Prelude

MATLAB ApplicationSimple Nonlinear Rainfall-Runoff Model (Appendage)

 

 The aim of this recent appendage to the original paper, Simple Nonlinear Rainfall-Runoff Model by Dr. H. A.

Basha, published in the Journal of Hydraulic Engineering,Vol. 5, No. 1, January 2000, is a brief revision of the proposed model in the light of a minor inconsistency that was recently detected in the analytical derivation of the differential equations governing the model.

 After adjusting the very equations to accommodate the missing parameter, a new estimation of all parameters

was conducted as the model was applied to an alternate subwatershed and rain-gauge during four single and multi-peaked storms. The following is a concise overview of the original methodology adopted by Dr. Basha to derive and implement his model, and an extensive demonstration of the procedure we went through to eliminate the inconsistency and apply the modified model to another shed of similar characteristics. A comprehensive conclusion was drawn to assess the influence of this modification and discuss the success versus uncertainty of this rainfall-runoff model, of which all contributing sources of error were reasonably justified.

 The model was first implemented into an Excel spreadsheet to serve as the guideline to the MATLAB program

defining the model. The spreadsheet contained an elaborate scheme of links that would allow a direct real-time visual manipulation of the governing parameters to yield the best graphical fit. Once we obtained our comparable set of computed runoff values and an initial estimate of the parameters, we embarked on writing the MATLAB program to provide us with similar corresponding results. An input program was devised to resemble the given storm data files and feed the mother program that would accordingly do the rainfall-runoff calculations. Finally, these initial estimates were further refined using nonlinear least-squares regression, which is a built-in MATLAB routine.

  

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Abstract  The model presented is a simple nonlinear conceptual simulation of rainfall-runoff phenomena. It is based on

the concept of a nonlinear storage outflow relationship, assumes a nonlinear loss module, and is a function of few parameters that can be estimated efficiently using nonlinear least-squares regression. The model has been applied to the watershed under study with considerable success.

       

   

Watershed Description 

             The nonlinear rainfall-runoff model has been applied to the Walnut Gluch experimental watershed in Arizona. The watershed is composed of various subwatersheds, of which the Lucky Hills subwatershed data were used.  The area of the Lucky Hills 106 (LH-106) subwatershed is 0.36 ha, and the climate is semiarid and subject to heavy rains in the summer period.(As evident from the runoff data, the baseflow is nonexistent.)             Concerning the rainfall data, it was obtained from the nearby gauges RG-83 and RG-384. Despite the fact that both gauges are not far off the area of interest (LH-106), a significant variation was realized in the distribution of the measured rainfall data obtained from the two gauges. Although the average rainfall distribution over LH-06 ought to be determined from the arithmetic or weighted mean of both gauges, the individual variations of the gauges significantly differed from each other in terms of total rainfall, peak intensity, and time to peak. Consequently, the rainfall distribution used for each storm event is taken from either one of the two gauges.  (Usually, the choice of the applicable gauge depends on its proximity to the area under study and the direction of the moving storm event.)             Originally, Dr. Basha focused his study on the other subwatershed, LH-102 (1.46 ha), along with the contribution from RG-384. To test the creditability of the proposed model, we considered the alternate subwatershed and gauge, as indicated in the figure below:  

  

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Abstractions   

          After flowing across the watershed surface, excess rainfall becomes direct runoff at the watershed outlet under the assumption of Hortanian overland flow. The difference between the observed total rainfall hyetograph and the excess rainfall hyetograph is termed abstractions, which are primarily water absorbed by infiltration with some allowance for interception and surface storage. The abstractions are assumed to be either constant over the time duration of the storm or exponentially decreasing.              The Φ-index method assumes a constant rate of abstractions that will yield an excess rainfall hyetograph with a total depth equal to the depth of direct runoff over the whole watershed. The value of Φ is determined by picking a time interval length Δt, judging the number of contributing rainfall intervals M, subtracting ΦΔt from the observed rainfall in each interval, and adjusting the values of Φ and M as necessary so that the depths of direct runoff and excess rainfall are equal:  

  

Where  is the observed rainfall in time interval m.               The abstractions can be also assumed to be exponentially decreasing over the time duration:  

  

Where  is the initial infiltration rate; and k is the decay constant. (k = 0 reduces the equation to a constant rate of

abstraction .) This equation has the form of Horton’s infiltration equation. It assumes the infiltration is occurring under ponding conditions, which isn’t necessarily true, particularly in semiarid conditions.               All in all, a constant rate of abstractions was assumed over the time duration of storm for this appendage. This conclusion was reached to limit the number of parameters to be estimated, and eliminate the inapplicability of  ponding-prevailing conditions in a semiarid region. The inconvenience of this assumption and its influential contribution to the total inaccuracy of the proposed model are tackled in depth later on in this document. 

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Nonlinear Reservoir Model               Most conceptual models are based on the continuity equation and a relationship between the storage and the outflow. The continuity equation is: 

 Where s = depth of storage over the watershed; i = rainfall intensity; f = infiltration rate; and q = rate of direct runoff at any given time t. The outflow is assumed to be nonlinearly related to the storage function: 

             The parameters a and x represent physical characteristics of the watershed, of which the latter quantifies the degree of nonlinearity of the watershed. Substituting the 2nd equation into the 1st yields: 

             This is a first-order nonlinear nonhomogenous ordinary differential equation. The nonlinearity is represented by the outflow being a power function of the storage, and the nonhomogeneity is represented by the excess rainfall term. An approximate solution can be obtained using a perturbation technique called the delta expansion.             Ultimately, after expressing the storage in the form of power series and taking its time derivative, one obtains a sequence of linear first order homogenous ordinary differential equations: 

 With the following solution for zero initial conditions: 

 And the runoff is then obtained as follows: 

 

 

(Where all  a‘s were mistakenly omitted from the original derivation. Accordingly, it was made certain that ln took dimensionless arguments in this appendage.) 

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Parameter Estimation                The model parameters were originally a, x, tr, and the infiltration parameters f0 and k (in the case of an exponentially decreasing rate of abstractions.) By assuming the infiltration to be constant throughout the storm interval, we’ve eliminated the latter parameter and reduced our estimation to the former four.              The parameters a and x represent physical characteristics of the watershed. The exponent x ranges from 1.2 to 1.6 (equal to one for linear systems) and corresponds to the type of storage element, and it can be used to quantify the degree of nonlinearity of the watershed. As to a, this is the major parameter under study in this appendage and is expected to relatively deviate from the previously set value (20 to 25 h-1) in the original publication, which erroneously disregarded the parameter’s presence in all ln arguments. After setting initial estimates to those parameters falling within the mentioned ranges, the estimates are then further refined using nonlinear least-squares regression.              The lag between the inflow and the outflow can be directly related to tr, the lag imposed by the linear channel, which resembles the difference in time between the first time reading of observed runoff on LH-06 and the first time reading of precipitation from RG-83. As mentioned earlier, the exponential argument k defines the time distribution of the infiltration. Settingk = 0, yields a constant infiltration rate, whereas as k increases, the infiltration in the initial stages increases. In reference to the Abstractions section of this paper, for a depth of runoff rd = Vt/A, where Vt is the total volume of observed runoff, the parameter f0 can be estimated by solving:  

  For a constant infiltration, the evaluation of this equation is straightforward and allows for the determination of one of the parameters f0.  

             All in all, the parameters are estimated such that: a) the residual between the computed and the observed runoff is minimum; and b) the depth of excess rainfall is equal to the depth of observed direct runoff over the watershed. The implementations of all estimation-related observations are to be demonstrated in the Application 

RAINFALL - RUNOFF MODEL

Main principles of the model

The rainfall - runoff model is an hydrological model. It determines the runoff signal which leaves the watershed basin from the rainfall signal recived by this basin. The model used here uses a lagragian approach. The originality of this overland model by direct simulation is that it integrates a maximum of data from satellite Earth observation and from hydrological radars.The following figure summarises the main principles of the model:

Inputs

o Rainfall

Spatial and temporal rainfall heterogeneities are taken into account by the model. 

This model takes into account the quantity of rainfall estimated from the hydrological radars of Nimes and Toulouse, belonging to the Aramis network of Meteo France, and from rainfall direct measurements realised by the DDE11. So, rainfall can be estimated spatially and temporally, but due to the distance between the radars and the observations, these measurements are not fully accurate.

     

o Digital Elevation Model (DEM)

Knowledge of ground topography by digital elevation models

The digital elevation model comes from the altimetric data base provided by IGN (BD ALTI ®). It covers the Aude department and is essential to determine the trajectory of all the masses of

water running over the basin.  

o Land use maps

Ground roughness evaluation using a land use map

In addition to the DEM and slopes, this information is essential. It makes it possible to calculate travel time of water particules from one point to the catchment area to another for the rainfall - runoff model and for the hydraulic model in the flood plain.

Outputs : flood hydrographs

The rainfall - runoff model provides flood hydrograph at the outlet of the area catchment (or at any point of the basin). This one makes it possible to evaluate some characteristics like maximum flow and elevation time. These data can be compared to the measurements realised by the DDE11. At this stage of the study, comparisons carried out using the few available data show a good agreement.

Another kind of relevant information can be obtained from the model : determination of the watershed boundaries. The map below shows the superimposition of the Repudre catchment area determined by the DDE11 (by a traditional but not automatic method) and by the model (by an entirely automated method).

              

  FLOOD PROPAGATION MODEL

Main principles of the model

Saint Venant's equations:

Barre de Saint Venant's equations (1887) describe free surface flow in shallow water. These equations provide water heights and velocity components surfacic fields along the vertical and versus time (for more details please contact IMFT).

Inputs

 

o Flood hydrographs in each point of the frontier

Relation between runoff on the uphill basin (rainfall - runoff relation) and flood downstream (Saint Venant in flood plain) by hydrographs in each point of the frontier.

o DEM

BD Alti :

Knowledge of ground topography by Digital Elevation Models.

The digital elevation model comes from the altimetric data base provided by IGN (BD ALTI ®). It covers the Aude department and is essential to determine travel time of all the masses of water running over the basin.

Influence of DEM quality on modelisation

o Problems linked to the use of DEM

Example of the Orbieu valley: problem as regards bed river determination.

The problem of the linear river profile acknowledgment from DEM carries to false runoff areas. These areas can be either very flat areas or areas for which DEM resolution and interpolations hide some altitude variations.

   

Consequences linked to the problem of the Orbieu river determination in flat area.

Chosen solution : "Dig" the DEM to make the river pass on the true place. Earth observation imagery makes it possible to know exact location of rivers.

o DEM processing

Necessity to filter the DEM

Example of linear river profile in the Aude river.

o DEM influence on hydraulic modelisation outputs

DIFFERENCES : Maximum flow : 800m3/s instead of

1000m3/s elevation time : 1h difference

COMMON POINTS :  Elevation start 

Total overflowed volume

DEM characteristics influence strongly hydraulic outputs. Observed data do not allow to choose the best DEM. (Note that subsidence is not

modelised here.)

Outputs

Example of flood expansion

The figure above shows water heights and velocity components fields near the confluence between the Aude and the Orbieu rivers.

TEST AREA OF CUXAC D'AUDE    

More precise data make it possible to better study this area which was dramatically touched by the events.  

Inputs :o BD TOPO

Topography

In this BD TOPO, we can find topographic data (DEM), but also rivers, dikes, channels, SNCF embankments... Modelised area has been limited to an elevation lower than 27 m IGN.

 

Grid for the model

Studied area is cut up into triangular meshes of variable density according to the proximity of rivers (about 20 m mesh closed to the river and 200 m far from it).

 

grid of 18711 nodes

o Embankment bursting at Salleles d'Aude (*)

When the flood reached its highest level, the embankment of Salleles did not support any longer the 2500 m3/s of water. The embankment bursted and the SCNF embankment also. Calculation leads to the simulation of this event (see animation page) and of the flood that inundated Cuxac d'Aude and its plain (see figure below).

 

When the dike   bursted, the SNCF embankment took off. We can see the railway of the "petit train du Minervois" at Salleles d'Aude.

Photo extracted from the special publication "l'apocalypse" by the dailies Midi Libre and Independant.

o Submersion of Cuxac d'Aude by overflow on the left bank (*)

This local view shows heights water and velocity components fields around Cuxac d'Aude. Overflow can be seen on the left bank of the river Aude.

 

(*)  These results correspond to the first calculation using a local model. Upstream flood hydrograph is not reproduced (constant flow at the maximum stage of the flood) but, in spite of the lack of parameter calibration, these results are interesting as they show globally the mechanisms observed.

 FLOOD EXPANSION AND EMBANKMENT BURSTING MODELISATION  AT SALLELES

D'AUDE

    

             

After a fixed view showing topography and grid, the simulation shows the arrival of a 2500 m3/s flow through the Aude river and its propagation through the basin downstream. Then, the embankment of the channel at Salleles d'Aude is partially destroyed, bringing out the SCNCF embankment dowstream. All the water, retained upstream by the embankment until this time, drains all over

Cuxac d'Aude plain downstream. Some images of the simulation superimposed to the p+xs image   .

EARTH OBSERVATION DATA     

Digital Elevation Model

DEMs can be generated using Earth observation optical or radar imagery. Nevertheless, as the study completion time was too short to use DEMs derived from remote sensing data, IGN DEMs were used (BD ALTI® et BD TOPO®).

Land use maps used to determine roughness coefficient for modelling

 

o Choice of Earth observation data versus area

 

o Land use maps with 50m resolution (the same as BD Alti DEM)

 

 

o Land use map of Cuxac d'Aude area

Land cover and elevation level maps

Flooded area

Comparison between interpreted flooded area photo and modelisation result.

     

Updated information plan

Earth observation data could be used for mapping recent urbanised areas of towns concerned by a PPR (Risk Prevention Plan). 

 

zoom realised on the Ikonos image

(1 meter resolution)

   

CONCLUSION    

Validity of the proposed approach

Flash flood phenomenon has serious human and economic impacts. Efforts have to concentrate on improving prevention and forecasting methods in order to help allevaite the impacts of these events.The proposed modelisation is directed towards operational use.

The numerous physical phenomenon which appear during a flash flood are complex. Modelisation is composed of two indissociable and complementary models : the rainfall - runoff model and the flood propagation model. As forecasting models and techniques based on a statistical approach are not very adapted to flash floods, we have tried to qualify modelling approaches based on available data in order to obtain relevant results even if they are not very precise. The whole approach has nevertheless been validated by comparing modelling results with available measurements. We could mention in particular the reconstitution of the flood hydrograph in Moussoulens and the comparison with flooded areas on left bank near Cuxac d'Aude .

Contribution of Earth observation

From the sue of Earth observation data, we have clearly shown that:

- The digital elevation model has a determinant role on the quality of the modelisation results. High resolution images are valuable data for modelling at a smaller spatial scale (z accuracy is lower than one meter);

- The available digital elevation models make it possible to obtain relevant information, but data processing has to be improved to obtain better slope measurement accuracy;

- The land use map allows runoff characterization which has significant impact on forecasting. Nevertheless further studies have do be carried out to evaluate better roughness coefficients;

- When waters have subsided (which can be seen on images several days after the flood) post crisis analysis and evaluation of the modelisation results quality are possible.

Applications to prevention and crisis management

In this field, observations are difficult and available measurements are not fully reliable. The tested modelisation provides results which have nearly the same level of reliability, but it makes it possible to anticipate or to test scenarios before crisis occurrence.

The rainfall - runoff model programming allows its use in real time both for forecasting and prevention activities. The part of the calculation which needs no information about the flood development can be done once for all. Rainfall forecassting is progressively introduced into the model to provide forecasting hydrographs in a few seconds.

In its last version, the flood propagation model at a large scale will need a huge calculation volume which will make its use in real time impossible for forecasting/alert activities due to a too long answer time (a few days) despite permanent increase in calculation power. The flood propagation model is of most interest for prevention applications as it reliably reproduces flood scenarios at large and small scales.

Limits and future developments

These models can be improved even if their results are nearly the same as those obtained from observations. Their study still belongs to the research field. Efforts have still to be done to transfer this methodology to other flood types and countries.

This study shows that space imagery techniques can be used for natural risk management in an

operational context.     

Hydrological Theory

Calculating Runoff

Rainfall Runoff Models

 From the MIDUSS Version 2 Reference Manual - Chapter 7

(c) Copyright Alan A. Smith Inc.

Rainfall runoff models may be grouped in two general classifications that are illustrated in Figures 7-16 and 7-17.  The first approach uses the concept of effective rainfall in which a loss model is assumed which divides the rainfall intensity into losses and an effective rainfall hyetograph.  The effective rainfall is then used as input to a catchment model to produce the runoff hydrograph.  It follows from this approach that the infiltration process ceases at the end of the storm duration. 

Figure 7-16 - A rainfall-runoff models using effective rainfall.

An alternative approach that might be termed a surface water budget model, incorporates the loss mechanism into the catchment model.  In this way, the incident rainfall hyetograph is used as input and the estimation of infiltration and other losses is made as an integral part of the calculation of runoff.  This approach implies that infiltration will continue to occur as long as the average depth of excess water on the surface is finite.  Clearly, this may continue after the cessation of rainfall.

Figure 7-17 – A rainfall-runoff model using a surface water budget

MIDUSS  allows you the option to use particular implementations of both these techniques.  The effective rainfall approach is employed in a convolution algorithm that uses response functions of different shape.  For the case of triangular or rectangular response functions the time base is computed by a kinematic wave equation which involves the intensity of the effective rainfall.

The convolution process is therefore nonlinear in that the response function changes throughout the storm but the principle of superposition is retained.  These two approaches are embodied in the ‘Rectangular’ and ‘Triangular SCS’ options of the Hydrology/Catchment command. 

The third convolution option uses a response function which is obtained by routing a rectangular input of duration  t and height A/t through a linear reservoir with a lag or storage coefficient KL=tc/2.  For this case, the time of concentration tc is computed using the kinematic wave equation [7-41] but for the maximum value of effective rainfall intensity.

An example of a surface water budget model is also made available in the form of the SWMM/RUNOFF algorithm and can be implemented by using the ‘SWMM method’ option.  It is important to note that if the ‘SWMM method’ option is to be employed it is necessary to use the Horton or Green and Ampt infiltration models to represent the rainfall losses.  The four options are described in the sections that follow.