in a stationary environment),then a plotting position formula and probability-plot would be sufficient forcapturing the frequency characteristics for all but the rarest flood events withinthe dataset.

The Computer simulation of multi-thousand years of flood annual maximaprovides a flood record analogous to the latter case described above. With thatin mind, the basic construct for the stochastic Computer simulation procedurecan be described as follows.

An extremely long record of 24-hour, 10 mi2precipitation annual maximais generated using Monte Cario sampling procedures (assuming stationaryclimate). A 72-hour general storm is developed for each of the 24-hourprecipitation annual maxima based on the probabilistic characteristics of the

temporal and spatial components of historical extreme storms. A storm date,end-of-month is selected for occurrence of the storm. Hydrometeorologicalparameters are then selected to accompany each storm based on the historicalrecord in a manner that preserves the seasonal characteristics and dependenciesbetween parameters. The general storms and all other hydrometeorologicalparameters associated with the storm events are then used to generate anannual maxima series of floods using rainfall-runoff modeling. Characteristicsof the simulated floods such as peak discharge, runoff volume and mximumreservoir level are ranked in order of magnitude and a non-parametric plottingposition formula and probability-plots are used to describe the magnitude-frequency relationships.

20.6.2. Details of the Simulation Procedure

One approach to conducting simulations would be to Computer genrate ahuge number of storms and floods, say I07evenls and compute the frequencycurves based on lliose lloods. Ilowcver, Ihe compululional effort to conductthis large number of flood simulations makes this approach impractical. With


25/145


727

current Pentium level (300 mhz) computational and storage power of personalcomputers, 25,000 flood simulations can be conducted in about 12 hours usingabout 3 gigabytes of storage.

If flood events more common than an Annual Exceedance Probability(AEP) of about 1:2500 are of interest, then 25,000 simulations of annualmaxima are adequate to develop the magnitude- frequeney curves. In manyapplications, there is a desire to estimate magnitude-frequeney curves for floodevents more rare than an AEP of 1:2500. In these cases, an altemative MonteCario sampling procedure is needed that allows development of the magnitude-frequeney curves for extremely rare floods while recognizing the practicallimits posed by computational power/storage constraints. This can beaccomplished using a piecewise approach (Barker et al3) that requires muchless computational effort. Magnitude-frequeney curves can be constructed bycomputing several simulation sets. Each simulation set is used to define adifferent portion of the frequeney curve - for example one to two log-eyeles ofannual exceedance probability (Fig. 20.14).

This approach can be best explained with an example. Consider the casewhere flood events with an AEP of 1:1,000 to 1:10,000 are of interest. Sincethe largest flood events in an annual maxima series (either historical orComputer generated) exhibit the greatest variability, a record length about 10times greater than the target recurrence interval (1/AEP) is appropriate to

reduce uncertainties due to sampling variability for the upper end of thisfrequeney range. Thus, a record length of 100,000 annual maxima would beused to develop probability - plots for making magnitude-frequeney estimatesin the target range of 1:1,000 to 1:10,000 AEP.

In a standard Monte Cario approach, annual maxima storm magnitudeswould first be sampled at random and then the hydrometeorological parameterswould be selected to accompany the storm. This approach would require thatthe fiill 100,000 sample set be generated. In the piecewise approach, it isrecognized that the smallest storms in the sample set are not going to generatethe largest floods. Sincc wc are only interested in the upper-most log-cycle(s)of extreme llood characteristics, we would simlate floods from the collection

of Ilie largor storms from a rcduccd sample set. For this example, we woulddevelop 1 he magniludc-frequcncy estimates based on the 25,000


26/145


72N

largest storms/floods from a record length of 100,000. This would provide asufficient number of floods to adequately define the magnitude-frequencyrelationship in the target range of 10"3to 10"4AEP.

Implicit in the piecewise approach is that that few or none of the75,0 smallest storms (storms more common than a 4-year event for thisexample) will produce floods sufficiently rare to reside within the 10~3to 104target zone for flood characteristics. Confirmation of this behavior can be madeby examination of the computer-generated floods and determining the range ofstorm magnitudes that are producing floods within the target zone.

To simplify generation of the 25,000 largest storms, a plotting positionformula is used to create a representative sample of extreme storms. The annualexceedance probabilities of the 24-hour 10 mi2 precipitation for assembly of72-hour general storms within each simulation set are defined by theGringorten7plotting position formula:

=1-0.44rexN

+ 0./2

Fig. 20.14. Example of Piecewise Assembly of Magnitude-Frequency Curve.


27/145


wherePexis the annual exceedance probability,Nis the number of years for therecord length being simulated, n is the actual number of simulations beingconducted (n out of N years), and i is the rank of the precipitation annualmaxima being simulated (ranges from 1 to n).

The resulting nfloods from each simulation set are ranked in descendingorder of magnitude and the Gringorten plotting equation is used to compute theannual exceedance probabilities for flood peak discharge, flood runoff volume,and mximum reservoir level.

20.7. CURRENT CONFIGURATION OF STOCHASTICMODEL

The stochastic inputs generation component of the flood model is currentlyconfigured to function with HEC-136, a lumped, event rainfall- runoff model. Inthis configuration, the lumped rainfall-runoff inputs common to HEC-1modeling are replaced by distributed rainfall-runoff inputs that are computedwithin the SEFM. However, the SEFM is not limited to the HEC family ofmodels. The stochastic inputs generation component of the flood model wasintentionally constructed separately to allow it to be used with other rainfall-runoff models. Fully distributed event rainfall-runoff models could also be

used to simlate the flood response of the watershed. This type of approach hasrecently been done in combining the SEFM stochastic inputs generator withWATFLOOD5 6,17a distributed rainfall-runoff model that can be operated inboth event and continuous modes. The SEFM approach could also beconfigured to operate in a pseudo-continuous mode using a resamplingapproach for the hydrometeorological components.

The remainder of discussion here will relate to the application of the SEFMin conjunction with the HEC-1 rainfall-runoff model.

20.8. SOFTWARE COMPONENTS

The general storm stochastic event flood model (SEFM) is comprised of sixsoftware components. These components include: data entry; input data pre-processor; HEC-1 template file; stochastic inputs generator; IlliC-l rainfall-runoff flood computation model; and an output data


28/145


Fig. 20.15. Flow Chart for Operation of the Computer Software for StochasticSimulation.

post-processor. Figure 20.15 depicts the sequence of actions required forcondueting the Computer simulations using the software components. Each ofthese components is briefly described in the following sections.


29/145


7U

20.8.1. Data Entry Software Component

Input parameters are entered via a Microsoft Excel 9 720 workbook. Atab/worksheet is allocated for each of the hydrometeorological components.Each tab contains a data entry screen and on-screen prompts to assist in entryof the required input parameters.

20.8.2. Input Data Pre-Processor Software Component

The input data pre-processor performs a variety of tasks, and operates within

Microsoft Excel 9/ using Microsoft Visual Basic for Applications10

.The firsttask conducted through the spreadsheets is to perform validity checks of thevales of the input parameters for each of the hydrometeorological variables.Vales that are found to be out of bounds are identified/flagged on thespreadsheet. The second task is to conduct preliminary Monte Cariosimulations for each of the hydrometeorological variables to allow examinationof the generated vales and to compute sample statistics of the generatedvales. This allows a basic confirmation of the validity of the generated valesand allows comparisons to be made with historical data. Lastly, the pre-processor is used to cali the stochastic inputs generator to conduct Monte Carioinput parameter generation for all hydrometeorological variables for use in theComputer flood simulations and to create the input files for the HEC-1

hydrologic model.

20.8.3. Stochastic Inputs Generator Software Component

The stochastic inputs generator is comprised of a series of Fortran modulescompiled as dynamic link libraries (dlls). These modules are used for MonteCario generation of the inputs that are required by HEC- I, the rainfall-runoffflood computation model. Interflow runoff, surface runoff, and snowmeltrunoff are also computed by these modules and Ihe runoff vales are passed toHEC-1 for transformation into hydrographs.

These modules are considered generic sinee they are general in nature anddo not contain fixed parameter vales that are applicable to a spccificwatershed(s). Where site-specific watershed conditions occur that are differentfrom the approach taken in the generic modules, a site- specific module can besubstituted for the generic module and the model operated as a site-specificapplication. This situation is mst likely to occur for larger watersheds (greaterthan nominal 500 mi2) where the spatial considerations of antecedentprecipitation, snowpack, and antecedent soil moisture may warrant a site-


30/145


specific solution.The stochastic inputs generator reads a HEC-1 input file, called a HEC-1

template file, that contains the Monte Cario input in 80 column card format(cards). The output from the routine is an ASCII text HEC-1 input file withthe Monte Cario cards replaced by HEC-1 cards that reflect the Monte Cariosimulated surface runoff, interflow, initial reservoir elevation, and initialstreamflow. A separate input file is created for each Monte Cario simulation offlood annual maxima. Thus, if 25,000 simulations (25,000 annual maxima) areperformed to define a frequency curve, then 25,000 HEC-1 input files will becreated by the routine.

20.8.3.1. Random Number Generation and Monte Cario Simulation

Selection of hydrometeorological inputs using Monte Cario samplingprocedures requires the use of a random number generator. The algorithm usedin the SEFM was originally developed by Lewis et al IN and programmed byHosking14. It is a multiplicative congruential generator with base 23l-l andmultiplier of 75. This algorithm has been thoroughly tested and is the generatorcommonly used at the IBM Research Divisin14,18. Each hydrometeorologicalinput is allocated a stream of random numbers on the interval [0-1]. Eachstream of random numbers is non-overlapping with streams for other

hydrometeorological components.Standard Monte Cario sampling procedures are used for selection of valesfrom probability distributions. The inverse transformation mcthod (Jain1, Salaset al26) is used extensively for generation of random variates.

20.8.3.2. Messages and Error Handl ing

Infonnational messages and error messages from the stochastic inputsgenerator are stored in an ASCII text file. This file is purged at the start ofeach simulation so it only contains messages from the most recent Computerrun. If a simulation fails or has problems, this file can be accessed to helpdetermine the cause of the problem.

20.8.4. HEC-1 Template FileThe SEFM program, specifically the stochastic inputs generator, reads theHEC-1 template file and replaces the Monte Cario cards with standard FIEC-1cards that simlate the desired hydrologic component. The HEC-1 templatefile is similar to a standard HEC-1 input file except that it is much shorter,because the precipitation and runoff calculations are being performed by the


31/145


SEFM program. Table 20.3 lists the HEC-1 Monte Cario cards that are readand replaced during the simulation.

Pable 20.3. Monte Cario Cards for HEC-1 Template File.

Incorporation of the surface runoff and interflow runoff components isaccomplished by replacing the HEC-1 basin runoff data cards by two sets ofHEC-1 cards. The surface runoff (precipitation minus infiltrated moisture) isrepresented by HEC-1 PIincremental precipitation cards. These are followedby UI unit graph cards for the surface runoff unit hydrograph(s). Since theSEFM program is performing the soil moisture und surface runoffcalculations, a uniform loss rate card LUis used with (he loss rate sel to /ero.The interflow for each sub-basin is entered into IIIC-I on Ql clirccl input

hydrograph cards since the interflow unit liydrographs uro ippliod by theSIIM program. The surface and inlorllow runolT liyili'onnipliN mv liteneombined.

MI20.8.5. HEC-1 Rainfall-Runoff Flood Computation Model

Software ComponentHEC-1 is executed in a batch mode utilizing only a few computational featuresof the HEC-1 model. HEC-1 is used primarily to: transform surface runoff intohydrographs for each sub-basin; conduct channel routing of sub-basinhydrographs through the stream network; and to route the inflow flood throughthe reservoir.

A batch file is created in the HEC-1 directory with each stochasticsimulation. The batch file executes the HEC-1 program reading each input filefrom the specified subdirectory. For each simulation, an HEC- 1 output punchfile is created that contains computed hydrographs at specific points of interestsuch as the inlet to the reservoir and outflow from the reservoir.

Monte Cario

CardCorresponding

HEC-1 Card

Purpose

MCIT IT Simulation Duration and Time-StepMCID ID Run Title for Project/StudyMCBA BA Surface And Interflow ComponentsMCBF BF

Initial Streamfiow and Base FlowRecessionMCRS RS Initial Reservoir Elevation


32/145


20.8.6. Output Data Post-Processor Software Component

The output data post-processor is used as a repository for the output from theflood simulations and is contained within a Microsoft Excel20 workbook.Vales of the hydrometeorological inputs are passed to the post-processor toallow examination of the inputs that produced a given output. Visual Basic20routines are used to read the hydrographs saved in the punch files, and extractthe maxima peak flow rate, runoff volume, and reservoir elevation, and toconstruct magnitude-frequency curves in Microsoft Excel20.Standard featuresof the Excel spreadsheet allow the output to be sorted and analyzed in anymanner desired to examine the hydrologic conditions that produce a givenmagnitude flood.

20.9. APPLICATIONS OF SEFM

The following sections present results from applications of the SEFM onmountain watersheds in the western US. Results from the Keecheluswatershed29 and AR Bowman watershed27,28 are presented. The Keecheluswatershed is a forested mountain watershed with a drainage area of 55 mi2located on the east slopes of the Cascade Mountains in Washington. Theclimate is humid with mean annual precipitation averaging 94 inches (Fig.20.2). The AR Bowman watershed is located in central Oregon on the Crooked

River (Fig. 20.7) and luis a drainage area of 2,300 mi 2. The climate is semi-arid to sub-luimid willi mean annual precipitation ranging from a low of 8inches, to a high of 32 inches on the mountain ridge tops. Land use consists ofirrigated agriculture in the valley bottoms, open rangeland over much of thewatershed, and forested hillsides and ridge tops where there is adequate waterand soils to support timber.

20.9.1. Seasonalities of Storms and Floods

Irrigation reservoirs in mountainous watersheds typically fill in the spring ofthe year from snowmelt runoff and reach their lowest levels in the fall of theyear following the irrigation season. This pattern of reservoir operation hasimplications for the seasonal frequencies of mximum reservoir levelsproduced by floods. In arid, semi-arid and sub-humid climates in the westernUS, the flood season may be separate from, or partially overlap, the seasonwhen the reservoir is at normal high levels. The SEFM is particularly useful inconducting flood analyses that allow partitioning out the seasonalities ofextreme storms, extreme flood peaks and runoff volumes, and mximumreservoir levels.

Figures 20.16 a,b,c depict seasonality histograms for the seasonal


33/145


frequencies of extreme storms, flood peaks, and mximum reservoir levels forAR Bowman Dam and reservoir on the Crooked River in central Oregon.

It is seen in Fig. 20.16a that there are two storm seasons, a winter andspring-summer season. Snowpack is at a mximum in the winter and earlyspring and conventional flood analyses had considered the winter period tooffer the greatest potential for rain-on-snow flood events. However, largefloods (Fig. 20.16b) were found to occur more frequently in the late-springwhen soils are fully wetted at the end of the spring snowmelt season, andsnowpacks lingered at the high elevations in the watershed. Likewise,mximum reservoir levels due to floods were found to occur more frequently

in the late-spring (Fig. 20.16c) when the reservoir was full or nearly full fromsnowmelt runoff and the frequeney of extreme floods was highest.

This type of analysis can be useful for evaluating how the seasonality offloods interaets with current reservoir operations to produce mximumreservoir levels. This type of information provides a logical starting point foroptimizing reservoir operations to meet multi- purposc goals and to reducehydrologic risk.


34/145


736

0.50

0.40

0.30

0.20

0.10

0.

00

SEASONALITY OF EXTREME STORMS0.50

AR BOWMAN WATERSHED0.40

>-O5 0 30

Z)20-20 tL

OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP END OF MONTH

OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP END OF MONTH

Fig. 20.16a,b,c. Seasonality of Extreme Storms, Flood Peak Dischargc, andMximum Reservoir Level for AR Bowman Watershed,Crooked River, Oregon.

SEASONALITY OF EXTREME FLOOD PEAK DISCHARGE-AR BOWMAN WATERSHED

i

:,w ,WM, 1 i


35/145

Stochastic Evcnt Flood Model (SEFM) / 20

7V7

AR B o w m a n D a m Reservoir Inflow

Vs. Predpitation

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 Watershed Average

Predpitation (in)

90.000

80.0

'S .70,000

60,000 1

50,000 !=

40,000

30,000

20,000 10,000

o

Fig. 20.17. Scatterplot of Flood Peak Discharge Versus Storm Magnitude for ARBowman Watershed, in Central Oregon.

25,0 annual maxima for the AR Bowman watershed. The AR Bowmanwatershed resides in a semi-arid to sub-humid watershed and ex h i bits thecharacteristically high variability between flood magnitude atld storm magnitudecommonly seen in these dry climatic settings.

20.9.2. Relationship Between Storm Magnitude and FloodMagnitudeAntecedent conditions can be an important factor in determining the floodresponse of a watershed. This is particularly true for watersheds in arid orsemi-arid climates where large soil moisture dficits are common. In theseclimatic settings, storm amounts are not large compared to those in humidclimates. Many storms fall on dry watersheds and limited runoff is produced.Thus, there is high variability in the relationship between storm magnitude andflood magnitude. Figure 20.17 shows the results of an SEFM simulation of


36/145


738

100000

RUNOFF VOLUME (acre-feet)

20.9.3. Relationship Between Flood Peak Discharge and FloodRunoff Volume

Historical long duration extreme storms in the western US exhibit highvariability in the temporal and spatial distributions of precipitation over largeareas. This high variability in storm characteristics results in high variability inthe shapes of flood hydrographs and in the relationship between flood peakdischarge and flood runoff volume. Figure 20.18 displays the results from asimulation of 25,000 annual maxima for the AR Bowman in central, Oregon.This was a site-specific study that included simulations for both 72-hourstorms as well as sequences of long-duration storms over a 15-day period. Thevariability of simulated flood peaks and runoff volumes is consistent with that

seen in historical streamflow data.

LUO

DC] = 0for i=\,...,n.The MAR(p)

model can be expressed as

0(B)Y(= et (21.70)

where Q(B)is a square matrix of polynomials in B which is defmed as

0(B ) = I - xB{- 02B^- - pBP(21.71)

in whichIis an (nXn)identity matrix; &/,j =1,..., p , are nxnparameter matrices;BJis a scalar difference operator such thatBJZt= Z and et

is an (xl) vector of normally distributed noise terms with mean 0 and

variance - covariance matrix G.The noises e, are independent in time but aredependent in space. Such spatially correlated noise can be modeled by


65/145

Stochastic Analysis, Modeling and Simulation (SAMS 2000) / 21

767

e,=Be i (21.72)

where e/is a (xl) vector of standardized normal variables independent in timeand in space andBis an (nXn)parameter matrix.

It may be shown that the variance-covariance matrices of the MAR(p)model are given by

K = 't


66/145

21 / ./. I ). Salas, W.L. Lae, D.K. F revert

7(H

matrix. This solution, however, requires that Gbe a positive defmite matrix.

21.3.5. Multivariate CARMA(p,q) Model

When modeling multivariate hydrologic processes based on the fullmultivariate ARMA model, often problems arise in parameter estimation. TheContemporaneous Autoregressive Moving Average (CARMA) model wassuggested as a simpler alternative to the full multivariate ARMA model (Salas,et al., 1980). In the CARMA model, both autoregressive and moving averageparameter matrices are assumed to be diagonal such that a multivariate modelcan be decoupled into component univariate models. Thus, the modelparameters OandO do not need to be estimated jointly, but, instead, they can be estimatedindependently for each single site by using the standard univariate ARMAmodel estimation procedures. This allows that the best univariate ARMAmodel can be identified for each single station. The CARMA(/?,/) model canbe expressed as

z,= t +e, - (21-78)j=i J=i

whereZtis a multi-dimensional vector of the normalized series with mean zero,e( is a multi-dimensional vector of normal noises (residuals) with mean zero

and variance-covariance matrix G, and 0 ,and 0 . arerespectively the autoregressive and the moving average diagonal parametermatrices. Consequently, Eq. (21.78) can be decoupled into the modelcomponents as

2f;>= y/ + ;> - (2i.79)y=i ' y=i

Thus, Eq. (21.79) is the expression of a univariate ARMA(/j,^) model for site i

such that the parameters


67/145


7ti'J

)1.It may be shown that the variance covariance matrix Gof the

cross-correlated noises etis equal to

G = E(f) = BBr (21.81)

Thus, a CARMA model implies that the cross-correlations between sites arecarried through the residuals.

Two methods can be used for estimating the Gmatrix:

(1) . The MLE estmate of Gmay be obtained by

& =-- ^ (21-82)

where, t= 1,...,Nare the residuals calculated from model (3.53).

(2) . The MOM estmate of G may be obtained as a function of theparameters 0and 0 and the cross-covariancesMu-ofZ,, i.e.,

G=f(0, 0, Mk) (21.83)

where k =0,..., max(p, q) -1. Further details on these estimation procedures

may be found in Salas et al. (2000.)

21.3.6. Multivariate Periodic Autoregressive, MPAR(p),Model

The MPAR(p) model can be expressed as

X(B)YVX= ^ (21.84)

where (/i )is a square diagonal matrix of periodic polynomials in B whichis dofmod lis


68/145

21 / J.D . Salas, W.L . Lae, D.K. F revert

770

X(B) = I -


69/145


771

matrix of the residuals Gx.

After estimating p and Gt as indicated above,B%can

be estimated from


70/145

21 / .1.1). Salas, W.L . Lae, D.K. F revert

772

case at hand) of the corresponding generated flow at a key station (or subkeystation) or, in temporal disaggregation, to ensure that the generated seasonalvales add exactly to the generated annual valu, three methods of adjustmentbased on Lae and Frevert (1990) are provided in SAMS. These methods willbe described in detail in the following sections.

21.3.7.1. Spacial Disaggregation

Valencia and Schaake Model

The disaggregation model can be expressed as (Valencia and Schaake, 1973)

Yt=AXt+Bet (21.90)

in which Yt is an (fx1) column vector with elements , Y' i= 1, ... , f;Xt is an(Tzxl) column vector with elements , Xt i = 1, ... , h where h and/areappropriate matrix dimensions. For example, in the key station to substationdisaggregation / and h represent the number of key and substations,respectively. et is an (fx 1) vector of normally distributed noise terms withmean 0 and the identity matrix as its variance - covariance matrix. The noises efare independent in both time and space. A and B are (fxh)and (hxh)parameter

matrices. The number of key stations/in the above equations can be more thanone so the above model can be used to disaggregate annual flows at severalkey stations to their corresponding flows at substations in a multivariate formwhich would be able to preserve the inter (cross) correlations among thestations.

The model parameter matricesAandBcan be estimated by using MOM as(Valencia and Schaake, 1973):


71/145


77

A= MYX)M^(X) (21.91)

BBT=M0(Y)-M0(YX)M^(X)M0(XY) (21.92)

and

M,(X) = F[XtXl],

MY)= 7^,7,

Mk(YX) = E[YtXlk], and = E[Xfi_k].

Equations (21.92) and (21.93) can be used to obtain estimates of Aand Bbysubstituting the population moments M0(X), M0(Y), M0(XYJ, andMg(YX)by their

corresponding sample estimates.

Mejia and Rousselle Model

This model can be expressed as

Yt=AXt+Bef+ CYt_x (21.93)

in which YnXp ep A, and Bare defmed in the same way as for the Valencia

and Schaake model and C is an additional (hxh)parameter matrix. As for theValencia and Schaake model, the number of key stations / in the aboveequations can be more than one so the above model can be used todisaggregate annual flows at several key stations to their corresponding flowsat substations.

The model parameter matricesA, B, and C can be estimated by usingMOM as:

A= {[M0(YX)-MX(Y)MQ'(Y) Mf(XY)]

[M0(XY) - Mx(XY) MQX(Y) Mf(XY)]-'}

C = [MX(Y)-AMx(XY)]M^(Y) (21.95)

RBT= M0(Y) - AM0(XY) - CM'((Y) (21.96)


72/145


73/145


y'=1 CX^2/'= 1

where

N

r=(\/N)%drt , (21.102a)/=i

I Pr, = ^, (21.102b)9tandNis the number of observations, nis the number of substations (orsubsequent stations), qtis the t-th observed valu at a key station (or

substation), c/'is the t-th observed valu at substationj(or subsequent

station), qtis the generated valu at the key station (or substation), cj'1

is the generated valu at substation i(or subsequent station), q*is the

adjusted generated valu at substation i(or subsequent station), and L(/ )and (/ )are respectively the estimated mean and standard deviation

ciof for site i.

21.3.7.2. Temporal Disaggregation Lane's Condensed Model

The model eun ho expressed as


74/145

21/ J.D . Salas, W.L . Lae, D.K. F revert

771

(21.104)

(21.105)

(21.106)

K,r = + ,,-1 (21-103)

in which Yvx is a (nX1) column vector with elements Y^'], i1, ...,n; Xvis

an (x 1) column vector with elements , / = 1, ... , n; evis a (nx 1) vector of

normally distributed noise terms with mean 0 and the identity matrix as its

variance-covariance matrix. The noises evare

independent in time and space and nis the number of sites.The model parameter matricesA, B, and C can be estimated

by using MOM as (Lae and Frevert, 1990):4 = {m*


75/145


777

&V,Tt=1

\(j2

/= i

Adjustment for temporal disaggregationThree approaches are also available for the adjustment of temporaldisaggregated data. They are: approach 1:

approach 2:

(21.110)

andapproach 3:

(21.111)

where cois the number of seasons,Qvis the generated annual

valu,

qvX is the generated seasonal

valu, q*ris the adjusted generated seasonal valu, flris the estimated mean of

qv%for season t, and Tis the estimated standard deviation of qvXfor season t.

21.3.8. Modeling SchemesIn modeling complex hydrologic systems involving many sites, although inprincipie it may be possible to model the hydrologic time series at all sitesjointly, it is generally more convenient to model them by combining a numberof univariate and multivariate models and concepts ofaggrcgation anddisaggregation techniques. Obviously for a complex syslcm lliere is not ;iuniquc way of combining various models.


76/145


77/145

21 / J.D. Salas, W.L . Lae, D.K. Frevert

77N

A particular combination of models and techniques is sometimes called amodeling scheme. SAMS uses two modeling schemes. We will Ilstrate themassuming that we want to model a river system involving key stations (thefurthest downstream site), substations (stations draining into a key station), andsubsequent stations (stations draining into a substation). In Scheme 1, theannual flows of all key stations are aggregated into an artificial index station.Such aggregated annual flows are modeled by using a stationary univariatemodel such as an ARMA(1,1) model. Then a spatial multivariatedisaggregation model is applied so that annual flows that are generated at theartificial station can be spatially disaggregated into the annual flows at the keystations. Similar spatial disaggregation models and procedures are applied todisaggregate the annual flows at the key stations into annual flows for thesubstations, and subsequently into the annual flows at the substations. Once theannual flows at all stations are generated then multivariate disaggregationmodels are applied so that the annual flows can be temporally disaggregated toobtain the corresponding seasonal flows. In Scheme 2, rather that creating anartificial station, the annual flows at the key stations are modeled (andgenerated) by using a stationary multivariate model such as the MAR(l) model.Then the remaining steps to model and generate the annual and seasonal flowsat all other stations are accomplished by following identical steps as in Scheme1. Further description on these modeling schemes are illustrated in Sections

21.5 and 21.6.21.3.9. Model Testing

The fitted model must be tested to determine whether the model complies withthe model assumptions and whether the model is capable of reproducing thehistorical statistical properties of the data at hand. Essentially the keyassumptions of the models refer to the underlying characteristics of theresiduals such as normality and independence.

21.3.9.1. Testing the properties of the residuals

Testing the residuals properties generally involves testing the normality and

the independence of the residuals. First, the residuals are obtained from thespecified models after the parameters are estimated. For instance, in the case ofthe univariate PARMA model of Eq. (21.58), the residuals are the numberse{vex2,exv ... that are derived from the


78/145


79/145


77'

model. On the other hand, in the case of the MPAR model of Eq. (21.84), the

residuals are the set of numbers ... i = 1,..., neach set icorresponding to each site or station. Testing the residual propertiescan be done in several ways depending on how the residuals are arranged.

Several tests are available for testing the normality of the residuals.Common normality tests include the skewness test, the chi-square goodness offt test, the Kolmogorov-Smimov test, and the product moment correlation test(Salas et al, 2000). For periodic-stochastic models, the normality tests shouldbe applied on a month-by-month basis. Often though the tests are applied

considering the entire sample of residuals. In the case of multivariate models,the normality tests should be applied for each set of data (site by site). InSAMS, the skewness test of normality is applied on a month-by-month basisand on a site by site basis.

Likewise, several tests are available for testing the independence of theresiduals. The Portmanteau lack of ft test and the Anderson test (Salas et al,1980) are commonly used for testing independence in time when the residualsare derived from stationary stochastic models. On the other hand, the cross-correlation t-test may be used for testing independence in time when theresiduals are derived from periodic- stochastic models such as those describedin the previous sections. The t-test is applied for the correlation between theresiduals of two successive months, i.e. twelve tests for monthly data.

However, the Portmanteau or Anderson tests may be also applied for testingthe independence of residuals derived from periodic-stochastic models, basedon the autocorrelation of the entire residuals series. In SAMS, the Portmanteautest of independence was applied. For testing the independence betweenresiduals of two different sites (independence in space), the usual test is basedon the cross-correlation t-test. Also this test should be applied for the cross-correlation between residuals of two sites on a season-by-season basis (twelvetests for monthly data), although the test can be applied based on the cross-correlation of the entire residual series for each pair of sites.21.3.9.2. TestingARMA modelparsimony

For a fitted ARMA(p,q) model, SAMS tests its model parsimony using Akaike

Information Criterion (AIC) (Salas, et al., 1980). For comparing amongcompeting ARMA(p,q) models, the following equation is used:

AIC(p, q)=N ln(G2e) + 2(p + q) (21.112)

where N is the sample size and cr is the mximum likelihood estimate of theresidual variance. Under this criterion the model which gives th minimumAIC is the one to be selected. SAMS computes AICs for the fitted model and


80/145


7H0

the models of both one step higher order and one step lower order forcomparison. For instance, for a fitted ARMA(1,1) model, SAMS will computethe AIC vales for ARMA(1,1), ARMA(2,1), ARMA(1,2), ARMA(1,0), andARMA(0,1) models for comparison. Besides, to test the assumption of whitenoise, the AIC of the ARMA(0,0) is also computed.

21.3.9.3. Testing the properties of the process

Testing the properties of the process generally means comparing the statistical

properties (statistics) of the process being modeled, for instance, the processYvrin Eq. (21.58), with those of the historical sample. In general, one wouldlike the model to be capable of reproducing the necessary statistics that affectthe variability of the data. Furthermore, the model should be capable ofreproducing certain statistics that are related to the intended use of the model.

If Yvz has been previously transformed fromXvT, the original non-normal process, then one must test, in addition to the statistical properties of Y,some of the properties ofX Generally, the properties ofYinclude the seasonal mean, seasonal variance, seasonal skewness, andseason-to-season correlations and cross-correlations (in the case of multisiteprocesses), and the properties ofX include the seasonal mean, variance,

skewness, correlations, and cross-correlations (for multisite systems).Furthermore, additional properties ofXVTsuch as thoserelated to low flows, high flows, droughts, and storage may be includcddepending on the particular problem at hand.

In addition, it is often the case that not only the properties of the seasonalprocesses Yv T and Av T must be tested but also the properties of thecorresponding annual processes AY and AX. For example, this case ariseswhen designing the storage capacity of reservoir systems or when testing theperformance of reservoir systems of given capacities, in which one or morereservoirs are for over year regulation. In such cases the annual propertiesconsidered are usually the mean, variance, skewness, autocorrelations, cross-

correlations (for multisite systems), and more complex properties such as thoserelated to droughts and storage.The comparison of the statistical properties of the process being modeled

versus the historical properties may be done in two ways. Depending on thetype of model, certain properties of the Y process such as the mean(s),variance(s), and covariance(s), can be derived from the model in cise form. Ifthe method of moments is used for parameter estimation, the mean(s),variance(s), and some of the covariances should be reproduced exactly,


81/145


however, except for the mean, that may not be the case for other estimationmethods. Finding properties of the Y process in cise form beyond the first twomoments, for instance, drought related properties, are complex and generallyare not available for most models. Likewise, except for simple models, findingproperties in cise form for the corresponding annual process A Y, is notsimple either. In such cases, the required statistical properties are derived bydata generation.

Data generation studies for comparing statistical properties of theunderlying process Y(and other derived processes such as A Y, XandAX)aregenerally undertaken based on samples of equal length as the length of thehistorical record and based on a certain number of samples which can giveenough precisin for estimating the statistical properties of concern. Whilethere are some statistical rules that can be derived to determine the number ofsamples required, a practical me is to generate say 100 samples which cangive an idea of the distribution of the statistic of interest say 6. In any case, thestatistics 0(i), i= 1,...,100 are estimated from the 100 samples and the mean 9and variance S2(0) are determined. Then, the mean deviation,MD(Q)

MD(0) = 6 - 6(J7) (21.113)

:md die relativo root mean square deviations,RRMSD(Q)

100X [6(0- 8(S/)P

are obtained in which 6( H)is the statistie derived from the historical sample(historical statistie). The statistics MD(Q) and RRMSD(Q) are useful forcomparing between the historical and model statistics derived by datageneration. In addition, one can observe where 6( H)falls relative to 6- S(0)and 6+ S(6).Also graphical comparisons such as the Box-Cox diagrams areuseful.

21.4. STOCHASTIC SIMULATION

In section 21.3 we have presented a number of stochastic models andmodeling schemes that can be used for simulating or generating artificialrecords. Generally stochastic simulation begins by generating uniform randomnumbers. Then normal random numbers must be obtained by usingappropriate transformations of the uniform random numbers. Several uniform


82/145


7N}

and normal random numbers generators are available in literature (see forinstance, Bradley, 1987 and Press et al. 1986). Subsequently, the normalrandom numbers must be incorporated into the stochastic model. Section21.4.1 summarizes a procedure to generate synthetic hydrologic time series byusing stochastic models. Section 21.4.2 discusses how stochastic simulationcan be used for forecasting.

21.4.1. Synthetic Generation of Hydrologic Data

Let us assume that our original monthly flow data denoted by have Xv xbeen

transformed into normal flows by using the logarithmic transformation, i.e.(21.115)

Then it has been further standardized seasonally as

Subsequently, we fitted a PARMA(1,0) model to the Z series, i.e.

(21.117a)

As described in section 21.3.3 this model assumes that evx is normallydistributed with mean 0 and standard deviation z(e).Therefore the generating

model can also be written as

ZV_T= $uZvr_l+c7r(e)VT, (21.117b)

in which eVT is normally distributed with mean zero and standard deviationone.

For generating monthly flows, the reverse procedure is followed. We start bygenerating standard normal random numbers ev T. Then Eq.

(21.117b) is used to generate the Zs. After generatingZvx then YVT

can be obtained by

Yvr = Yz+ SX(Y)ZVT (21.118)

andXVTcan be generated by applying the appropriate inverseIransformation to the Yv Tdata. In our case, sinceXv Twas transformed


83/145


7N}

by using Eq. (21.115), then the Zs are generated by applying thein verse transformation

= exP(K,r) ~ ar (21.119)

Note that for generating the autocorrelated seriesZvr,the warm-upprocedure is followed. Suppose that we want to generate N years of monthlyflows. In this procedure, we start by generating ^ ,, from Eq.(21.117b) in which the previous valuZ012is assumed to be equal to

zero (i.e. the mean of the process, which is zero in this case). Thus, Z ,.... Z 2, Zj_[,..., Z2 2, ZN J r L X 2 are generated (and subsequently(he Y 's andthe Zs) whereL is the warm-up length required to remove(lie elToct of Ihc initial assumption. In SAMS L is arbitrarily chosen equal lo50. The dvanlagc of (lie warm up procedure is that it can be


84/145


7M-I

used for low order and high order stationary and periodic models while exactgeneration procedures available in the literature apply only for stationaryARMA models or the low order periodic models. Generation based onmultivariate models is carried out in a similar manner except that vector ofstandard normal random numbers must be generated.

21.4.2. Forecasting Based on Stochastic Simulation

The basic times series models are perfectly adequate for generating data forplanning studies, where the main concern is not the immediate next one or two

years. The generation of data for planning studies is usually performed using arandom set of starting conditions that have nothing what-so-ever to do with thecurrent flows, the immediate past flows or any available forecasts. However,in real time operations, the main concern is what happens in the next fewmonths or at the most the next few years. In this case, the current state of thesystem and all associated forecasts or variable that can help to forecast the nextfew months or years is indeed important. In order to make use of thisinformation, the time series models are either used differently or modifled tobetter use the available information. These changes will in most cases onlyslightly alter the generated flows and then only for a short time. However,minor changes can be of relatively large importance terms of the safety and

efficiency of operations. Besides the immediate past flows, current forecastscould be important as would be variables such as the ocean temperatures orother variables of potential valu in forecasting future flows.

Three ways will be discussed as to how the models may be adopted forstochastic forecasting. The flrst is simply to utilize the time series models butmaking use of the recent past flows. Rather than a random start, the models arestarted with the most recent flows or their corresponding transformed andstandardized counterparts. For example, consider a simple annualautoregressive model of order one. This model has but one lagged term. Ratherthan use a random term for the lagged flow when starting the generation, it iseasy to inser the present years flow when generating flows traces. Often thelagged flow term is in a transformed and standardized form in the model andsome simple calculations are required to modify the actual flow into the corredform for the model. For autoregressive models of any order, Ihis approach is


85/145


very simple. For ARMA models, the process is more complicated as one hasno idea what the correct vales are for the random terms for the recentlyexperienced years. Generally, they are easiest approximated by frst solving

the model equation for the most recent random term. By successivesubstitution an equation can then be developed which gives the random term atany time as an infinite series consisting only of past flows. Of course, only afew terms are usually needed to adequately estimate the vales. The advantageof this approach is that it makes use only of the time series models and, afterestimating the initial conditions, the generation of data is the same process asnormally used for planning studies. The disadvantage of this approach is that itdoes not include some other information, which might be of help.

The second approach is to expand upon the first case by additionallyadding terms to the time series model to represent current forecasts or valesof forecasting variables. This is easily done, however the parameter estimationmay be complicated and care must be taken to avoid pitfalls. Least squares

parameter estimation my in fact be the least prone to problems. The mainadvantage is that all current and past knowledge is now utilized. A majordisadvantage is that the model now has many more sets of parameters. Forexample, using a monthly model as an example, the model for generating Mayflows is dependent upon the starting time. If the generation is started inJanuary, the model for the May flows has different parameters than if thegeneration has started in say March. Further, the model parameters forgenerating the first May llows is not the proper set for generating May flows ayear henee. If the goal is to generate flows for the next 18 months, 18 sets ofparameters (or equivalent if more than one month is generated at one time) areneeded for each of the 12 starting times or a total of 18 times 12 or 216 sets of

parameters. If the goal is to generate 36 months into the future, lliree times asmany are needed.The third approach is where an entire series of variables are available into

the future and the time series model is modified to include these exogenousvariables. In cases where the future variable vales are nccurate into thefuture, the time series model may only one set of parameters. However, if theaccuracy changes with distance into the future, the same approach is needed asfor the second approach.

21.5. DESCRIPTION OF SAMS

In section 21.5.1, a general description of SAMS is presented in whichdifferent operations undertaken by SAMS are briefly explained. Then, eachoperation is explained and illustrated more thoroughly in sections 21.5.2,21.5.3and 21.5.4.

21.5.1. General Overview

SAMS is a Computer software package that deais with the stochastic analysis,


86/145


7N(>

modeling, and simulation of hydrologic time series. It is written in C andFortran and runs under modera windows operating systems such asWINDOWS NT and WINDOWS 98. The package enables the user to choosebetween different options that are currently available. SAMS performs thesemain functions: 1) Statistical Analysis of Data, 2) Fitting a Stochastic Model(includes parameter estimation and testing) and 3) Generating Synthetic Series.

SAMS has the capability of analyzing single site and multisite annual andseasonal data and the results of the analysis are presented in graphical ortabular forms or are written on output fdes. The current versin of SAMS canbe applied to annual and seasonal data, such as quarterly and monthly data.

The Statistical Analysis of Data module consists of data plotting,checking the normality of the data, data transformation, and data statisticalcharacteristics. Plotting the data may help detecting trends, shifts, outliers, orerrors in the data. Probability plots are included for verifying the normality ofthe data. The data can be transformed to normal by using differenttransformation techniques. Currently, logarithmic, power, and Box-Coxtransformations are available. SAMS determines a number of statisticalcharacteristics of the data. These include basic statistics such as mean, standarddeviation, skewness, serial correlations (for annual data), season-to-seasoncorrelations (for seasonal data), annual and seasonal cross-correlations formultisite data, and drought, surplus, and storage related statistics. These

statistics are important in investigating the stochastic characteristics of thedata.

The second main application of SAMS Fitting a Stochastic Modelincludes parameter estimation and model testing for altcrnative


87/145


univariate and multivariate stochastic models. The following models areincluded: (1) univariate ARMA(p,q) model, where p and q can vary from 1 to10, (2) univariate GAR(l) model, (3) univariate periodic PARMA(p,q) model,(4) univariate seasonal disaggregation, (5) multivariate autoregressive MAR(p)model, (6) contemporaneous multivariate CARMA(p,q) model, where p and qcan vary from 1 to 10, (7) multivariate periodic MPAR(p) model, (8)multivariate annual (spatial) disaggregation model, and (9) multivariatetemporal disaggregation model. Two estimation methods are available, namelythe method of moments (MOM) and the least squares method (LS). MOM isavailable for most of the models while LS is available only for univariate

ARMA, PARMA, and CARMA models. For CARMA models, both themethod of moments (MOM) and the method of mximum likelihood (MLE)are available for estimation of the variance-covariance (G) matrix. Regardingmultivariate annual (spatial) disaggregation models, parameter estimation isbased on Valencia- Schaake or Mejia-Rousselle methods, while for annual toseasonal (temporal) disaggregation Lane's condensed method is applied.

For stochastic simulation at several sites in a stream network system adirect modeling approach based on multivariate autoregressive and CARMAprocesses are available for annual data and multivariate periodicautoregressive process is available for seasonal data. In addition, two schemes

based on disaggregation principies are available. i

;

or this purpose, it isconvenient to divide the stations into key stations, substations, andsubsequent stations.Generally the key stations are the farthest downstreamstations, substations are the next upstream stations, and subsequent stations arethe next further upstream stations. In the lirst scheme, the annual flows at thekey stations are added creating an annual flow data at an artificialor indexstation. Subsequently, a univariate ARMA(p,q) model is ftted to the annualflows of the index station. Then, a spatial disaggregation model relating theannual flows o" the index station to the annual flows of the key stations isfitted. further, a statistical disaggregation model relating the annual flows ofthe key station to Ihose of the substations and another disaggregation model

relating the annual llows ol'lhc substations and the subsequent stntions, arelitlod. In taet, this is a Ihroe-levcl (spatial) disaggregration procedure. In thesecond scheme a multivariate AR(p) model is fitted to the annual data of thekey stations, then the rest of the model relating the annual flows at the keystation, substations, and subsequent stations are conducted in a similar manneras in the first scheme. Furthermore, if the objective of the modeling exercise is


88/145


89/145


7N*

critical valu, the hypothesis of normality of the data cannot be rejected. Onthe other hand, if the sample skewness coefficient is greater than the tablevalu, the hypothesis of normality is rejected. In addition, for the specifiedseason, the normal probability plot for the transformed seasonal data and thecomparison of the theoretical generated distribution and the sampledistribution for that season are also displayed.


90/145


nlw Stolion B: fT

, - |

t ' ; |7~ | Xianarfonnationi j ave | Diipey j TtaoriennftIS e |

; ;!, v .. . j~~ blatas| scepiTians(matiOn| | PioycjsMnnu{

Fig. 21.2. Annual data transformation result.

If the data at hand is not normal, one can check whether it can benormalized by a certain transformation function. The user can choose any typeof transformation by simply clicking on the corresponding button. Three typesof transformations are available: logarithmic, power, and Box-Coxtransformations. The transformation can be done all at once for all seasons oron a season-by-season basis. Figure 21.3 shows an example of seasonaltransformation results.

In the event that the user wants to model site 1 data with an ARMA (p,q)model. Then, the ARMA model will be ftted to the transformed data and notthe original data.

A save option allows the user to save the transformation parameters in aspecial file. To understand this feature of SAMS, suppose that a usertransformed the data and ftted the PARMA (1,1) model to the data.Subsequently, the user wants to fit a different model to the transformed data.Instead of doing the transformation process over again, the user can simplyopen the transformation file, which was saved previously.

Type ofTransformation: Logarlthmlc Y(D = In(X(l) * a) where X(t) : Original Serles. Y(t):Transformed Serles a: ParameterofTransformation

Skewness Test of Normallty- C omputedValu : -0.017 Table Valu (10% significantelevel): 0.544 Result: Hypothesls of Normalitynot Rejected


91/145

Stochastic Analysis,Modeling and Simulation(SAMS 2000) / 21

7'i

Skewness Test of NormalityOMAt wmmwhnm m,testihs: 2ySite 1 - KEECHELUS_RESERVOIR

2 Check normality of

data and usetransformation options:


92/145


72

'OisTiHijn'finN orv

Site 1 Season 1

Itarofomalions ave fiiplay Tr ansfoim Ai Sites | :rilSlation#: |T

EnterSeason#:

|T

(IX: SAMftK _ HR

Site 1 Season 1

Enter the valu ofa:. JO.O V 'la.'.IBWUco | Statjff | 6cceptTiarofoima>ion| Eiinl |

Comp.Val.0.0004

0.0004-0.0022

-0.0021

-0.0078-0.0361-0.0151-0.0050-0.01430.0004-0.04380.0070

Tab.Val.(10%

level)0.54360.54360.54360.54360.54360.54360.54360.54360.54360.54360.54360.5436

Skewness Test ofNormalit)Trans. Co-eff.a Co-. Log 8.0000 0 0000Log

3.5000

0.0000

Log 1.7000 0.0000Log -1.7000 0.0000Log -7.3000 0.0000Log 40.0000 0.0000Log 120.0000 0.0000Log 80.0000 0.0000Log -1.4000 0.0000Log 0.0000 0.0000Log -1.1000 0.0000Log 2.5000 0.0000

Type of Transformation: Logarithmic Y(t) = In(X(t) + a) where X(t): Original Series, Y(t):Transformed Series a: Parameter ofTransformation

NORMAL PPBABIUTY PAER NORMAL PRBABILITV PAPER

10 30 50 70 %NON-EMCEEOANCE PRBABIUTV

Satriple Theotelicai

NON-EACEEDANCE PPOBABILITY


93/145


72

21.5.2.3. Statistical Character istics of Time Series

A number of statistical characteristics can be calculated for the original andtransformed data. They can be available in graphical and tabular formats andcan be saved in an output file. These are summarized below.

-For Annual Data:

1. Basic statistics such as mean, standard deviation, skewnesscoefficient, coefficient of variation, mximum, and mnimumvales.

2. Serial correlation coeffcients.3. Cross-correlation coeffcients for multisite data.4. Drought, surplus (flood), and storage related statistics.

-For Seasonal Data:1. Basic statistics such as seasonal means, standard deviations, skewness

coeffcients, coeffcients of variation, mximum, and minimum vales.2. Season-to-season correlation coeffcients.3. Season-to-season cross-correlation coeffcients for multisite data.4. Drought, surplus (flood), and storage related statistics.

Graphs also display the 95% limits. If a correlation coefficient liesbetween these two lines, it means that the correlation is not statisticallysignifcant.

21.5.3. Fitting a Stochastic Model

The LAST package included several programs to perform several objectivesregarding stochastic modeling of time series. The basic procedure involvedmodeling and generating the annual time series using a multivariate AR(1) orAR(2) model, then using a disaggregation model to disaggregate the generatedannual flows to their corresponding seasonal flows. In contrast, SAMS has twomajor modeling strategies which are direct and indirect modeling. Direct

modeling means fitting an stationary model (univariate ARMA or multivariateAR or CARMA) directly to the annual data or fitting a pcriodic (seasonal)model (univariate PARMA or multivariate PAR) directly lo the seasonal dala


94/145


of the system at hand. Annual to seasonal disaggregation modeling on theother hand is an indirect procedure since the modeling of seasonal datainvolves also modeling of the corresponding annual data as well. Regardless ofwhether the input data available is annual data or seasonal (for examplemonthly data) the user must select on the annual button if the final objectiveof the modeling exercise is to generate annual flows only. Otherwise, if theobjective is to generate monthly quantities then the seasonal button must beselected.

The following specific models are currently available in SAMS under each

category:For Annual Modeling:

1. Univariate ARMA(p,q) model.2. Univariate GAR(l) model.3. Multivariate AR(p) model (MAR).4. Contemporaneous ARMA(p,q) model (CARMA).

Multivariate annual (spatial) disaggregation.For Seasonal Modeling:

1. Univariate PARMA(p,q) model.2. Univariate seasonal disaggregation.3. Multivariate PAR(p) model (MPAR).4. Multivariate seasonal disaggregation.

Standardization implies that not only the mean will be subtracted but inaddition the data will be further transformed to have a standard deviation equalto one. For example, for the season 5 data, the mean for season 5 will besubtracted from each data point, then each observed data point for that seasonwill be divided by the standard deviation of the 5 th season. As a result, themean and the standard deviation of the standardized data of the 5 thseason willbecome equal to zero and one, respectively. Then, the order of the model to befitted can be selected. Subsequently, the method of estimation of the model

parameters must

cap20 hidro.docx

Documents