Simplification of a Comprehensive Hydrological Model for Scenario Analysis

Working Paper

SIMPLIFICATION OF A COMPREHENSIVE

HYDROLOGICAL MODEL FOR SCENARIO ANALYSIS

P.E.V. v a n Walsum Y. Nakamori

December 1985 WP-85-92

International Institute for Applied Systems Analysis

A-2361 Laxenburg, Austria

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

SIMPLIFICATION OF A COMPREHENSIYE

HYDROLOGICAL MODEL FOR SCENARIO ANALYSIS

P.E.V. van Walsum Y. Nakamori

December 1985 WP-85-92

Working Papers are interim r e p o r t s on work of t h e International Institute f o r Applied Systems Analysis and have r e c e i v e d only lim- ited review. Views o r opinions e x p r e s s e d h e r e i n d o not neces- s a r i l y r e p r e s e n t t h o s e of t h e Institute o r of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

PREFACE

R e s e a r c h by t h e R e g i o n a l Water P o l i c i e s Project of IIASA i s focused on t h e design of decision s u p p o r t systems t o a s s i s t in t h e analysis of r a t i o n a l water policies in regions with intense a g r i c u l t u r e a n d with open-pit mining activities. One direction of t h i s r e s e a r c h i s aimed at t h e elaboration of sim- plified models of i n t e r r e l a t e d groundwater p r o c e s s e s , c r o p growth p r o c e s s e s , basing o n available comprehensive and o t h e r models.

One of t h e methods used by t h e p r o j e c t f o r t h i s p u r p o s e i s based on t h e development and application of t h e I n t e r a c t i v e Modeling S u p p o r t System f o r Model Simplification d e s c r i b e d r e c e n t l y by Y. Nakamori et a l . in WP-85-77.

This p a p e r outlines a c o n c r e t e application of t h i s system in t h e c o n t e x t of t h e study f o r t h e S o u t h e r n P e e l region in t h e Netherlands

S e r g e i Orlovski P r o j e c t Leader

Regional Water Policies P r o j e c t

SIbfPLIFLCATION OF A COMPREHENSIVE

HYDROI.OGICAL MODEL FOR SCENARIO ANALYSIS

P.E.V. van Walsum and Y. Nakamori

1. INTRODUCTION

Intense a g r i c u l t u r a l development in many regions of t h e world puts a n increasing p r e s s u r e on t h e environment both by consuming water r e s o u r c e s and by discharging pollutants t h a t a r e hazardous t o t h e population and t o n a t u r a l ecosystems. Apart from being a r e s o u r c e t h a t i s vital f o r socio- economic development and f o r t h e evolution of n a t u r a l ecosystems, a regional water system i s a basic medium through which local human i n t e r - ventions p e n e t r a t e t o and are "felt" in o t h e r p a r t s of a region. I t i s t h e l a t t e r a s p e c t t h a t lends regional water systems t h e i r complexity. This gives r i s e t o a demand f o r t h e design of decision s u p p o r t systems t h a t can help regional decision makers in formulating policies aimed at providing a satis- f a c t o r y balance between t h e a g r i c u l t u r a l development on t h e one hand and t h e development of t h e environment on t h e o t h e r .

Using a n example region in t h e Netherlands. a prototype of such a decision s u p p o r t system h a s been developed within t h e framework of IIASA's Regional Water Policies P r o j e c t (RWP, in press). Methodologically t h e sys- tem i s based on t h e use of a two-stage decomposition, with s c e n a r i o analysis

in t h e f i r s t s t a g e and policy analysis in t h e second. The s c e n a r i o analysis s t a g e i s d i r e c t e d towards generating scenarios of t h e potentially rational development of t h e regional system, as seen from t h e regional perspective.

A set of coupled "comprehensive" models t h a t a r e state-of-the-art mathematical descriptions of relevant socio-economic and environmental p r o c e s s e s i s t h e b e s t tool f o r evaluating s c e n a r i o s in t e r m s of regional objective function values (e.g. income from a g r i c u l t u r e , nitrogen concen- t r a t i o n of groundwater). However, due t o t h e i r complexity and high compu- tational demand, comprehensive models a r e not suitable f o r screening ana- lyses using mathematical programming and i n t e r a c t i v e methods f o r multi- objective choice. F o r t h i s reason it i s necessary t o develop r e d u c e d models of t h e same processes. The comprehensive and reduced models a r e t h e n combined into a h i e r a r c h i c a l system, with a n integrated s e t of reduced models on t h e f i r s t level and coupled comprehensive ones on t h e second.

In t h e mentioned study t h e choice h a s fallen on t h e use of l i n e a r reduced models in o r d e r t o t a k e advantage of t h e f a c t t h a t l i n e a r mathemat- ical programming techniques a r e vastly b e t t e r developed than nonlinear ones. For developing one such reduced model from an existing comprehen- sive model of a regional hydrologic system, use w a s made of t h e Interactive Modeling Support System (IMSS) t h a t w a s developed by Nakamori et a1.(1985). The model simplification p r o c e d u r e t h a t w a s followed i s t h e sub- ject of t h i s p a p e r .

For regions hydrologically similar t o t h e example region in t h e Nether- lands, t h e Southern Peel Region, t h e comprehensive ("second level") model FEMSATP has been developed (Querner & Van Bakel, 1984). This model i s based on a finite-element approximation of t h e p a r t i a l differential equation describing t h e regional hydrologic system. Coupled t o FEMSATP i s t h e c r o p production model SIMCROP (Querner & Feddes, in p r e s s ) , which p r e d i c t s t h e e f f e c t s of s o l a r radiation and t h e availability of moisture and nitrogen on t h e a c t u a l c r o p production.

After having given a s h o r t description of t h e example region in t h e Netherlands, with t h e emphasis on those a s p e c t s t h a t a r e of r e l e v a n c e h e r e , namely those pertaining t o water quantity p r o c e s s e s , we p r o c e e d by giving a brief outline of t h e models FEMSATP and SIMCROP and t h e i r application t o

t h e S o u t h e r n P e e l Region. Subsequently a specification is given of some of t h e c h a r a c t e r i s t i c s t h a t t h e reduced model should have

-

t h i s specification follows from t h e intended way of implementing and using t h e reduced model.

This specification is t h e n followed by t h e description of t h e a c t u a l modeling e x e r c i s e and t h e validation of t h e reduced model as a component of t h e s c e n a r i o module.

2. THE SOUTHERN PEEL REGION

The S o u t h e r n P e e l i s a n undulating a r e a of a b o u t 30.000 h a in t h e south of t h e Netherlands. The lie of t h e land v a r i e s in altitude between 17 and 35 m above sea level.

A l a r g e p a r t of t h e area used t o b e c o v e r e d by a l a y e r of p e a t t h a t grew as a consequence of extremely high groundwater levels. Most of t h e p e a t h a s been delved and used f o r heating. The remaining p e a t areas are now p r o t e c t e d from exploitation, because of t h e i r value as r e c r e a t i o n o r n a t u r e areas. These n a t u r e areas can only k e e p t h e i r value if high enough groundwater levels are maintained.

Roughly half of t h e land i s used as p a s t u r e f o r d a i r y c a t t l e ; t h e remaining area is used f o r growing a v a r i e t y of c r o p s , of which maize is t h e most important one, followed by s u g a r b e e t s , potatoes and cereals. F a r m e r s t r y t o r e d u c e moisture deficits by subirrigation and s p r i n k l e r i r r i g a t i o n . Subirrigation i s t h e infiltration of water into t h e bottoms of ditches, t h e r e b y raising t h e groundwater level under t h e neighboring fields; t h i s i n c r e a s e s t h e availability of moisture f o r capillary r i s e t o t h e rootzone.

Sprinkling is a more d i r e c t way of supplying moisture t o t h e soil. Water f o r sprinkling i s pumped from t h e groundwater o r t a k e n from t h e s u r f a c e water supply system. This pumping from groundwater a f f e c t s a g r i c u l t u r a l produc- tion in o t h e r p a r t s of t h e region and a l s o t h e conditions in n a t u r e areas. In t h e S o u t h e r n P e e l t h e s u r f a c e water supply system coincides with t h e d r a i n a g e system. I t consists of some l a r g e r canals and a network of d i t c h e s and b r o o k s with a varying density (Figure 1).

2 and 3 -a-m- 'Peelrand' fault 4

0 1 2 3 4 5 k r n Built-up area

I

-

⁵

^a

Nature area

Figure 1: surface water system of the Southern Peel

3.1. _FEMSATP

FEMSATP is a finite-element model t h a t i s quasi three-dimensional (i.e i t uses a schematization into purely v e r t i c a l flows and purely horizontal flows). F o r advancing through time i t uses a Crank-Nicholson implicit calcu- lation scheme, meaning t h a t t h e flows are calculated using t h e a v e r a g e of t h e hydraulic heads"' at t h e beginning and a t t h e end of a time-step ( Q u e r n e r & Van Bakel, 1984). Using t h e recommended time-step of one week, FEMSATP r e q u i r e s f o r a one-year r u n about 20 min of CPU time on a VAX 11/780 u n d e r Unix.

In FEMSATP t h e s a t u r a t e d groundwater flow i s schematized into purely v e r t i c a l flow in flow-resisting l a y e r s (aquitards) and purely horizontal flow in permeable l a y e r s (aquifers). The p h r e a t i c l a y e r in t h e S o u t h e r n Peel i s modeled as a n a q u i t a r d (Figure 2).

The f i r s t aquifer i s p r e s e n t in both t h e E a s t e r n and Western p a r t of t h e region, but d i f f e r s in thickness. In t h e E a s t e r n p a r t t h i s a q u i f e r lies on t h e hydrological basis t h a t s e r v e s as t h e lower boundary of t h e groundwater flow system. This lower boundary i s p r e s e n t at a much shallower depth in t h e E a s t e r n p a r t t h a n in t h e Western p a r t due to a geological fault t h a t r u n s through t h e middle of t h e region. In t h e Western p a r t a second a q u i t a r d is

*For t h e convenience o f t h e hydrologically non-informed r e a d e r , a glossary o f t e r m s i s provided:

aquifer

-

a geological l a y e r w i t h a r e l a t i v e l y high permeability, t h u s w i t h a low r e s i s - t a n c e t o t h e flow o f groundwater through t h e pores between t h e subsoil particles.

aquitard

-

a geological l a y e r w i t h a r e l a t i v e l y low permeability, t h u s w i t h a r e l a t i v e l y high r e s i s t a n c e t o groundwater flow.

evapotranspiration

-

t h e combined evaporation f r o m t h e soil s u r f a c e and f r o m t h e sur- f a c e s o f crop l e a v e s ; b y potential evapotranspiration i s meant t h e evapotranspiration t h a t would t a k e place under optimal conditions o f m o i s t u r e supply t o t h e soil; b y actual evapotranspiration i s meant t h e amount t h a t o c c u r s under t h e actual m o i s t u r e supply con- d i t i o n s

-

t h e actual value i s lower t h a n t h e potential one.

hydraulic head

-

t h e potential e n e r g y o f w a t e r ; w a t e r f l o w s i n t h e d i r e c t i o n o f t h e s t e e p e s t (downward) gradient o f t h e hydraulic head.

iqfrastructure

-

t h e combined outlay o f canals, hydraulic s t r u c t u r e s e t c . phteattc layer

-

t h e geological l a y e r i n which t h e groundwater t a b l e is.

solar radiation

-

t h e amount o f e n e r g y i n sun r a y s .

layers

groundwater level approx.

network of \ depth

phreatic layer + ^{0 m}

(first aquitard)

. . .

. . .

"

firstaquifer -> . : : : : : : :

.*::::::::: -

. . .

second aquitard 3

/

^--

geolo#cal fault

'wEmERN PART EASTERN PART

Figure 2: Hydrogeological schematization of t h e Southern Peel

followed by a second a q u i f e r t h a t r e a c h e s t o a n a v e r a g e depth of about 325 m below ground level.

For different a s p e c t s of a regional hydrologic system, FEMSATP uses different aggregation levels. A region i s divided into subregions, e a c h with relatively homogeneous soil p r o p e r t i e s and hydrogeological schematization.

The description of t h e water movements in a second-level model r e q u i r e s a n a c c u r a t e r e p r e s e n t a t i o n of t h e geohydrological situation. T h e r e f o r e t h e subregions a r e subdivided into t r i a n g u l a r finite-elements. The Southern P e e l h a s been divided into 31 subregions and into 748 finite-elements.

A s u b r e g i o n i s a l s o subdivided into areas c h a r a c t e r i z e d by d i f f e r e n t t y p e s of land use. These t y p e s of land use are h e r e t e r m e d "technologies".

A p a r t from a g r i c u l t u r a l technologies, t h e model allows f o r t h e specification of built-up areas. n a t u r e r e s e r v e s a n d f o r e s t s . The typification of a n a g r i - c u l t u r a l technology includes among o t h e r t h i n g s w h e t h e r i t involves s p r i n - kling o r not. Of e a c h technology t h e area h a s only t o b e known as a p e r c e n - t a g e of t h e s u b r e g i o n , e i t h e r from collected d a t a a b o u t t h e c u r r e n t state o r from a t a r g e t s c e n a r i o t h a t i s g e n e r a t e d by a " s c e n a r i o module". The model a b s t r a c t s , however, f r o m t h e geometrical position(s) of a technology within a subregion: t h e t o t a l area of a technology may in r e a l i t y b e p r e s e n t as numerous p o r t i o n s of land s c a t t e r e d o v e r a s u b r e g i o n .

Figure 3: Schematization of flows in a s u b r e g i o n

The v a r i o u s w a t e r t r a n s p o r t a n d s t o r a g e p r o c e s s e s are simulated by t h r e e d i f f e r e n t submodels. They r e p r e s e n t t h e s a t u r a t e d zone, t h e unsa- t u r a t e d zone, a n d t h e s u r f a c e w a t e r system. The v a r i o u s w a t e r movements allowed f o r within t h e schematization of a s u b r e g i o n a n d between t h e t h r e e

submodels are shown in Figure 3. In this figure t h e summer situation is shown, with subirrigation and a supply of water towards t h e subregion.

3.2. S M C R O P

SIMCROP is a c r o p growth model t h a t r e q u i r e s as input d a t a t h e actual evapotranspiration d a t a from FEMSATP and t h e nitrogen application values from t h e nitrogen submodel of t h e scenario module (that is not described h e r e ) . Data of s o l a r radiation are also needed. Output of t h e model is in t e r m s of d r y matter production. If c e r t a i n economic d a t a a r e also provided (yield p e r kg d r y matter, and fixed cost p e r unit of a r e a ) , then t h e model also supplies t h e monetary yields of t h e c r o p s and t h e totals of income f o r t h e subregions and for t h e whole region.

4. SCENARIO ANALYSIS USING FEMSATP-SIMCROP

4.1. Scenario analysis procedure

The s c e n a r i o analysis p r o c e d u r e t h a t h a s been described in RWP(in p r e s s ) i s schematically depicted in Figure 5. The "scenario requirements"

t h a t t h e "user" h a s to specify pertain to t h e requirements on multi- objectives f o r t h e t a r g e t s c e n a r i o of regional development. The used pro- cedure for multi-objective choice consists simply of asking t h e u s e r to specify bounds on N-1 of t h e N objectives. An integrated set of (linear) models coupled to t h e linear programming system GEMINI-MINOS (LebedevJ984) then optimizes t h e N-th objective, provided t h a t t h e

N-1 requirements are feasible. The N-th objective h a s been taken as t h e sum of t h e investments t h a t would b e r e q u i r e d t o instantaneously t r a n s i t f r o m t h e c u r r e n t state to t h e t a r g e t scenario. These investments are minim- ized, because t h e l e s s t h e r e q u i r e d investments, t h e higher t h e probability t h a t t h e s c e n a r i o i s r e a c h a b l e through taking policy measures.

A f t e r obtaining a n "optimized" s c e n a r i o a r u n i s made with t h e second- level models, in o r d e r to obtain a m o r e a c c u r a t e estimate of t h e s c e n a r i o obtained a t t h e f i r s t level. Of special i n t e r e s t to t h e u s e r are of c o u r s e t h e objective function values obtained at t h e second level.

Figure 4: S c e n a r i o analysis p r o c

USER

^{\ 1}

( "RPMA

^"

)

^{1 \}

In t h e subsequent sections, descriptions are given of various types of variables t h a t play a r o l e in using FEMSATP-SIMCROP f o r s c e n a r i o analysis and t h a t a l s o a r e of r e l e v a n c e f o r t h e reduced model.

Interactive system with colour graphcs

Database of generated

scenarios Scenario

requirements

\ /

\

\

First _-level models

- - -

GEMINI-MINOS

^/

Ioptimized' scenario

\v

\

/ /

\ 4

Second-level models

Simulated' scenario

^{. \ /}

4.2. Fixed parameters and control variables

Of t h e regional c h a r a c t e r i s t i c s some a r e rigidly fixed, and a r e not modifiable by a regional authority (i.e. o u r "User"). Fixed p a r a m e t e r s a r e f o r instance t h e a q u i f e r permeabilities; a l s o t h e i n f r a s t r u c t u r e f o r s u r f a c e water supply t o t h e subregions i s considered t o b e non-modifiable. Condi- tions t h a t can b e modified, h e r e denoted by "control variables", are f o r instance t h e s u r f a c e water supply t o t h e region as a whole and t h e alloca- tion t o t h e d i f f e r e n t subregions (which i s , however, s u b j e c t t o t h e con- s t r a i n t imposed by t h e s u r f a c e water supply i n f r a s t r u c t u r e ) . The following control v a r i a b l e s a r e r e l e v a n t h e r e :

-

a r e a p e r c e n t a g e s of technologies, z ( j , r , k ) ;

-

capacities of sprinkling from s u r f a c e water and groundwater, s, ( r ) and g , ( r > ;

-

allocation of s u r f a c e water supply t o a subregion. S , ( r ) ;

The index k indicates whether a technology involves sprinkling ( k = l ) o r not (k =O). By capacities of sprinkling a r e meant t h e available flow-rate capacities of sprinkler-cannons and accompanying pumps. An i n c r e a s e of t h e s e capacities in comparison t o t h e capacities in t h e c u r r e n t s t a t e . r e q u i r e s of c o u r s e a c e r t a i n amount of investments.

The listed c o n t r o l v a r i a b l e s a r e s u b j e c t t o constraints t h a t d e r i v e from t h e fixed p a r a m e t e r s . These constraints a r e described in Kettun e t al. (in press).

4.3. State variables

The p a i r of models FEMSATP-SIMCROP compute a whole host of s t a t e variables f o r e a c h time-step; f o r t h e Southern P e e l a time-step of one week h a s been used. Various operating r u l e s a r e included in t h e model, like:

-

a soil moisture threshold f o r applying sprinkling;

-

a water-level threshold f o r supplying surface water t o a subregion.

(When t h e soil moisture depletes t o below t h e threshold value, sprinkling is applied; when t h e water level d r o p s below t h e threshold value s u r f a c e water supply is activated.) Since t h e r e s u l t s obtained from FEMSATP were found t o b e not very sensitive t o t h e c r i t e r i a used in t h e operating rules, t h e optimi- zation of t h e s e r u l e s is not considered here.

Only a limited number of s t a t e variables a r e of d i r e c t i n t e r e s t in t h e described scenario analysis procedure. These a r e :

-

c r o p evapotranspirations and c r o p productions;

-

volumes of sprinkling water e x t r a c t e d from s u r f a c e water and groundwa- t e r ;

-

volumes of sprinkling water applied t o a r a b l e land and grassland;

-

amount of subirrigation by infiltration of s u r f a c e water in t h e ditches;

-

groundwater levels in n a t u r e a r e a s at t h e end of summer.

The evapotranspirations a r e needed f o r t h e interpretation of t h e results;

t h e c r o p productions a r e also needed f o r this purpose, but t h e main reason f o r needing them is of course f o r computing t h e income from agriculture.

The volumes of sprinkling water and subirrigation water are of i n t e r e s t f o r interpretation and also f o r o t h e r aspects of t h e s e t of models describing t h e whole regional system. The groundwater levels in n a t u r e areas at t h e end of summer are required as objective functions in t h e model: a pro- cedure has been developed f o r interpreting t h e s e levels in t e r m s of t h e i r effect on natural ecosystems of t h e type p r e s e n t in t h e Southern Peel.

4.4. Uncontrollable variables: method of dealing with uncertainty Lastly, t h e r e a r e t h e "uncontrollable" variables, namely t h e meteoro- logical conditions. Owing t o t h e s e uncontrollable variables, in t h e (inter- mediate) formulation of t h e mathematical problem t h a t is t o b e solved in looking f o r a scenario, t h e r e a r e chance constraints f o r t h e agricultural income and t h e groundwater levels in t h e nature a r e a s . These are dealt with by means of t h e so-called deterministic equivalent approach t o chance

c o n s t r a i n t s containing s t o c h a s t i c variables. This implies t h a t when we use t h e reduced models f o r scenario-analysis, t h e values of u n c e r t a i n parame- ters a r e fixed p r i o r t o a c t u a l running of t h e mathematical programming algorithm (in t h i s c a s e ' t h e Simplex algorithm f o r l i n e a r programming). S o t h e simplified models only have t o b e l i n e a r in c o n t r o l variables; t h e coeffi- c i e n t s of t h e s e v a r i a b l e s may however b e functions of t h e uncontrollable variables, because p r i o r t o a r u n with t h e s c e n a r i o module t h e values of t h e s e uncontrollable v a r i a b l e s are fixed, t h u s making t h e model l i n e a r after all.

5.

MODEL SIMPLIFICATION

5.1. Introduction

F o r t h e derivation of a reduced model from t h e comprehensive model FEMSATP-SIMCROP, w e make use of a computer-assisted modeling p r o c e d u r e called t h e I n t e r a c t i v e Modeling S u p p o r t System (IMSS) t h a t was developed by Nakamori et a1 (1985). IMSS i s implemented on a micro-computer; t h e p r e s e n t version consists of 5 0 subprograms and r e q u i r e s f o r s t o r a g e more t h a n 600 KB computer memory. I t combines a l g e b r a i c and g r a p h - t h e o r e t i c a p p r o a c h e s to e x t r a c t a trade-off between human mental models and regression-type models based on t h e use of numerical d a t a . The modeling p r o c e s s of IMSS consists of t h r e e s e p a r a t e s t a g e s of dialogues. The f i r s t s t a g e i s f o r p r e p a r a t i o n of t h e modeling, including input of measurement d a t a and t h e initial v e r s i o n of t h e cause-effect relation on t h e set of vari- a b l e s , transformation of v a r i a b l e s , d a t a screening, and refinement of t h e cause-effect relation. The second s t a g e i s devoted to finding a trade-off between t h e measurement d a t a and t h e modeler's knowledge a b o u t depen- dencies between t h e v a r i a b l e s . The t h i r d s t a g e dialogue i s r e l a t e d to sim- plification or e l a b o r a t i o n of t h e model obtained a t t h e second s t a g e .

P r i o r t h e a c t u a l use of IMSS, a decision had to b e made with r e s p e c t to t h e way of dealing with t h e uncontrollable v a r i a b l e s , i.e. t h e meteorological conditions, and based on t h i s decision a s e r i e s of simulation experiments were performed with t h e comprehensive model. These experiments could not b e performed on t h e micro-computer, and had t h e r e f o r e to b e done o n

t h e mini-computer t h a t t h e model is now resident in (VAX 11/780 of IIASA).

And b e f o r e t h e simulation d a t a could b e t r a n s f e r r e d from t h e mini- t o t h e micro-computer, a decision had t o made with r e s p e c t t o t h e time-step t o be used in t h e reduced model, which determined t h e temporal aggregation t h a t is applied t o t h e r e s u l t s of t h e simulation b e f o r e t h e y got t r a n s f e r r e d t o t h e micro-computer.

Lastly, since w e w e r e dealing with a large-scale hydrological system, i t w a s n e c e s s a r y t o decompose in t h e system into smaller components; o t h e r - wise t h e modeling system IMSS could not "digest" t h e masses of d a t a t h a t even remain a f t e r temporal aggregation; such a decomposition a l s o h a s advantages with r e s p e c t t o t h e interpretability of t h e r e s u l t s t h a t a r e pro- duced when t h e developed model gets used. In e f f e c t , t h i s decomposition is a s t r u c t u r i n g of t h e reduced model. This i s in line with t h e emphasis t h a t t h e system IMSS places on s t r u c t u r a l considerations.

5.2. Preparation of data for LMSS

5.2.1. Treatment of uncontrollable variables

Since t h e values of uncontrollable variables a r e fixed b e f o r e making a r u n with t h e s c e n a r i o module, i t would b e possible t o use a different reduced model f o r e a c h possible combination of uncontrollable variables.

The method would, however, r e q u i r e t h e construction of a l a r g e number of such models, which is time-consuming and not v e r y p r a c t i c a l . Also, such a s e r i e s c a n not provide answers t o "questions" with r e s p e c t t o meteorologi- c a l conditions t h a t o c c u r r e d a f t e r t h e construction of models w a s com- pleted. S o h e r e t h e choice was made t o c o n s t r u c t a single model.

5.2.2. Design of simulation experiments with FEMSATP-SRdCROP For designing a s e r i e s of simulation experiments with FEMSATP- SIMCROP, we used t h e following procedures.

Because t h e s e r i e s of available d a t a were judged t o b e too s h o r t f o r t h e purpose of deriving a reduced model t h a t i s valid o v e r a wide r a n g e of conditions, t h e available "real" d a t a f o r 12 y e a r s were expanded t o a s e r i e s f o r 33 y e a r s . This w a s done by p e r t u r b i n g t h e "real data" by adding random

variables with normal distributions; if a n extremely unrealistic value hap- pened t o be obtained, i t w a s discarded.

For t h e controllable variables, pseudo-random numbers were gen- e r a t e d within t h e constraints t h a t derive from t h e fixed parameters. The complete description of t h e algorithm used f o r generating t h e s e numbers i s given in a s e p a r a t e publication (Kettun e t al., in press).

Although FEMSATP-SIMCROP distinguishes a number of different a r a b l e land use technologies. only one w a s used f o r t h e derivation of t h e reduced model, namely potatoes. The justification f o r this i s t h a t t h e a r a b l e land technologies differ mainly in t h e length of t h e growing season. For comput- ing c r o p productions in SIMCROP, t h e r a t i o between a c t u a l and potential evapotranspiration i s t h e main determining f a c t o r ; t h i s r a t i o is, however, not very sensitive t o t h e length of t h e growing season, because both t h e a c t u a l and potential evapotranspiration i n c r e a s e if t h e length of t h e season i s increased. S o in o u r experiments with FEMSATP-SIMCROP, we used only t h e following f o u r control variables ( p e r subregion) f o r a r e a s of technolo- gies:

-

z ( r ,l,O) : a r e a of 'arable land, non-sprinkled;

-

z ( r ,1,1) : a r e a of a r a b l e land, sprinkled;

-

z ( r ,2,0) : area of grassland, non-sprinkled;

-

^z ( r ,2,1) : area of grassland, sprinkled;

The notations f o r t h e remaining control variables a r e

-

^{s, ( r ) and g, ( r ) :} capacities of sprinkling from s u r f a c e water and ground- water;

-

S , ( r ) : allocation of s u r f a c e water t o a subregion

The notations used f o r t h e s t a t e variables mentioned in Section 4. a r e ( j =1 f o r a r a b l e land, j =2 f o r grassland, k =O f o r non-sprinkled, k =1 f o r sprin- kled) :

-

^e, ( r , j ,k ) and cp ( r , j ,k ) : c r o p evapotranspirations and productions:

-

² ( r ) : volume of sprinkling from s u r f a c e water;

-

i ( r ) : volume of sprinkling from groundwater;

-

i,, ( r ) : volume of sprinkling on a r a b l e land;

-

i g , ( r ) : volume of sprinkling on grassland;

-

^{sf ( r ) :} amount of subirrigation;

-

h , ( r ) : groundwater levels in n a t u r e areas at t h e end of summer ( r =10,16,27).

5.2.3. Choice of time step for uncontrollable variables

Though t h e problem of model simplification can b e viewed as a p r o c e s s in t h e c o u r s e of which t h e most a p p r o p r i a t e time-step f o r t h e r e d u c e d model is chosen and t h e n iteratively adjusted till a point h a s been r e a c h e d where t h e simplification by increasing t h e time-step (cf. FEMSATP's seven days) i s in "balance" with simplification through o t h e r means, w e h e r e h a v e chosen t h e time-step p r i o r t o o t h e r s t e p s of model simplification. W e simply s p l i t t h e y e a r into two halves: t h e winter half preceding a growing season, t a k e n from 1st October till 1st April, and t h e growing season itself. For t h e

"uncontrollable" v a r i a b l e s t h i s t h e n gives:

-

precipitation during winter, p l , and during summer, p 2 ;

-

potential e v a p o t r a n s p i r a t i o n during winter, e p , l , and during summer, %,2.

5.2.4. Decomposition of the regional system

F o r t h e p u r p o s e of decomposing t h e regional system into a set of sub- systems t h a t are connected t o e a c h o t h e r through t h e a q u i f e r system in t h e subsoil, w e defined t h e following i n t e r m e d i a t e state variable:

where gk ( r )

-

intermediate v a r i a b l e

ig ( r )

-

volume of sprinkling from groundwater during one summer Lk(r)

-

volume of "leakage" from t h e p h r e a t i c l a y e r t o t h e f i r s t aquifer.

The defined intermediate variable g k ( r ) c a n b e s e e n as t h e "impact" t h a t a subregion h a s on t h e regional system: ig ( r ) i s a n e x t r a c t i o n from t h e f i r s t aquifer, and Lk ( r ) i s a flow t o t h a t aquifer. S o [iQ ( r ) -Lk ( r ) ] i s t h e com- bined (negative) e f f e c t of iQ ( r ) and Lk ( r ) on t h e (summer) water balance of t h e p a r t of t h e f i r s t a q u i f e r t h a t i s directly beneath a subregion. The leak- a g e Lk(r), however, i s not only influenced by t h e activities in a subregion itself, b u t a l s o by t h e activities in t h e surrounding subregions: a n i n c r e a s e of t h e values of g k ( r ) in t h e surrounding subregions will "induce" a l s o a l a r g e r value of t h e leakage due t o t h e "sucking away" of water caused by t h e increased (negative) impacts on t h e water balances. A l a r g e r value of t h e leakage t h e n means a lower g k ( r ) in t h e c a s e t h a t i t i s positive, o r a more negative one in t h e case t h a t i t is negative (a change of sign is of c o u r s e a l s o possible). S o in t h e reduced model, t h e relationships describ- ing t h e g k ( r ) ' s should h a v e t h e form:

where v r , * , i = l , n a r e t h e v a r i a b l e s (of all types) describing t h e s u b r e - gional system; n, i s t h e number of subregions

-

in t h e Southern P e e l t h e number is 31. The s e t of n, equations of this type t o g e t h e r provide t h e

"linking" of t h e subregional systems.

5 - 2 5 L i n e a r i t y r e q u i r e m e n t f o r reduced m o d e l

Since t h e s c e n a r i o analysis p r o c e d u r e r e q u i r e s t h e a g r i c u l t u r a l income t o b e a Linear function of t h e control variables, t h e c r o p produc- tions and evapotranspirations must be in volumes and not in volumes p e r unit a r e a : In t h e latter case t h e values would have t o be multiplied by t h e r e s p e c t i v e areas in t h e objective function, which would lead t o a quadratic form. So, in o r d e r t o avoid this, t h e evapotranspirations and t h e c r o p pro- ductions obtained from FEMSATP-SIMCROP are f i r s t multiplied by t h e

r e s p e c t i v e areas, and only t h e n p r e s e n t e d t o t h e modeling system IMSS as state v a r i a b l e s f o r which a r e d u c e d model h a s t o b e derived. Since t h e state v a r i a b l e s are in volumes, t h e values of uncontrollable v a r i a b l e s should not only b e in volumes p e r unit a r e a , b u t a l s o in volumes as possible explanatory v a r i a b l e s f o r t h e system IMSS t o use. Each of t h e f o u r uncon- trollable v a r i a b l e s t h u s g e t s expanded t o 5 values ( p e r subregion):

-

t h e value p e r unit a r e a (which i s t h e s a m e f o r a l l subregions);

-

t h e value p e r unit a r e a , times t h e area x ( r ,l,O), t h e value times x ( r . l , l ) ,

-

t h e value times x ( r 3 . 0 ) and t h e value times x ( r 3 . 1 )

.

5.3. Application of IMSS

5.3.1. Introduction

The system IMSS includes t h e submodules shown in Figure 5. These modules are implemented in a n i n t e g r a t e d manner on a microcomputer with a color g r a p h i c a l display.

The system includes facilities f o r

-

d a t a transformation;

-

s t r u c t u r a l analysis;

-

l i n e a r modeling;

-

model verification and validation.

The modeling p r o c e s s of using IMSS consists of t h r e e d i f f e r e n t b u t inter- dependent s t a g e s of dialogues as shown in Figure 6. Of t h e facilities men- tioned above, s t r u c t u r a l analysis is used in all t h r e e s t a g e s , and i s t h e most emphasized f e a t u r e of t h e system.

The first s t a g e diaLogue i s r e q u i r e d f o r p r e p a r a t i o n of t h e modeling, including input of measurement d a t a and t h e initial version of t h e cause- e f f e c t relation on t h e set of variables, transformation of v a r i a b l e s , d a t a screening, and refinement of t h e cause-effect relation.

DATA TRANSFORMATlON

0

CAUSAL INFORMATION

0

LINEAR MODELING

0

I

DIGRAPH MODELING

MODEL ELABORATION

0

Figure 5: Submodules of IMSS

The s e c o n d s t a g e diaLogue i s devoted t o finding a trade-off between t h e measurement d a t a and t h e modeler's knowledge about dependencies between variables. Based on t h e measurement d a t a and t h e initial version of t h e cause-effect relation, using a n option of t h e regression method, t h e computer finds a model t h a t is linear in estimated coefficients. The model is, however, not necessarily linear in t h e variables themselves: they could have been transformed in t h e f i r s t stage. Then t h e corresponding digraph models a r e drawn in o r d e r t o facilitate t h e understanding and elaboration of t h e obtained model. If t h e s t r u c t u r e of t h e model i s modified, t h e affected p a r t s of t h e model a r e again tested by means of regression methods. A s e r i e s of r e c i p r o c a l considerations and calculations by t h e analyst and t h e computer are repeated until t h e s t r u c t u r e of t h e model becomes satisfactory in t h e eyes of t h e analyst.

The t h i r d s t a g e d i a l o g u e is r e l a t e d t o model simplification and ela- boration. Model simplification is based on t h e use of t h e equivalence r e l a - tion, and model elaboration is a n application of r e g r e s s i o n analysis includ- ing t h e hypothesis testing on estimated coefficients, and examinations of t h e explanatory and predictive powers of t h e model.

5 -3.2. Structural considerations

One of t h e main advantages of using t h e system IMSS i s t h e facility f o r t h e s t r u c t u r i n g of r e d u c e d models; t h i s c o r r e s p o n d s to "causal information"

and "digraph modeling" modules in Figures 5 and 6. F o r o u r p u r p o s e of find- ing a simple l i n e a r model, t h e s t r u c t u r i n g of t h e system is mathematically redundant. This is because t h e s t a t i s t i c a l closeness between t h e comprehensive and r e d u c e d models i s t h e dominant requirement on t h e solu- tion t o o u r problem. But in systems analysis, mathematical redundancy i s certainly not synonymous t o uselessness : One of t h e g r e a t benefits of s t r u c - t u r a l consideration i s t h a t i t provides a learning e x e r c i s e about t h e under- lying system (which i s h e r e equated with t h e available comprehensive model of it). The complexity and ambiguity of a system i s in t h e e y e of t h e beholder. P u t differently, t h e complexity and ambiguity t h a t i s p e r c e i v e d depends on t h e quality of t h e mental model t h a t t h e p e r c e i v e r uses f o r understanding a system, which in t h i s case is a comprehensive model.

Digraph modeling c a n provide a visualization t h a t a s s i s t s t h e construction of such a mental model. S o t h e t r a c i n g of causation with t h e aid of a digraph model is a g r e a t help f o r understanding a comprehensive model and t h u s a l s o f o r obtaining a simple model t h a t i s suitable f o r implementation within t h e framework of a s c e n a r i o analysis p r o c e d u r e . A s a byproduct i t can even sometimes help to r e f i n e t h e original comprehensive model itself.

Let us denote t h e set of variables by

The s t r u c t u r a l consideration of t h e reduced model i s important f o r verify- ing whether t h e model behaves grossly in t h e fashion w e intend i t to. By t h e s t r u c t u r e of t h e model i s meant t h e cause-effect relation between variables.

To introduce t h e cause-effect relation, t h e adjacency matrix

cause-ef f ect .relation

system variables basic statistics

measurement data pre-calculations

modification of transformation

A

data screening

JI

model building, hypothesis testing, residual plots, mult i coll ineari ty checking, extrapolation

Figure 6: The interactive modeling support system

A =(aij), i , j =1,2,

...

,m. i s p r e p a r e d ; t h e e n t r i e s a r e defined by 2 if zi certainly a f f e c t s zj

aij

= 1

^{1 if zi} possibly a f f e c t s zj

.

0 if zi n e v e r a f f e c t s zj

To fill in e n t r i e s of t h i s matrix is sometimes quite difficult because t h e s t a t e variables (including t h e intermediate ones) often influence each o t h e r in such a manner t h a t i t i s difficult t o s e p a r a t e causes and effects. S o t h e work r e q u i r e s a d e e p insight into t h e comprehensive model and t h e r e a l world under study. The burden of entering t h e adjacency matrix i s reduced, however, by initially assuming t h e validity of transitive inference. I t i s t h e n possible t o subsequently check t h e resulting adjacency matrix by drawing a digraph corresponding t o i t and modifying i t if necessary.

Mathematically t h e p r o c e s s of deriving a digraph i s as follows.

Let B b e t h e binary relation on S X S defined by ( z i , zj ^{) E} B if and o n l y if a i j # O .

We introduce a digraph D = ( S , B ) where t h e elements of S a r e identified as v e r t i c e s and those of B as a r c s . The v e r t i c e s are r e p r e s e n t e d by points and t h e r e i s a n a r c heading from zi t o z j if and only if (zi , z j ) i s in B. If t h e r e is a path from zi t o z j , we s a y zj i s r e a c h a b l e from zi

.

Apparently t h e digraph D i s t r a n s i t i v e , i.e., if z, i s r e a c h a b l e from zi and zk i s r e a c h - a b l e from z j , then zk i s r e a c h a b l e from zi. T h e r e f o r e , w e c a n r e d u c e D t o t h e condensation digraph DC by grouping mutually r e a c h a b l e v a r i a b l e s and selecting a so-called p r o z y v a r i a b l e in e a c h group. Such a v a r i a b l e

"represents" itself and t h e o t h e r v a r i a b l e s belonging t o t h e group.

Finally, w e obtain a skeleton digraph DS by removing a r c s as long as t h e reachability p r e s e n t in Dc i s not destroyed. If t h i s digraph DS is still complicated, format amendments c a n b e c a r r i e d out t o facilitate i n t e r p r e t a - tion. Those amendments include replacement of v e r t i c e s , pooling of v e r - t i c e s of t h e same level and contraction of v e r t i c e s between adjacent levels.

This digraph DS i s usually highly aggregated and less informative, but visu- alizes t h e system s t r u c t u r e in a c l e a r manner. However, because t h e

cause-effect r e l a t i o n i s not necessarily t r a n s i t i v e , w e often have t o modify t h e digraph DS and t h e corresponding e n t r i e s in t h e adjacency matrix A . I t should b e noted t h a t if t h e digraph DS i s highly condensed, w e should look at t h e original d i g r a p h D s o t h a t t h e modification' of A c a n b e done as w e intend.

After s e v e r a l i t e r a t i o n s , t h e s t r u c t u r e of a subregional model w a s drawn as shown in Figure 7, where t h e full lines indicate t h e unconditional influences in t h e d i r e c t i o n of t h e arrowheads and t h e dotted lines indicate t h e conditional influences. The digraph indicates f o r instance t h a t t h e amount of infiltration of s u r f a c e water s, depends o n t h e "pool" of meteoro- logical v a r i a b l e s a n d o n [S,

-

^s,]

-

being t h e s u r f a c e water supply capacity minus t h e c a p a c i t y r e q u i r e d f o r supporting t h e sprinkling f r o m s u r f a c e water t h a t c a n t a k e place. By "conditional influence" i s meant t h a t f o r instance t h e size of area z ( r .LO) affects only e, ( r ,LO), etc. (The evapo- t r a n s p i r a t i o n s are t a k e n in volumes as explained e a r l i e r . )

The c r o p productions and groundwater levels in n a t u r e areas are not shown in t h e diagram in o r d e r t o avoid i t being c l u t t e r e d . Crop productions are assumed to depend o n both t h e a c t u a l and potential e v a p o t r a n s p i r a - tions; groundwater levels in n a t u r e areas are assumed t o depend on t h e values of t h e intermediate v a r i a b l e s g k ( r ) in subregions surrounding a n a t u r e area, and a l s o on t h e meteorological variables.

The u s e of intermediate v a r i a b l e s in t h e s t r u c t u r e s e r v e s not only t h e p u r p o s e of making full use of t h e s t r u c t u r a l modeling f e a t u r e s of IMSS, b u t a l s o t o provide what one could c a l l stepping-stones f o r t h e r e g r e s s i o n modeling t h a t t a k e s p l a c e in t h e second and t h i r d s t a g e dialogues. Such stepping-stones help d e a l with non-linearities in t h e comprehensive model:

I t i s e a s i e r t o d e r i v e l i n e a r equations f o r two slightly non-linear relation- ships t h a n f o r a v e r y non-linear relationship t h a t is t h e composite of t h e two slightly non-linear ones.

In t h e d i g r a p h of Figure 7 one would p e r h a p s e x p e c t t h e t o t a l i r r i g a - tion capacity [s,

+

g,

1

t o a p p e a r . However, t h i s i s not n e c e s s a r y because t h e information with r e s p e c t t o t h e size of t h i s total capacity i s a l r e a d y contained by t h e sum of t h e sprinkled areas [ z ( r , l , l ) + z ( r ,2,1)]: t h e sum of

t h i s a r e a multiplied by t h e sprinkling capacity p e r unit a r e a yields t h e t o t a l capacity. Since t h i s information is a l r e a d y contained in t h e mentioned sum, t h e method of r e g r e s s i o n modeling "finds t h i s out" when a s e a r c h is made f o r explanatory variables.

Figure 7: S t r u c t u r e of r e d u c e d model. The notation used is t h e same a s intro- duced in s e c t i o n 5.2.2.1 5.2.2.

5.3.3. F i n d i q trade-off structures

The p u r p o s e of t h i s s t e p , which is t h e main objective of t h e "second s t a g e dialogue", i s t o find a trade-off s t r u c t u r e between t h e computer model and t h e mental model. F i r s t a reduced subregional model i s obtained by t h e methods of stepwise o r all-subset regression. (See f o r instance Mosteller &

Tukey, 1977). Then t h e corresponding d i g r a p h s are drawn t o f a c i l i t a t e t h e understanding and e l a b o r a t i o n of t h e obtained model. If t h e s t r u c t u r e of t h e model is modified, t h e affected p a r t s of t h e model are again t e s t e d by t h e r e g r e s s i o n methods. A s e r i e s of r e c i p r o c a l considerations and calcula- tions by t h e analysts and t h e computer are r e p e a t e d until t h e s t r u c t u r e of t h e model becomes s a t i s f a c t o r y with r e s p e c t t o t h e c u r r e n t problem. This p r o c e s s i s summarized in t h e following.

Let us define two s u b s e t s

s,"

and St of S f o r e a c h zi : S,"

=

[ z j ; aji=Z

1,

Sf'

=

f z j ; aji =I

1.

Following t h e terminologies in s t a t i s t i c s , w e call Sf t h e core v a r i a b l e set and S t t h e o p t i o n a l v a r i a b l e set f o r x i . The elements of S," are always chosen as t h e explanatory v a r i a b l e s f o r zi and t h o s e of Sf' are c a n d i d a t e s .

For e a c h zi , if S,"uSf+$ , t h e n t h e coefficients of t h e equation:

a r e identified using t h e simulation d a t a and a r e g r e s s i o n method. The c r i - t e r i o n of goodness of f i t used h e r e i s t h e controlled d e t e r m i n a t i o n c o e m - c i e n t , i.e., t h e s q u a r e of t h e modified coefficient of multiple c o r r e l a t i o n :

where

Zik

is estimates of t h e kth d a t a zU, of t h e v a r i a b l e z i ,

zi

t h e sample

mean of x i , n t h e number of d a t a points and p t h e number of selected explanatory v a r i a b l e s ( x j f s ) . The set of selected v a r i a b l e s (which in any c a s e includes all t h e core variables) includes t h e combination of c a n d i d a t e variables t h a t yields t h e value of R' n e a r e s t t o unity.

Table 1. An example of a subregional model ( f o r r = 2 4 ) . The notation is t h e same as introduced in Section 5.2.2.

Su

=

1.2503D+03

+

1.2916D+01*(Sc -sc)

+

1.7803D+OO*e sub p.2

-

3.0936D+OO*p sub 2

An example of subregional model i s shown in Table 1. The individual i n t e r p r e t a t i o n of t h e coefficients is quite difficult o r impossible. There- f o r e , at t h i s s t e p , w e should check by t h e digraph whether t h e s t r u c t u r e of t h e model i s suited f o r t h e p u r p o s e of s c e n a r i o analysis. The p r e s e n t e d r e s u l t shown in Table 1 is actually t h e one t h a t i s obtained a f t e r s e v e r a l r e p e t i t i o n s of t h i s s t e p and intensive discussions t o modify t h e model s t r u c - t u r e using t h e d i g r a p h s . In a subsequent section more will b e said about t h e n a t u r e of t h e relationships given in Table 1.

5.3.4. Model validation

In t h i s s t e p t h e explanatory and predictive powers of t h e subregional model are examined by t h e following statistics:

-

s t a n d a r d e r r o r s of estimated coefficients,

-

t-ratios of estimated coefficients,

-

s t a n d a r d deviation of residuals,

-

F-ratio against a null hypothesis,

-

controlled deterministic coefficient,

-

c o r r e l a t i o n coefficients, and

-

residuals and predictions.

Although t h e simulation experiment with FEMSATP-SIMCROP w a s designed carefully so t h a t t h e values of of c o n t r o l v a r i a b l e s had a low c o r r e l a t i o n with e a c h o t h e r , i t i s possible t h a t some intermediate v a r i a b l e s are highly c o r r e l a t e d with e a c h o t h e r because of t h e p r o p e r t i e s of t h e comprehensive model. Also, i t i s possible t h a t t h e c o r r e l a t i o n could have been introduced by t h e transformation of t h e c o n t r o l variables; e.g. multi- plication of a l l x ( r , j , k ) with t h e summer precipitation p 2 introduces c o r r e l a t i o n between t h e 4 newly c r e a t e d explanatory variables. If t h e p r e s e n c e of c o r r e l a t i o n means t h a t in t h e relationships f o r which c e r t a i n v a r i a b l e s are t h e e x p l a n a t o r y v a r i a b l e s w e can eliminate some of them as long as t h e reduction does not d e s t r o y t h e cause-effect r e l a t i o n s t r u c t u r e n e c e s s a r y f o r t h e intended use of t h e model. This means t h a t r e l a t i o n s t h a t

in t h e eyes of t h e analyst "must" b e t h e r e but t h a t are not strongly sup- p o r t e d by t h e "statistics", n e v e r t h e l e s s g e t r e t a i n e d in t h e model.

Table 2. Example of a l i n e a r relationship and s o m e s t a t i s t i c s

Subregion 24 Regressand

== >

e, (1,O) Equation No. 1

variable coefficient s t a n d a r d e r r o r t-ratio c o r r e l a t i o n

Sc -c 0.2855D+02 0.2675D+02 0.1067D+01 -.0461

gk 0.7617D+00 0.6178D+00 0.1233D+01 -.3140

ep,2*x(l,0> 0.3571D+00 0.4305D-01 0.8295D+01 0.9827

p2*x(1,0) 0.5097D+00 0.5241D-01 0.9726D+01 0.9848

constant 0.9766D+03

Degrees of Freedom

=

2 3 Adjusted R-Square =0.9932 S.D. of Residual

=

0.6465D+03 F-Ratio=0.9810D+03

T(23,0.05)

=

2.0687 F(4,23,0.05)=2.7955

Table 3. Illustration of t h e predictive powers of relationship given in Table 2.

Subregion 24 Regressand

== >

^{e, (1.0) Equation} No. 1 Case Number Measurement Prediction S t a n d a r d E r r o r

No. 29 0.1566D+05 0.1533D+05 0.6784D+03

No. 30 0.3930D+04 0.4975D+04 0.7164D+03

No. 31 0.9383D+03 0.5715D+03 0.7572D+03

No. 3 2 0.1914D+04 0.1704D+04 0.6988D+03

No. 33 0.1436D+05 0.1328D+05 0.6683D+03

The Number of Cases

=

5 Correlation (meas,pre) =0.9821 Mean S q u a r e E r r o r

=

0.5090D+06 Mean Absolute E r r o r =0.1725D+00

An example of a l i n e a r relationship and some r e l e v a n t s t a t i s t i c s are given in Table 2. F o r a complete description of t h e meaning of t h e s t a t i s t i c a l indicators, t h e r e a d e r i s r e f e r r e d t o Nakamori et al. (1985). Table 3. gives a n example of t h e p r e d i c t i v e powers of t h e derived relationship. The wide

r a n g e of values of t h e evapotranspiration is due to t h e fact t h a t t h e values in mm have been multiplied by [ R of a g r i c u l t u r a l a r e a ] ; t h e area of s u b r e - gion 24 i s 2175. ha. The a c t u a l values of explanatory v a r i a b l e s a r e , how- e v e r , not those obtained through using o t h e r d e r i v e d relationships t h a t t o g e t h e r comprise t h e whole model, but values t a k e n from t h e data. This leads t o a t o o f a v o r a b l e impression of t h e predictive powers, because t h e r e is of c o u r s e a c e r t a i n cumulation of e r r o r s upwards through t h e model h i e r a r c h y .

In t h e derived model a l a r g e amount of c o r r e l a t i o n is p r e s e n t due t o t h e f a c t t h a t t h e model s t r u c t u r e w a s more based on "human e x p e r t i s e " than on "statistical evidence". This w a s done in o r d e r t o obtain a model t h a t could b e a p r o t o t y p e f o r o t h e r regions and not just f o r t h e considered one : i t t u r n e d out t h a t f o r t h e considered region t h e evapotranspiration of non- sprinkled land could v e r y well b e explained just by t h e potential evapotran- s p i r a t i o n and precipitation

-

t h i s c a n b e suspected when one sees how high t h e r e s p e c t i v e c o r r e l a t i o n coefficients are as given in Table 2. (both corre- lation coefficients are h i g h e r t h a n 0.98)

5.3.5. Implementation o f reduced model in the s c e n a r i o module

Before implementing t h e subregional models, t h e equations given in Table 1 were o r d e r e d in such a manner t h a t they form a lower t r i a n g u l a r matrix

-

t h e numbers in t h e column 0 indicate t h e o r d e r . These equations c a n t h e n b e solved by means of forward substitution, which c a n v e r y easily b e done using t h e "matrix g e n e r a t o r g e n e r a t o r " system GEMINI, t h a t w a s developed by Lebedev (1984). The equations connecting t h e subregional models, t h e equations f o r g k ( r ) , c a n not b e o r d e r e d in such a way, however.

S o in t h e LP constraint-matrix t h e s e equations w e r e implemented as equality c o n s t r a i n t s containing in t o t a l 31 unknown gk ( r ) ' s . Since, however, t h i s set of equations w a s d e r i v e d f o r c e r t a i n r a n g e s of gk(r)-values, t h e s e vari- a b l e s are not l e f t completely f r e e in t h e model. Instead, lower and and u p p e r bounds are introduced t h a t are derived from r e s p e c t i v e l y t h e minimum and maximum values ( p e r subregion) p r e s e n t in t h e d a t a set t h a t w a s supplied t o IMSS. In o r d e r t o leave t h e model some freedom t o "extra- polate" t h e d e r i v e d relationships beyond t h e r a n g e s for which t h e y were

derived, t h e r a n g e s of t h e ^{g k} ( r ) ' s were extended by 20% on both t h e lower and u p p e r end; s o in total the r a n g e w a s broadened by 40%.

The equations giving t h e a c t u a l evapotranspirations (which are in v o l u m e s ) a l s o include i n t e r c e p t s . These i n t e r c e p t s imply t h a t even if t h e area of a technology is z e r o , t h e r e is still some evapotranspiration. This p a r a d o x i s explained by t h e fact t h a t w e are h e r e dealing with a s t a t i s t i c a l model, t h a t h a s "maximum validity" f o r t h e a v e r a g e value of t h e area of a technology f o r which t h e evapotranspiration equation w a s derived. Since t h e simulation experiments with FEMSATP-SIMCRIP were done using pseudo- random numbers using f o u r technologies ( a r a b l e land non-irrigated, a r a b l e land i r r i g a t e d , grassland non-irrigated, and grassland i r r i g a t e d ) t h e s e a v e r a g e values are roughly 25% of t h e a g r i c u l t u r a l area. The question a r o s e what t o do with t h e i n t e r c e p t s when one h a s 9 a g r i c u l t u r a l technolo- gies (each with a non-irrigated subtechnology and a n i r r i g a t e d one) instead of 1. The decision was made t o divide t h e i n t e r c e p t s by 9, because t h e a v e r - a g e values obtained by s c e n a r i o analysis can b e e x p e c t e d t o a l s o b e p r o p o r - tionally less. A similar p r o c e d u r e was applied f o r t h e 3 grassland technolo- gies. I t should, however, b e noted t h a t t h i s p r o c e d u r e w a s only applied t o t h e explicit i n t e r c e p t s given in Table 1, and not t o t h e implicit ones t h a t are obtained through t h e forward substitution.

A comparative sample of r e s u l t s obtained with t h e comprehensive model . and with t h e r e d u c e d model are given in Table 4.

5.3.6. Conclusion

Though t h e advantages of using IMSS f o r model simplfication only become fully a p p a r e n t a f t e r having actually used t h i s i n t e r a c t i v e system personally, i t is hoped t h a t t h i s p a p e r will have given t h e r e a d e r a complete enough description of i t s use in o r d e r t o a p p r e c i a t e t h e following main advantages :

-

t h e data-screening f e a t u r e s provide a powerful tool f o r debugging t h e data-set;

Table 4. Validation of t h e reduced model through comparison with comprehensive model

Evapotranspiration and Crop Production (subregion 24)

=

Evapotranspiration

*

^qa

=

Crop Production

=

Reduced Model

*

C

=

Comurehensive Model

- -

E a c t (mm) Qact (mm)

R C R C

potatoes (non-irrigated) ( i r r i g a t e d ) g r a s s land (non-irrigated)

( i r r i g a t e d )

Water Quantity Subregional Variabls (mm) (subregion 24)

R C

sprinkling f r o m groundwater f r o m s u r f a c e water subirrigation

-

t h e s t r u c t u r a l modeling f e a t u r e s are helpful f o r organizing one's thinking with

r e s p e c t to t h e r e d u c e d model and a l s o to t h e comprehensive itself.

-

i t enables r a p i d a c c e s s to t h e set of relationships comprise a r e d u c e d model;

-

i t enables r a p i d validation of t h e reduced model using input d a t a t h a t w e r e not

used f o r t h e modeling itself;

-

i t makes possible t h e e a s y refinement of t h e r e d u c e d model.

REFERENCES

Kettunen, J., Nakamori, Y. a n d Van Walsum,P.E.V. (forthcoming). Application of IMSS t o a r e g i o n a l hydrologic svstem with shallow g r o u n d w a t e r t a b l e s . WP-85-77, IIASA, Laxenburg, Austria.

Lebedev, V.Ju. (1984). System GEMINI ( G E n e r a t o r of MINos input) f o r gen- e r a t i n g MPS-files from fourmula-like d e s c r i p t i o n s of LP-problems.

IIASA S o f t w a r e L i b r a r y S e r i e s , IIASA, Laxenburg, Austria.

Mosteiler, F. a n d Tukey, J.H. (1977). Data analysis a n d R e g r e s s i o n : a second c o u r s e in s t a t i s t i c s . Addison-Wesley, Reading, Massachusetts.

Nakamori, Y ., Ryobu, M., Fukawa, H. a n d Sawaragi,

Y.

(1985). An I n t e r a c t i v e Modeling S u p p o r t System. WP-85-77, IIASA, Laxenburg, Austria.

Q u e r n e r , E . P . a n d Van Bakel, P . J . T. (1984). Description of Second Level Water Quantity Model, Including Some P r e l i m i n a r y Results. Nota 1586.

ICW, Wageningen, The Netherlands.

Q u e r n e r , E. P . a n d Feddes, R.A. (in p r e s s ) . Description of t h e c r o p produc- tion model SIMCROP. Nota x x x , ICW, Wageningen, The Netherlands.

RWP (forthcoming). R e p o r t of IIASA's Regional Water Policies P r o j e c t .