APPLICATION OF THE TOPKAPI MODEL WITHIN THE DMIP 2 PROJECT Gabriele Coccia(1), Cinzia Mazzetti(2), Enrique A. Ortiz(3) and Ezio Todini(1) (1)University of Bologna, Bologna, Italy (2)ProGea Srl, Bologna, Italy (3)HidroGaia, Paterna (Valencia), Spain

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: INTRODUCTION

TOPKAPI MODEL: A DISTRIBUTED AND PHYSICALLY BASED HYDROLOGIC MODEL Physically meaningful parameters whose values can be retrieved from thematic maps (DEM and soil type and land use maps) PHYSICALLY BASED MODEL

Easy calibration Ungauged catchments Continuous simulations (climate change, water resources, â€Ś)

DISTRIBUTED MODEL

1D outputs (flow, water balance, ect.) everywhere on the catchment 2D maps containing information on soil moisture, snow, evapo-transpiration

REDUCED COMPUTATIONAL TIMES AND PARSIMONIOUS PARAMETRIZATION

Real-time flood forecasting systems Real-time coupling with hydraulic models

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: INTRODUCTION

MODEL HYPOTHESES 1. The precipitation is assumed to be constant over the single cell. 2. The entire precipitation infiltrates into the soil until it is saturated (Dunne Mechanism). 3. The slope of the water table is assumed to coincide with the slope of the ground, unless the latter is very small (smaller than 0.01%); this constitutes the fundamental assumption of the Kinematic wave approximation in De Saint Venant equations, and it implies the adoption of a Kinematic wave propagation model with regard to horizontal flow, or drainage, in the unsaturated area. 4. The local transmissivity, like local horizontal subsurface flow, depends on the integral of the water content profile in the vertical direction (vertical lumping). 5. The saturated hydraulic conductivity is constant with depth in the surface soil layer but much larger than that of deeper layers.

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: INTRODUCTION

THE MODEL COMPONENTS

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: INTRODUCTION

KINEMATIC APPROACH HYPOTHESIS TOPKAPI is based on the hypothesis that sub-surface flow, overland flow and channel flow can be approximated using a Kinematic wave approach.

Sub-surface flow

Overland flow

∂η = a − bη c ∂t

Channel flow

The integration in space of the non-linear Kinematic wave equations representing subsurface flow, overland flow and channel flow results in three ‘structurally-similar’ non-linear reservoir equations.

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: SOIL, SURFACE AND CHANNEL COMPONENTS

SUBSURFACE, SURFACE AND CHANNEL FLOWS Soil parameters

Non-linear reservoir equation for SOIL component

ϑr = residual soil moisture content ϑs = saturated soil moisture content L = thickness of the surface soil layer [m] ksh = horizontal saturated hydraulic conductivity [ms-1] αs = parameter which depends on the soil characteristics Surface parameters:

Lks tan(β ) ∂Vs Cs X α s 2 u u = pX + Qo + Qs − 2α Vs Cs = α α s ( ) ∂t ϑ − ϑ X s r L

(

)

Non-linear reservoir equation for SURFACE component

Co Wo ∂ ( XWo ho ) αO ( ) = ro XWo − XW h o o ∂t ( XWo )α o

tan(β ) Co = n0

1

2

Non-linear reservoir equation for CHANNEL component

no = Manning’s friction coefficient for the surface roughness

B ⋅y

x c0 4 s0 (senγ ) 3 ∂Vc Ax = u 3 = rc + Qc − 2 Vc 2 4 1 ∂t 3 3 3 2 n (tan γ ) X C x = Bx (1 + tan γ )

(

)

2

Channel parameters: -1/3 nc = Manning’s friction coefficient for the channel roughness [m s]

Cross Section Geometry Parameters

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: EVAPO-TRANSPIRATION COMPONENT

EVAPO-TRANSPIRATION The Evapo-Transpiration is estimated as a function of the air temperature. An empirical equation, that relates the reference potential evapo-transpiration, ET0m, to the compensation factor Wta, to the mean recorded temperature of the month T and the maximum number of hours of sunshine N of the month, was developed. The reference potential evapo-transpiration is computed on a monthly basis using one of the available simplified expressions such as for instance the one due to Thornthwaite and Mather (1955).

T (i ) ET0 m (i ) = 16a(i ) 10 m b n (i ) N (i ) a (i ) = b= 30 12

c

i=1,12 months

T (i ) ∑ i =1 5 12

Monthly reference potential evapo-transpiration Thornthwaite

1.514

c = 0.49239 + 1792 × 10−5 b − 771 × 10 −7 b 2 + 675 × 10 −9 b 3

The relation used, which is structurally similar to the radiation method formula (Doorembos et al., 1984).

ET0 m = α + β NWtaTm

Monthly reference potential evapo-transpiration Doorembos

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: EVAPO-TRANSPIRATION COMPONENT

EVAPO-TRANSPIRATION Once coefficients α and β have been found, the equation can be used to obtain actual evapotranspitation values for any crop culture at any time step ∆t according to the following equations:

ET0 = K c (α + βNWtaT∆t ) ETa = 0 V ETa ET = 0 Vsat ETa = ET0

∆t 30 ⋅ 24 ⋅ 3600

Potential evapo-transpiration

V ≤ β1Vsat

β1Vsat ≤ V ≤ β 2Vsat

Actual evapo-transpiration

V > β 2Vsat

Evapo-transpiration parameters: Kc = crop factor β1,β2 = percentages of the saturation volume Landuse map

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: SNOW ACCUMULATION AND MELTING COMPONENT

SNOW ACCUMULATION AND MELTING The snow accumulation and melting module of the TOPKAPI model is driven by a radiation estimate based upon the air temperature measurements. At each model pixel the mass and energy balances are computed. 1. Net solar radiation estimation;

Rad = λET + H Net solar radiation

Sensible heat

Latent heat flux

The estimation of the radiation is performed by re-converting the latent heat, assumed to be equal to the potential evapo-transpiration, in the radiation and assuming the sensible heat to be equal to the latent heat:

C er = 606 . 5 − 0 . 695 (T − T0 ) Latent heat flux

λET = Cer ⋅ ET0 Sensible heat:

ηal = efficiency factor for albedo; ηrad = radiation efficiency factor

Rad = 2 ⋅η alη rad ⋅ [606 .5 − 0 .695 (T − T0 )]⋅ ET 0

H = λET 89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: SNOW ACCUMULATION AND MELTING COMPONENT

2. Computation of the Solid and Liquid Percentage of Precipitation; Threshold temperature

Ts σ = 0.6

F (T ) =

1 1+ e

−

Ts = 0

T −Ts

σ

3. Estimation of the water mass and energy balances based on the hypothesis of zero snowmelt; Water mass balance

Z t*+ ∆t = Z t + P Energy balance

[

]

Et*+ ∆t = Et + Rad + C siT ⋅ [1 − F (T )]⋅ P + CsiT0 + Clf + C sa (T − T0 ) ⋅ P ⋅ F (T )

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: SNOW ACCUMULATION AND MELTING COMPONENT

4. Comparison of the total available energy with that sustained as ice by the total available mass at 273 °K;

C si Z t*+ ∆t T0 ≥ Et*+ ∆t The available energy is not sufficient to melt part of the accumulated snow

R sm = 0 ∗ Z t + ∆t = Z t + ∆t Et + ∆t = Et∗+ ∆t

5. Computation of the snowmelt produced by the excess energy; and updating the water mass and energy budgets.

C si Z t*+ ∆t T0 < Et*+ ∆t

(

The available energy is sufficient to melt part of the accumulated snow

)

(

)

C si Z t*+ ∆ t − R sm T0 = E t*+ ∆ t − C si T0 + C lf R sm Snowmelt parameter: Ts = temperature threshold for snow melting

Et∗+ ∆t − CsiT0 Z t∗+ ∆t Rsm = Clf ∗ Z t + ∆t = Z t + ∆t − Rsm ∗ Et + ∆t = Et + ∆t − (C siT0 + Clf )Rsm

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: PERCOLATION COMPONENT

PERCOLATION TO DEEPER SOIL LAYERS For the deep aquifer flow, the response time caused by the vertical transport of water through the thick soil above this aquifer is so large that horizontal flow in the aquifer can be assumed to be almost constant with no significant response on one specific storm event in a catchment (Todini, 1995). Nevertheless, the TOPKAPI model accounts for water percolation towards the deeper soil layers even though it does not contribute to the discharge. The percolation rate from the upper soil layer is assumed to increase as a function of the soil water content according to an experimentally determined power law (Clapp and Hornberger, 1978. Empirical Equations for some soil hydraulic properties; Liu et al., 2005).

v Pr = k sv ν sat

αp

Percolation

Percolation parameters: ksv = vertical saturated hydraulic conductivity [ms-1] αp = parameter which depends on the soil characteristics

89TH AMS Annual Meeting, Phoenix, January 2009

Soil type map (pedology)

MODEL APPLICATION: BASINS DESCRIPTION

MODEL APPLICATION: DMIP 2 SIERRA-NEVADA BASINS TOPKAPI model has been applied within the DMIP 2 to the North Fork American River Basin and the East Fork Carson River Basin

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL APPLICATION: PRE-PROCESSING DATA

DEPITTING THE DEM AND CREATING THE DRAINAGE NETWORK 1) Defining the basin boundary, namely the model application area; 2) Defining the resolution of the model (cell dimension); 3) Depitting the DEM: eliminating false outlet and sinks. Each TOPKAPI cell can have up to three input cells and only one output cell; FALSE OUTLETS: cells conveying water outside of the basin boundary

4) 5) 6) 7)

SINKS: cells without any output direction

Defining the connection of each cell with the surrounding ones; Defining the drainage network; Organizing the drainage network using Strahlerâ€™s orders: Comparing the model drainage network to the real river network.

Elevation [m] High : 3468.3

Low : 1462.26

Treated DEM of the North Fork American River

Treated DEM of the East Fork Carson River

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL APPLICATION: PRE-PROCESSING DATA

SETTING THEMATIC MAPS - SELECTING INITIAL PARAMETER VALUES

SOIL TYPE and LAND USE

ϑr = residual soil moisture content ϑs = saturated soil moisture content L = thickness of the surface soil layer ksh = horizontal saturated hydraulic conductivity αs = parameter for subsurface flow ksv = vertical saturated hydraulic conductivity αp = parameter for percolation

no Kc nc B γ

= Manning’s friction coefficient for the surface roughness = crop factor = Manning’s friction coefficient for the channel roughness = Width of the rectangular channel = slope of channel sides (for triangular sections)

β1, β2 = evapo-transpiration parameters Ts = temperature threshold for snow accumulation-melting

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL APPLICATION: HYDRO-METEROLOGICAL DATA

HYDRO-METEOROLOGICAL DATA Blue Canyon Huysink

Gridded hourly precipitation data Gridded hourly temperature data #

Observed hourly streamflow data at North Fork Dam for the American River, and at Gardnerville and Markleeville for the Carson River

North Fork Dam

1/10/1987 1/10/1988 WARM UP

1/10/1997 CALIBRATION

31/12/2002 VALIDATION

* #

* #

GardnerVille

Markleeville

Spratt Creek !(

Observed daily Snow Water Equivalent data at Blue Canyon and Huysink for the American River, and at Poison Flat, Ebbett Pass, Blue Lake and Spratt Creek for the Carson River.

Blue Lake !(

Ebbett Pass !(

Poison Flat !(

1/10/1989 1/10/1990 WARM UP

CALIBRATION

1/10/1997

31/12/2002 VALIDATION

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL APPLICATION: CALIBRATION RESULTS, AMERICAN RIVER, LONG-TERM PERIODS

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL APPLICATION: CALIBRATION RESULTS, AMERICAN RIVER, FLOOD EVENTS

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL APPLICATION: CALIBRATION RESULTS, AMERICAN RIVER, OVERALL AND EVENT STATISTICS

North Fork Dam - American River – Event Statistics Calibrated Uncalibrated N° Events 13 13 ER [%] 13.43 78.77 EP [%] 17.18 101.45 ET [h] 4.00 7.31

YEAR

Peak [m3s-1]

r

rmod

NS

EV

1989 1990 1991 1992 1993 1994 1995 1996 1997 ALL

334 89 507 164 487 129 751 738 1818 1818

0.95 0.90 0.93 0.86 0.92 0.82 0.92 0.91 0.98 0.94

0.92 0.71 0.78 0.62 0.88 0.47 0.80 0.83 0.97 0.89

0.87 0.47 0.68 0.11 0.84 -0.90 0.83 0.83 0.95 0.88

0.90 0.69 0.79 0.47 0.84 -0.15 0.85 0.83 0.95 0.89

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL APPLICATION: CALIBRATION RESULTS, CARSON RIVER, LONG-TERM PERIODS

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL APPLICATION: CALIBRATION RESULTS, CARSON RIVER, OVERALL AND EVENT STATISTICS YEAR 1991 1992 1993 1994 1995 1996 1997 ALL

Gardnerville – Carson River Peak [m3s-1] PB [%] r rmod NS 38 88.2 0.85 0.47 -0.72 27 137.8 0.94 0.44 -2.12 84 -1.7 0.92 0.83 0.81 31 79.3 0.92 0.50 -0.46 167 -19.2 0.95 0.81 0.86 221 3.5 0.93 0.86 0.86 527 0.8 0.90 0.72 0.80 527 13.9 0.89 0.81 0.79

Gardnerville - Carson River – Event Statistics Calibrated Uncalibrated N° Events 4 4 ER [%] 6.26 329.48 EP [%] 38.34 699.20 ET [h] 1.50 2.75 Markleeville – Carson River YEAR Peak [m3s-1] PB [%] r rmod NS 1991 39 102.3 0.84 0.44 -1.28 1992 27 160.5 0.93 0.40 -3.51 1993 74 3.5 0.92 0.79 0.79 1994 32 110.0 0.94 0.47 -1.20 1995 168 -7.7 0.96 0.92 0.91 1996 200 19.2 0.93 0.91 0.84 1997 385 13.5 0.94 0.83 0.88 ALL 385 28.0 0.91 0.91 0.79 Markleeville - Carson River – Event Statistics Calibrated Uncalibrated N° Events 5 5 ER [%] 24.75 211.43 EP [%] 30.28 382.71 ET [h] 7.20 7.80

89TH AMS Annual Meeting, Phoenix, January 2009

EV -0.23 -0.55 0.81 0.02 0.89 0.86 0.80 0.80

EV -0.46 -1.02 0.79 -0.23 0.92 0.86 0.88 0.82

MODEL APPLICATION: CALIBRATION RESULTS, CARSON RIVER, SNOW WATER EQUIVALENT

Blue Lakes – 2456 m

Ebbets Pass– 2672 m

Poison Flats – 2358 m

Spratt Creek – 1864 m

89TH AMS Annual Meeting, Phoenix, January 2009

CONCLUSION

CONCLUSION 1) The TOPKAPI model well reproduces the streamflow series in basins with a complex hydrological regime, it can reproduce both flood and low water events with good accuracy; 2) The TOPKAPI model was born for low elevation catchments, where the snow melting component is not much significant and its snow accumulation and melting module is simple; however, its application in high elevation basins is feasible and the results are not extremely precise, but good on the whole; 3) The experience in calibrating the model and the comparison between calibrated and uncalibrated simulations show that literature parameter values are usually too small for the superficial soil layer conductivity; using the TOPKAPI model in ungauged basins, it is necessary to account for that; 4) As shown, in the application in the Carson River, by the results at Markleeville using the calibration performed at Gardnerville, the TOPKAPI model is capable to produce optimal simulations in ungauged points of the basin.

89TH AMS Annual Meeting, Phoenix, January 2009

THANK YOU FOR YOUR ATTENTION!

89TH AMS Annual Meeting, Phoenix, January 2009

EVALUATION INDEXES N

PB =

∑ (S

i

− Oi )

N

∑O

Percent Bias

× 100

i =1

i

i =1

r =

N

N

N

i =1

i =1

i =1

N ∑ S iOi − ∑ S i ∑ Oi 2 2 N N 2 N N 2 N S S N O O − − ∑ i ∑ i ∑ i ∑ i i =1 i =1 i =1 i =1

rmod = r

min {σ sim , σ obs } max {σ sim , σ obs } N

NS = 1 −

∑ (S

Correlation Coefficient

Modified Correlation Coefficient

− Oi )

2

i

i =1 N

∑ (O

i

−O

Nash-Sutcliffe Efficiency

)

2

i =1

1 N ∑ (S i − Oi ) − N ∑ (Si − Oi ) i =1 EV = 1 − i =1 N 2 ∑ (Oi − O ) N

2

Explained Variance

i =1

89TH AMS Annual Meeting, Phoenix, January 2009

EVALUATION INDEXES

N

ER =

∑

NY avg

∑

Q p , i − Q ps ,i × 100

i =1

NQ

∑T

p ,i

− T ps , i

i =1

Absolute Peak Time Error N

N

RMSE =

Percent Absolute Peak Error [%]

p , avg

N

ET =

Percent Absolute Event Runoff Error [%]

× 100

i =1

N

EP =

Bi

∑ (S

− Oi )

2

i

Root Mean Square Error

i =1

N

89TH AMS Annual Meeting, Phoenix, January 2009

4th MODEL HYPOTESIS L

∫( ) ~

T = kϑ(z) dz = Transmissivity,

if

( )

~ ~ k ϑ(z) = ks ⋅ϑ(z)α ⇒ T = ks

~

∫ϑ(z) dz ~ α

0

0 α

L 1 ~ α ~ 1 ~ ~α 1 ~ Θ= ∫ϑ(z)dz ⇒ Θ = ∫ϑ(z)dz ≠ ∫ϑ(z) dz L0 L0 L0 L

se

L

L

89TH AMS Annual Meeting, Phoenix, January 2009

where:

ϑ(z) −ϑr ϑ(z) = ϑs −ϑr

MODEL DESCRIPTION: SOIL COMPONENT

SUBSURFACE FLOW

~ ∂Θ ∂q (ϑ s − ϑ r )L + = p ∂t ~ ∂x q = tan (β )k s L Θ α

p Qo Ground surface

Qs

Continuity equation Dynamic equation

Non-linear reservoir equation for SOIL component

Lks tan(β ) ∂Vs Cs X α s 2 u u C = = pX + Qo + Qs − 2α Vs s s (ϑs −ϑr )α Lα ∂t X

(

Qs

)

Soil parameters

ϑr = residual soil moisture content ϑs = saturated soil moisture content L = thickness of the surface soil layer [m] ksh = horizontal saturated hydraulic conductivity [ms-1] αs = parameter which depends on the soil characteristics

89TH AMS Annual Meeting, Phoenix, January 2009

Soil type map (pedology)

MODEL DESCRIPTION: SURFACE COMPONENT

OVERLAND FLOW The input to the surface water model is the precipitation excess (r0) resulting from the saturation of the surface soil layer.

Overland watersurface

∂qo ∂ho = r − o ∂t ∂x 1 5 1 qo = (tan β ) 2 ho 3 = Co hoα o no

Continuity equation

Dynamic equation

Non-linear reservoir equation for SURFACE component ro

Co Wo ∂ ( XWo ho ) ( XWo ho )α O = ro XWo − αo ∂t ( XWo )

tan(β ) Co = n0

Surface parameters:

no = Manning’s friction coefficient for the surface roughness [m-1/3s] Land use map

89TH AMS Annual Meeting, Phoenix, January 2009

1

2

MODEL DESCRIPTION: CHANNEL COMPONENT

CHANNEL FLOW: KINEMATIC WAVE

(

)

∂Vc u r Q = + − qc c c ∂t 2 s0 sin γ 3 8 3 qc = yc 2 5 2 3 nc tan γ 3

TRIANGULAR cross-section

Continuity equation

Dynamic equation

Non-linear reservoir equation for CHANNEL component

B ⋅y

x c0 4 s0 (senγ ) 3 ∂Vc A = u x = rc + Qc − 2 Vc 3 2 4 1 ∂t 3 3 2 n (tan γ ) 3 X C x = Bx (1 + tan γ )

(

)

2

Rectangular cross-section parameters: nc = Manning’s friction coefficient for the channel roughness [m-1/3s] B = Width of the rectangular channel [m] γ = Slope of the channel sides Strahler’s orders

89TH AMS Annual Meeting, Phoenix, January 2009

MODEL DESCRIPTION: CHANNEL COMPONENT

CHANNEL FLOW: MUSKINGUM-CUNGE-TODINI It is possible to use the Muskingum-Cunge-Todini routing method, as an alternative to the Kinematic non-linear reservoir, for channels with slope smaller than 0.1%. Oˆ t +∆t = C1 I t + ∆t + C 2 I t + C3Ot

C1 =

C2 =

C3 =

− 1 + Ct* + Dt*

Ct* =

c t ∆t β t ∆x

Dt* =

Qt β t BS 0 ct ∆x

1 + Ct*+∆t + Dt*+∆t 1 + Ct* − Dt* 1 + Ct*+∆t + Dt*+∆t 1 − Ct* + Dt*

β=

ct At Qt

Ct*+∆t

1 + Ct*+∆t + Dt*+∆t Ct*

q11 = c1 ⋅ q01 + c 2 ⋅ q00 + c3 ⋅ q10

Todini E., 2007. A mass conservative and water storage consistent variable parameter Muskingum-Cunge approach. Hydrol. Earth Syst. Sci., 11:1645–1659.

89TH AMS Annual Meeting, Phoenix, January 2009