A constrained median statistical routing method and the Pecos River case

A constrained median statistical routing method and the Pecos River case

A. Charnes

Center for Cybernetic Studies, University of Texas at Austin, Austin, TX, USA

R. Dickinson and J. Heaney

Center for Water Resources, University of Florida at Gainesville, Gainesville, FL, USA

S. Duffuaa

Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

This paper develops a mathematical model of constrained nonlinear form for river routing. The data recorded at the gauges are usually estimates of the actualflow. Thus the model automatically makes best-fit adjustments for this data so as to preserve physical and hydrological conditions on the river basin. This best fit is accomplished by minimizing a heavily penalized sum of absolute deviations between the recorded and the adjusted data. The model simultaneously estimates new losses, reservoir storage, spill, and flood inflows. The new estimates conform to physical and hydrological conditions, such as mass balance. The model is applied to the case of the Pecos River and compared to previous studies performed on the river. The results show that a substantial amount of water is owed to Texas. On the average the water owed to Texas per year is 282,000 acre-feet, above the amount proposed by New Mexico by 60,000 acre-feet.

Keywords: median, robust estimation, linear programming, mass balance

Introduction

The paper develops a new mathematical model of nonlinear constrained median form for river routing. The statistical estimates obtained from the model are robust and satisfy hydrological and/or physical conditions, such as conservation of mass. A river routing is a mathematical abstraction of a river which numerically estimates the availability of water in the river at given points and times under specified or assumed conditions. The routing model estimates all gains and depletions on the river in all reaches. The depletions result from such items as reservoir evaporation, channel losses, irrigation diversions, and domestic and industrial water usages. Gains are from such items as river tributaries, flood inflows, and underground con-

Address reprint requests to Dr. Duffuaa at the Department of Sys- tems Engineering, King Fahd University of Petroleum and Minerals, Dhahran-31261, Saudi Arabia.

Received 5 August 1988; accepted 26 January 1989

tributions. In general, all natural and man-made activ- ities are considered in a routing study. The routing begins at the uppermost reach (a reach is a region in the river; see Figure I) and descends to the next lower reach, adding up all gains and losses and at the same time keeping mass balance and all other relevant hydrological conditions. The times used for the estimated braces are whole periods, typically months.

To the best of our knowledge, there have been only two previous routing studies applied to the Pecos River in the legal dispute between Texas and New Mexico. The first routing study’ is contained in Senate document number 109 (SD 109) and the second one* is the so-called Review of Basic Data (RBD). Both routing studies employed the least square method for curve fitting, and they both violate basic hydrological conditions. Both studies exhibit negativef7ood inflow. Flood inflow is calculated in both studies as the residual or the difference between total gains and total losses. Our new method will correct the statistical methodology and ensure conformance to hydrological and physical conditions.

330 Appl. Math. Modelling, 1989, Vol. 13, June 0 1989 Butterworth Publishers

Median statistical routing method: A. Charnes et al.

The new model consists of three major parts:

(4

(b)

(cl

Hydrological adjustment for the estimates. The estimates will be adjusted automatically to conform to hydrological conditions; Mass balance equations for all gauges are to be satisfied. The routing is given in the terms of the new adjusted estimates. Yearly balance satisfaction, that is, the estimated monthly data should add up to or be as close as possible to the yearly recorded data.

The best estimates are obtained by using the robust least absolute value criterion instead of sensitive least squares.3-8 The routing is performed simultaneously for the whole river.

Data recorded at river gauges are usually estimates of flow. In the model the monthly recorded data and estimates developed by using this data are adjusted to conform to hydrological and/or physical conditions.‘y9 The adjustments to monthly recorded data are achieved by minimizing a heavily penalized sum of absolute deviations from the monthly data. Weights are utilized in the objective function in order to force the estimated inflows or outflows to be as close as possible to the monthly or the yearly recorded inflows or outflows. The routing process is sequential in time as well as in reaches, and each run develops all monthly routing for a year simultaneously. For purposes of computation only the nonlinear functions in the model are approximated by piecewise linear ones in order to utilize a linear programming code which can accommodate the large size of this model when applied to the Pecos River.

The model for routing studies was employed in the legal dispute between Texas and New Mexico over the water of the Pecos River. The objective was to deter- mine New Mexico’s obligation for water deliveries at the Texas-New Mexico border. Comparisons of the routing results with the old routings were performed and indicated that substantial amounts of water were owed to Texas by New Mexico.

The body of this paper is organized as follows. The second section presents the new model for a general hypothetical river. In the third section the model is applied and stated for the Pecos River case, taking adequate account of computational considerations. The last section presents conclusions and suggests direc- tions for further research.

Description of a hypothetical river

Consider a hypothetical river with H reaches, R reservoirs, Pgauges, and G springs. The reaches are num- bered beginning at the head of the river and proceeding downward. We only have irrigation demands in some of the reaches. The inflow-outflow functions are developed only for K reaches out of all the reaches, and they are monotone and convex. The fitting or best fit is done for gauges in reaches where we have inflow- outflow functions; it could be done for all the gauges. Figure I depicts the river.

* gage 1

T Reach 1

Reach k-l

spring gain

irrigation retu-

flood inflow-e Reach k

Reach H

Figure 1. Hypothetical river

Formulation and statement of the model for the hypothetical river

Let

&,p) Okp)

observed inflow at gauge p in month i estimated outflow from gauge p in

month i using observed inflow and the inflow-outflow functions p;,, esti-

.fh&,P))

mated reach by reach OkP)

y(ilp) estimated inflow at gauge p in month i estimated outflow at gauge p in

month i x(i,p), y(i,p) new adjusted estimates for inflow and

outflow which satisfy hydrological and physical conditions

w(p), w(i,p) weights on the goals

Note: Inflow at gauge p is the same as what leaves gauge p if p is not at a reservoir or at a diversion point. Outflow at gauge p is what leaves gauge p minus what is lost until it reaches gauge p + 1.

We can express our best-fit objective and constraints as

Min x c yG,p) + z 2 4i,p)lx(i,p) - &,p)l P i P i

subject to

(a) PipMi,P)) - yG,P) 5 0 p= l,...,K i= 1 7 * * . 3 12

(b) mass balance equations for gauge p, p = 1,. . . ,P,

Appl. Math. Modelling, 1989, Vol. 13, June 331

Median statistical routing method: A, Charnes et al.

(c) 5 x(i,p) - 5 Z&p) = 0 i=l i=l

,$ Y(LP) - i{ W,p) = 0 p= l,...,K

To insure feasibility always, one can goal the yearly requirements instead of having them hold exactly as in (c). The pi’s are convex functions, and the spill, leakage, and evaporation can be expressed or approximated by convex functions. Hence, we have a convex programming model. This can be solved by any nonlinear code. In our case we have used piecewise linear approximation and a linear programming code of Ali and Kennington. lo

Pecos River case

In this section we present a general description of the river’s legal framework and the development projects. The model developed in the previous subsection is then specialized to the Pecos case, and necessary computational considerations are brought forward.

Legal framework and major developments on the river

The Pecos River is an interstate stream which rises in north central New Mexico and flows in a southerly direction through New Mexico and into Texas. The stream flows through semiarid regions where demand for water exceeds supply and rainfall ranges between 11 to 14 in/yr. The flow of the river is quite variable. The area of the basin is 35,000 mi2, about 20,000 in New Mexico and the rest in Texas.

The controversy over the allocation of Pecos River water is more than half a century old. During this pe- riod, development on the river has extended over many years. The question of at which stage of development one should base the allocation of the river water supply has been answered legally in the Water Compact of 1947. It stipulated that the water should be allocated between the two states at the stage of development on the river at the beginning of the year 1947. This is referred to as the 1947 Condition. The 1947 Condition is defined as the situation in the Pecos River Basin which produced in New Mexico the man-made depletions from the stage of development existing at the beginning of the year 1947, and the augmented Fort Sumner and Carlsbad acreage.

The major irrigation projects on the Pecos River are the Fort Sumner project of about 5221 acres and the Carlsbad irrigation district of about 22,368 acres. Both are irrigated by diversions from the river. The augmented area for Fort Sumner is 6500 acres, for Carls- bad 25,055 acres. There is a huge pumpage activity between Acme and Artesia in the Roswell area. an irrigated area of 91,744 acres. Reach-by-reach description is section.

See maps in Fig& 2. given in the next sub-

332 Appl. Math. Modelling, 1989, Vol. 13, June

Development of the river routing

The map in Figure 2 shows the area from Alamo- gordo to the New Mexico-Texas border. We developed the routing study sketch in Figure 3, from which we have four reaches:

(1) Alamogordo reservoir to Artesia (2) Artesia to McMillan dam (3) McMillan dam to Carlsbad (4) Carlsbad to state line

We also have four reservoirs:

(1) Alamogordo reservoir of capacity 132,200 acre feet; (2) McMillan reservoir of capacity 38,600 acre feet; (3) Avalon reservoir of capacity 6000 acre feet; (4) Red Bluff reservoir 8 miles south of the state line

of capacity 310,000 acre feet

We have eight flow gauges at

(1) Alamogordo dam (2) Fort Sumner project (3) Acme (4) Artesia (5) McMillan dam (6) Avalon dam (7) Carlsbad (8) Red Bluff

The sketch of the river routing network is in Figure 3. The inflow-outflow functions are developed for reach 2 and 4 and a part of reach 1 from Alamogordo to Acme. Losses from Acme to Artesia are considered implicitly in this model. For reach 3 a loss of 7% (i.e., an outflow of 93%) is assumed for the Carlsbad main canal. All the water for this reach is released through that canal.

The inflow for reach 1 is from the gauge past Sum- ner, for reach 2 is from the Artesia gauge, and for reach 4 is from the Carlsbad gauge. The outflow for these functions is measured at Acme gauge, McMillan gauge, and Red Bluff gauge, respectively. These functions are monotone and convex.

To apply the model we developed, let us define the following variables and parameters for each year:

x(i,P), Y(&P), G,P), O(Lp), w(i,p), w(p)

C(r) Ckr) Fkh) PM&h)

are as defined before leakage from reservoir r in month i release from reservoir r in month i spill from reservoir r in month i evaporation from reservoir r in month i evaporation coefficient in reservoir j,

month i, year y capacity of reservoir r storage at end of month i in reservoir r flood inflow in reach h in month i pumpage from the river in reach h in

month i spring gains in reach h in month i irrigation demand in reach h in month i irrigation return in reach h in month i Artesian gain in month i

] NEW MEXICO !-“7

TEXAS

t

Figure 2. The Pecos River Basin with detail of New Mexico portion

Sr(*) K,(e) v,

spill function in reservoir r leakage function in reservoir r auxiliary parameters for ranking priority in

theobjective V,_, < Vr,t = 1,. . . ,7


evaporation function in reservoir r; all are functions of content

base inflow-outflow function in month i from gauge p to gauge p + 1

auxiliary variable for content in order to compute spill in reservoir r in month i

auxiliary variable for content in order to compute evaporation or leakage in reservoir r in month i

NW

Mb9

auxiliary variables for content in order to compute leakage in reservoir r in month i

auxiliary variables for content in order to compute evaporation in reservoir r in month i

an auxiliary variable used to piecewise lin- earize the content leakage relationship

an auxiliary variable used to piecewise lin- earize the content surface area relationship

JW,p ,A

fYi,p)

W,p)

P(P)

N(P)

P(P)

N(P)

an auxiliary variable used to piecewise lin- earize the inflow outflow functions

positive deviation from the recorded monthly inflow

negative deviation from the recorded monthly inflow

positive deviation from the recorded yearly inflow

negative deviation from the recorded yearly inflow

positive deviation from the yearly estimated outflow

negative deviation from the yearly estimated outflow

The gauges for which we have inflow-outflow functions are (1) to (l), (2) to (3), (4) to (5), (7) to (8); (1) to (1) is done for completion and accuracy.

The model stated for the Pecos River case

Min c c ykp) + 2 c w(i,p)(P(i,p) + N(i,p)) + x dp)(P(p) + N(p)) P i P i P

+ c w(P)(P(P) + N(P)) + i 5 5 V,N(i,r,t) + 5 5 i V,M(i,r,t) P r=li=lt=1 r=li=lf=l

subject to

x(i,l) -y(i,l) 5 0

.fi*MiJ)) - y(iJ) 5 0

fi4Mi,4)) - YGA) 5 0

.fiMi,7)) -yG,7) 5 0 x(i,l) - R(i,l) - Z&l) + C(i - 1,l) = 0

Z(i,l) - S(i,l) - E(i,l) - C(i,l) = 0

S,(Z(i,l)) - S(i,l) = 0

Z(i, 1) - S(i, 1) - T(i,l) = 0

&T&l) + K(i - 1,l) - M(i,l) = 0

-E(i,l) + E,(M(i,l)) = 0 R(i,l) + S(i,l) - R(i,2) + B(i,l) + x(i,2) = 0



y(i,2) - x(i,3) = 0

x(i,3) + A(i,l) + F(i,l) - PM(i,l) - x(i,4) = 0

y(i,4) + F(i,2) - R(i,2) - Z(i,2) + C(i - 1,2) = 0

Z(i,2) - S(i,2) - E(i,2) - K(i,2) - C(i,2) = 0

&(Z(i,2)) - S(i,2) = 0

Z(i,2) - S(i,2) - T(i,2) = 0

1T(i,2) + JC(i - 1,2) - iV(i,2) = 0

-K(i,2) + K,(iV(i,2)) = 0

-K(i,2) + N(i,2) - M(i,2) = 0

-E(i,2) + E,(M(i,2)) = 0

R(i,2) + S(i,2) + K(i,2) + G(i,3) + F(i,3) - R(i,3) - Z(i,3) + C(i - 1,3) = 0

Z(i,3) - S(i,3) - E(i,3) - K(i,3) - C(i,3) = 0

S,(Z(i,3)) - S(i,3) = 0

tC(i - 1,3) + &T(i,3) - N(i,3) = 0

--K&3) + K3(N(i,3)) = 0

-K(i,3) + N(i,3) - M(i,3) = 0

-E(i,3) + E3(M(i,3)) = 0

R(i,3) + S(i,3) + K(i,3) - x(i,7) = 0

y(i,7) + F(i,4) - x(i,8) + G(i,4) = 0

R(i,l) - D(i,l) 2 0

monthly goals:

x(i,p) - Z(i,p) + P(i,p) + N(i,p) = 0

yearly requirements:

2 x(i,p) - $J Z(i,p) + P(p) + N(p) = 0 i=l i=l

Other computational details

Spill. Spill is the maximum of the water available in a month in the reservoir after irrigation release minus the reservoir capacity. Thus the spill in month i from reservoir r is

S,(Z(i,r)) = S(i,r) = max (Z(i,r) - C(r), 0)

S,(Z(i,r)) = S(i,r)

= &[JZ(i,r) - C(r)1 + Z(i,r) - C(r)]

To allow the model to compute spill automatically, we

Ft. Sumner

Flood m-b

Routing Representation

Flood Inflow

Flood bd-h

Flood Innow

SPiII Channel Irrigation Loa.9 Pumping Alemogordo- ” Acme Salt Ceder0 +

Channel Lo88 Arteiie-McMillan Seepege Evaporation

Almg. - Alemogordo

Ft.S. - Fort Sumner

Figure 3. Routing representation


broke the Z(i,r) into a spill part and a nonspill part. We assigned priority coefficients in the objective to insure that the nonspill part comes into the optimal solution first. The spill part will not come in unless the nonspill is equal to the reservoir capacity. Thus we will add the following to the model:

McMillan:

objective + 2 2 5 V,Z(i,r,t) r=li=lf=l

K(i,2) = 1.69N(i,2,1) + 0.32N(i,2,2) + 0.2iV(i,2,3)

+ O.l3N(i,2,4) + O.l3N(i,2,5)

+ O.l2N(i,2,6) + 0.1 lN(i,2,7)

with 0 5 N(i,2,t) 5 5, t = 1, . . . , 6, 0 5 N&2,7) 5 8.5, i = 1, . . . , 12, and IX:=, N&3,1) = N(i,3).

Avalon:

and

Z(i,r) - Z(i,r,t) - Z(i,r,2) IO

Z(i,r,t) 5 c(r), forr= 1,2,3 i= l,..., 12

An optimal solution for the model will give Z*(i,r,2) as spill; i.e., S(i,r) = Z*(i,r,2) for month i, reservoir r. We note that after irrigation release comes spill, then leakage, and then evaporation.

K(1,3) = 0.5N(i,3,1) + 0.6N(i,3,2) + 0.4N(i,3,3)

+ 0.4N(i,3,4) + 0.4N(i,3,5) + 0.3N(i,3,6)

with 0 5 N(i,3,t) 5 t = 1, . . . , 6, i = 1, . . . , 12, and C$=, N(i,3,t) = N(i,3).

We assign priority coefficients in the objective function to insure that lower parts of the content in the reservoir come in first in an optimal solution.

Leakage. Since the Alamogordo reservoir does not leak and the Red Bluff reservoir is south of the state line, no content-leakage relationship is developed for them.

A content-leakage relationship is developed for McMillan and Avalon reservoirs, using the data in Refs. 1 and 9. The relationships are then piecewise linear- ized, and they have the following form:

Evaporation. To allow the model to compute evaporation, we express surface area as a function of content. Then we use the monthly evaporation coefficients per unit surface to compute monthly evaporations because the available evaporation coefftcients are in terms of surface area.

The available data for Alamagordo dam’ were used to develop content-surface-area relationship. With this relationship the monthly evaporations are

E(i,l) = e(l,i,y)[W,l)l = e(l,i,y)[O.O525M(i,l,l) + O.O38M(i,1,2) + O.O346M(i,1,3)

+ O.O338M(i, 1,4) + O.O2586M(i,lS) + 0.03 18M(i, 1,6) + O.O25lM(i, 1,711

withOSM(i,l,t)S20,t = 1,. . . , 6,OrM(i,l,7) 5 12.2, andZ:=l M(i,l,t) = M(i,l)where e(1 ,i,y) = monthly


evaporation coefficient for month i, year y, and reservoir 1. Available data’ were used to develop content-to-surface-area relationships for McMillan dam. The monthly

evaporation for the McMillan is

E(i,2) = eCLi,y)[W,2)1 = e(2,i,y)[O.478M(i,2,1) + O.l26M(i,2,2) + O.l06M(i,2,3) + O.O74M(i,2,4)

+ O.O78M(i,2,5) + O.O74M(i,2,6) + O.O83M(i,2,7)]

with 0 I M(i,2,t) I 5, c = 1, . . . , 6, 0 I M(i,2,7) 5 8.5, and Cj,, M(i,2,t) = M(i,2). Available data’ were used to develop the content-to-surface-area relationship for the Avalon dam. The monthly

evaporation for Avalon is

E(i,3) = e(3,i,y)[ S(i,30)1

= e(3,i,y)[0.425M(i,3,1) + O.l95M(i,3,2) + O.lM(i,3,3)

+ O.O5M(i,3,4) + O.l35M(i,3,5) + O.O8M(i,3,6)1

with 0 5 M(i,3,t) 5 1, t = 1, . . . , 6, and X7=, M(i,3,5) = M(i,3).

Red Bluff evaporation is not considered in the model, since Red Bluff is 8 miles south of the state line. Again we assign priority coefficients in the objective function to insure that the lower part of the content comes in first in an optimal solution.

Flood inflow. Flood inflow is computed as a residual in the mass balance equations. They are constrained to be nonnegative, which is insured because the model

automatically adjusts all other estimated quantities to conform to this reality.

Base inflow-outflow functions. The base inflow-outflow functions are developed by using a least absolute value criterion,9 and the data in Refs. 2 and 11. They are then approximated in a piecewise linear fashion. The following functions are used for the first subreach (Alamogordo to Acme):



For January, February, December, and November,

fi2(x(i,2)) = 0.0x(&2,1) + 0.915x&2,2) + 0.983x(&2,3) + 0.989x(i,2,4)

+ 0.991x(i,2,5) + 0.993x(i,2,6) + 0.995x(i,2,7) + x(i,2,8)

0 5 x(i,2,1) 5 1 0 5 x&2,2) 5 9 0 5 x(i,2,j) I 10 j=3,...,7

0 5 x&2,8) 5 500 5 x(i,2,j) = x(&2) i = 1,2,11,12

For March,

&*(x(3,2)) = 0.0x(3,2,1) + 0.978x(3,2,2) + 0.993x(3,2,3) + 0.996x(3,2,4)

+ 0.997x(3,2,5) + 0.998x(3,2,6) + 0.998x(3,2,7) + x(3,2,8)

0 4 x(3,2,1) I 1 0 5 x(3,2,2) 5 9 0 5 x(3,2,j) 5 10 j=3,...,7

0 <x(3,2,8) 5 500 i x(3,2,j) = x(3,2) j=l

For July, August, and September,

$,&x(&2)) = O.Ox(i,2,1) + 0.825x(i,2,2) + 0.943x(i,2,3) + 0.943x(i,W

+ 0.962x(i,2,5) + 0.97x&2,6) + 0.976x(i,2,7) + x(i,2,8)

0 5 x(i,2,1) 5 1 0 5 x(i,2,2) 5 9 0 9 X(i,2,j) 5 10 j= 3,...,7

0 I x&8,1) I 500 5 x(i,2,j) = x(i,2) i = 7,8,9 j=l

j=l

For October,

.f102(x(i,2)) = 0.0x(10,2,1) + 0.738x(10,2,2) + 0.864x(10,2,3) + 0.896x(10,2,4)

+ 0.913x(10,2,5) + 0.923x(10,2,6) + 0.931x(10,2,7) + x(10,2,8)

0 4 x(10,2,2) 5 1 0 %x(10,2,2) 59 0 5 X(10,2,]) 5 10 j= 1,...,7

0 5 x(6,2,8) 4 500 5 x(10,2,j) = x(10,2) j=l

For April,

&x(4,2)) = 0.0x(4,2,1) + 0.89x(4,2,2) + 0.954x(4,2,3) + 0.969x(4,2,4)

+ 0.975x(4,2,5) + 0.979x(4,2,6) + 0.982x(4,2,7) + x(4,2,8)

Orx(4,2,1)< 1 01x(4,2,2)59 Olx(4,2,j)s 10 j=3,...,7

0 9 x(4,2,8) I 500 i x(4,2,j) = x(4,2) j=l

For May,

&(x(5,2)) = 0.0x(5,2,1) + 0.819x(5,2,2) + 0.919x(5,2,3) + 0.941x(5,2,4)

+ 0.952x(5,2,5) + 0.959x(5,2,6) + 0.964x(5,2,7) + x(5,2,8)

05x(4,2,1)1 1 05x(5,2,2)59 Osx(5,2,j)s 10 j=3,...,7

0 I x(5,2,8) 5 500 5 x(5,2,j) = x(5,2) j=l

For June,

&(x(6,2)) = 0.0x(6,2,1) + 0.699x(6,2,2) + 0.823x(6,2,3) + 0.862x(6,2,4)

+ 0.882x(6,2,5) + 0.895x(6,2,6) + 0.904x(6,2,7) + x(6,2,8)

05x(6,2,1)5 1 05x(6,2,2)59 Osx(6,2,j)c: 10 j = 3, . . .., 7

0 5 x(6,2,8) 5 500 5 x(62,j) = x(6,2) j=I

in the second reach, Artesia to McMillan, one base inflow-outflow relation was used for all months:

336 Appl. Math, Modelling, 1989, Vol. 13, June


~i4(X(8,4)) = O.O~(i,4,1) + 0.661~(i,4,2) + 0.787x(&4,3) + 0.827x(i,4,4)

+ 0.849x(i,4,5) + 0.863x(i,4,6) + 0.87x(i,4,7) + x(i,4,8)

0 5 x(i,4,1) 5 1 0 5 x&4,2) I9 0 I x(i,4,j] 5 10

0 5 x(i,4,8) 5 500 5 x(i,4,J’) = x(i,4) i= 1, j=l

j=3,...,7

. . . 9 12

For the third reach we take as an outflow 93% of inflow. In the fourth reach, Carlsbad to New Mexico- Texas state line, a yearly inflow-outflow relationship is developed. Then the monthly distribution of the losses is used to get the outflow on a monthly basis. This method is used instead of monthly base inflow-outflow functions. The percentage distribution of losses over the various months is presented in Ref. 1, Appendix 20.

Systems software. The system software used to run a yearly routing study consists of a matrix generator which is developed to take the data for a year and prepare it in the format required by the linear programming code SMULP. Then SMULP solves the linear program. The size of the linear program run for each year is 1392 variables with 580 constraints. The computer utilized is a Cyber 175. The routing study was run for 29 years from 1919 to 1947. The computation time for each run averaged 2.70 min, of which, on the average, 1.3 min is used by the matrix generator and the printing of the results. The other 1.4 min is the

computation time needed to obtain the optimal solution of the linear program.

Results and conclusions

Prior to the development of the median routing study method, two routing studies had been applied to the Pecos River. The first one is in Ref. 1, referred to here as SD 109. The second one is Ref. 2, referred to as RBD. Table 1 is a summary of the results of the three routing studies. For each study, two columns show the three-year moving average outflow of the border and the corresponding three-year moving average of index inflow. The index inflow for RBD and SD 109 is the sum of all the flood inflows in the four reaches plus the routed inflow at Alamogordo reservoir. For the median routing study the index inflow is the sum of all flood inflows in all four reaches plus the adjusted inflow at the Alamogordo reservoir.

In the SD 109 routing studies, negative flood inflows are set to zero. In the RBD, the unreal negative flood inflow quantities are employed in their routing study

Table 1. Three-year moving average, 1919-1947

Median SD109 RBD

Year Index inflow Outflow Index inflow Outflow Index inflow Outflow

1921 633.19 464.97 1922 395.83 257.8 1923 437.48 276.5 1924 331.01 197.5 1925 379.94 222.97 1926 398.76 234.82 1927 383.01 243.8 1928 339.65 215.81 1929 260.37 169.34 1930 299.04 188.99 1931 342.34 198.27 1932 405.15 257.67 1933 356.2 239.9 1934 296.69 219.52 1935 253.15 157.33 1936 258.06 153.02 1937 459.91 303.27 1938 456.44 304.81 1939 467.84 303.17 1940 266.99 158.12 1941 630.13 643.91 1942 705.74 728.55 1943 714.98 740.93 1944 329.06 259.96 1945 231.12 159.95 1946 236.11 143.92 1947 218.64 119.96 Average 394.4 282.0

557.8 370.3 392.3 268.4 300.1 318.7 325.9 307.2 250.2 275 294.4 377.2 342.2 292.0 223.6 227.4 367.1 388.5 392.2 269.0 267.1 859.7 859.3 337.4 234.8 201.2 -

357.2

412.3 259.9 259.1 156.3 178.0 200.6 203.9 187.5 150.2 168.8 189.2 251.7 236.0 191.9 136.0 127.8 243.5 253.1 256.3 151.1 639.8 732.3 746.2 246.2 139.0 121.0 -

263

576.3 376.3 358.8 184.6 383.2 186.3 293.8 116.9 295.5 124.1 304.4 139.3 309.1 136.4 320.0 137.3 292.2 121.1 296.9 125.6 280.5 125.9 347.2 178.2 346.3 178.3 306.4 152.1 238.8 101.9 226.7 93.5 386.6 223.1 409.1 232.6 417.7 237.1 284.3 120.8 778.4 625.7 861.8 710.8 866.0 720.9 351.5 202.5 237.5 103.0 215.2 83.9 176.8 66.6 376.1 214



calculations as if the mass balance equations applied to nonreal possibilities. The water estimates in SD 109 are deficient in that no objective statistical estimation method is provided to justify setting estimates at zero.

The RBD negatives, which ignore hydrological and physical reality, effectively subtract water from that which must have been in the river to produce the observed time sequential data in a year. Thus both these studies estimate less water in the river than does our median routing study.

Quantitative comparison of the delivery of the three studies has been made in terms of average outflow at the border over the 29 years for which the routing studies were performed. The distortions produced by the ad hoc method in SD 109 and the negatives of the RDB method have the median routing study averaging 7% higher outflow than SD 109 and 32% higher than the RBD.

Such substantial differences should demonstrate the need to employ the more complicated type of method we have developed to insure valid estimates in routing studies. Our software is available to anyone, subject only to reimbursing us for copying, materials, and transportation expenses.

The developed model could be employed in esti- mating water allocations on the Nile River, where its water has to be divided between Egypt, Sudan, Ethi- opia, Zaire, Kenya, and Uganda. The model also can be utilized to estimate cash and labor flow between countries.

One direction of research is to explore modeling refinements which might greatly improve the computational speed and capability to handle larger problems. We intend in further research to try to go from a general linear programming model to a (generalized or pure)

network format which can provide two orders of mag- nitude improvement in speed and size of the problem which can be handled.

Another direction of extension of our results might be in more explicit introduction of stochastic elements into routing studies: e.g., the use of probability distri- butions for flood inflow computations, satisfaction of irrigation demands, and so on.

References

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Engineering Advisory Committee to the Pecos River, Pecos River Compact; Senate Document 109. Unpublished report Subcommittee on Review of Basic Data, Data Submitted to Engineering Advisory Committee, Pecos River Commission, on January 30, 1966. Unpublished report Bassett, G. and Koenker, R. Asymptotic theory of least absolute error regression. J. Am. Stat. Assoc., 1972, 73(343), 618-622 Bickel, P. J. and Doksum, K. A. Mathematical Statistics:Basic Ideas and Selected Topics. Hodlen-Day, 1977 Charnes, A. and Cooper, W. W. Management Models and Industrial Applications of Linear Programming, Vols. I, II. Wiley, New York, 1961 Charnes, A., Cooper, W. W., and Ferguson, R. Optimal estimation of executive compensation by linear programming. Management Sci. I 1955, 2, 138-151 Chow, V. T. Handbook of Applied Hydrology. McGraw-Hill, New York, 1964 Gentle, J. E. Least absolute value estimation: an introduction. Commu-Statist-Simula, Comouta B 1977. 6(4), 313-328 National Resources Planning Board, June 1942. The Pecos River Joint Investigation of the Participating Agencies Ali. A. Iabal and Kenninaton. J. J. SMU-LP: an in-core primal simplex dode for solving linear programs, Version 1. Technical Report OR 80015 Erickson, J. R., Hale, D. P., Vandertulip, J. J., Slingerland, C. L., and Anderson, J. C. Report of Review of Basic Data. Submitted to the Engineering Advisory Committee to the Pecos River. Unpublished report


Documents

A constrained median statistical routing method and the Pecos River case