Utilisation des outils numériques d'aide à la décision pour la gestion de l'eau

Le succès d'une gestion des écosystèmes naturels requiert une connaissance approfondie des différents processus qui interviennent et de leurs échelles de temps et d'espace particulières. Pour cette raison, les décideurs ont besoin d'analyser une vaste gamme de données et d'informations géographiques. Les modèles mathématiques, les systèmes d'informations géographi-ques et les systèmes experts sont capables de produire cette analyse, mais seule une minorité de gestionnaires les utilise actuellement. Cet article identifie quelques unes des raisons à l'origine de l'hésitation des gestionnaires à adopter de tels outils d'aide à la décision pour la gestion des ressources naturelles et propose une structure qui pourrait faciliter leur utilisation pour le processus de prise de décision. Cet exercice est réalisé à l'intérieur du contexte de la gestion intégrée par bassin. Une revue des systèmes d'aide à la décision est également présentée.Many methods of integrated or watershed management exist which account for the necessary biophysical and socio-economic factors at the watershed level. Some of these approaches are ecosystem oriented while others are socio-economically oriented. Whatever the definition, water management at the watershed level needs to account for a plenitude of variables related to the air, water, soil, biology, and economy. The successful management of natural ecosystems requires a thorough understanding of their characteristic time and spatial scales. Because of this, decision makers need to analyze a wide range of data and geographic information. Mathematical models, geographic information systems and expert systems are capable of performing this analysis, but only a minority of managers are currently using them. This paper identifies some of the reasons why ecosystem managers have been slow to adopt such decision support tools in natural resources management and proposes a framework to facilitate their use in the decision making process. This is done in an integrated watershed management context. A review of related decision support systems is also presented.Four types of decision-support tools are introduced : mathematical models, expert-systems, geographical information systems (GIS) and decision support systems (DSS). Mathematical models have long been used for simulation, prediction, and forecasting, however, they are often task specific and were rarely developed for management uses. GIS are more and more commonly being used for decision support as they become more affordable and user-friendly and are very well-suited for managing resources at a spatial scale. There exist many kinds of software ranging from a simple viewer used for cartographic purposes to complex GIS oriented toward spatial analysis and modelling. Expert systems are also interesting for decision support when specific goals are being considered. Finally, DSS are perhaps the digital tools most applicable to management purposes, often integrating one or more models, a GIS or expert system functionalities. There are two types of DSS : 1. Environmental Information Systems (EIS), and 2. Integrated Modelling Systems (IMS) EIS can be very user- friendly, relying heavily upon GIS and statistical functions.IMS also use GIS capabilities, but integrates several mathematical models as well. The level of integration between models varies considerably and the complexity of IMS are generally high.Two questions underlie the operational use of digital technologies for decision support. The first is whether or not such technology should be used at all, while the second is why such tools take time to be adopted by government and management agencies. The use of digital technologies is often required when the problem is complex and where there are a wide range of factors involved with different spatial and temporal scales. Three major constraints towards the implementation of decision support tools can be pinpointed :1. technology, 2. data, and 3. working organization. Technological constraints include cost, lack of user friendliness, and hardware problems, among other factors. Data constraints are mostly related to availability, cost, heterogeneity and volume. Finally, organization constraints pertain mostly to the manager's perception of the tool and the structural integration of the tool within the decision process.This paper proposes a 4-step approach to optimize the use of decision-support tools. The first step requires that managers and decision-makers clearly define their project, goals and budget, as well as, decide whether to use an integrated watershed management approach or a more discrete approach. This leads directly to the second step, which consists of choosing the most appropriate digital support tool. This requires communication between managers and scientists, and at this point, data gathering and integration should begin. The third phase consists of the development of a new tool or adaptation of an existing one within the context of the agency's management structure. The final step is the operational use of the decision support tool by the agency, following an initial trial period. The successful use of a decision support tool for management purposes depends on proper planning that accounts for all factors related to management needs, budget, data, ease of use, and organization integration

La modélisation hydrologique et la gestion de l'eau

Cet article brosse un portrait de différents types de modélisation hydrologique développés à ce jour. Nous passerons donc en revue l'hydrologie, à l'érosion hydrique des sols, au transport et aux transformations des polluants et à la qualité de l'eau en rivière. Ce bref survol, nous amène à conclure que si le développement de la modélisation hydrologique s'est fait jusqu'ici essentiellement en affinant la description des processus et en considérant des échelles spatiales et temporelles plus fines, l'étape suivante passe par l'intégration de ces divers modèles. Cette intégration permettra dès lors de considérer un ensemble de problématiques directement liées aux aspects de gestion environnementale.This paper presents an overview of physically-based hydrological modeling approaches and a look at the future of hydrological modeling within the context of water management. It extends beyond classical hydrological modeling by surveying the modeling of water contaminants transport in porous media and surface waters, as well as soil erosion.Increasing concerns in predicting the impacts of land use management on the hydrological cycle have led researchers to construct two types of physically-based distributed models. The first type of model views the watershed as an ensemble of inter-connected reservoirs and mimics water routing with various types of discharge expressions and conceptual models (e.g., the infiltration models of Green and Ampt (1911), Holtan (1961) or Smith and Parlange (1978); the unit hydrographs of Sherman (1932) and Dooge (1973) and the geomorphological unit hydrograph of Rodriguez-Iturbe and Valdes (1979); the ground water discharge model of Beven and Kirby (1979); etc...). It is noteworthy that the pioneering Stanford Watershed Model of Crawford and Linsley (1966) led to the development of many currently used hydrological models including HBV (Bergstršm and Forsman, 1973), SLURP (Kite, 1978), TOPMODEL ( Beven and Kirby, 1979) and CEQUEAU (Morin et al., 1981), to name a few. The second type of model discretizes the watershed into an ensemble of control volumes and mimics water routing using combinations of partial differential equations for mass and momentum conservation and phenomenological models (e.g., Darcy's (1856), Dupuit's (1863), Boussinesq's (1904) and Richards (1931) equations for unsaturated and saturated flow in porous media; Saint-Venant's (1871) and Manning's (1891) equations for overland and open channel flows). Hydrological models such as SHE (Abbott et al.,1986a, b), IHDM (Calver, 1988), KINEROS (Woolhiser et al., 1990), THALES (Grayson et al.,1992) and HYDROTEL (Fortin et al., 1995), among others, represent classical examples of this type of modeling. It is noteworthy that recent advances in remote sensing and in digital elevation modeling have greatly facilitated and simplified the use of most of the hydrological models.On another front, the adverse effects of agricultural, industrial and urban runoff on surface and ground waters have motivated the development and application of different approaches to predict the fate and transport of various water contaminants in the environment (i.e., eroded soil particles, adsorbed and dissolved nutrients and pesticides as well organic matter).In soil erosion modeling, these concerns have led researchers to construct nonpoint source pollution models for evaluating the impacts of alternative land management practices on water quality. Based on the empirical Universal Soil Loss Equation (Wischmeier and Smith, 1978), the first nonpoint source models included CREAMS (Knisel et al., 1980), AGNPS (Young et al., 1987) and SWRRB (Williams et al., 1985). However, the lack of physical realism in these empirical formulations prompted the development of physically-based erosion models such as GUEST (Rose et al., 1983; Hairshine and Rose, 1992a, b), WEPP (Nearing et al., 1989), LISEM (De Roo et al., 1994) and EUROSEM (Morgan et al., 1992). The advantage of these models over the USLE resides in their ease of integration with physically-based hydrological models. Because of its close ties with the hydrological cycle and the soil erosion process (adsorbed and dissolved contaminants), the development of physically-based models for nutrient and pesticide transport benefited directly from advances in soil erosion modeling, soil chemistry and soil physics. The modeling of nitrogen transport is a representative example of this. Early modeling efforts involved the coupling of first-order kinetics models for the nitrogen cycle (Mehran and Tanji, 1974) with two types of mass conservation equation in porous media: the convection-dispersion equation and the capacity transport equation. Well known soil nitrogen dynamics models include NCSOIL (Molina et al., 1983), SOILN (Johnsson et al, 1987), EPIC (Sharpley and Williams, 1990), LEACHN and LEACHA (Hutson and Wagenet, 1991, 1992, 1993), DAISY (Hansen et al., 1991) and AgriFlux (Banton et al., 1993).The first attempt to model surface water quality goes back to the work of Streeter and Phelps (1925) who studied the impacts of a municipal waste water discharge on dissolved oxygen (DO) and biological oxygen demand (BOD) of an Ohio river. To predict DO and BOD dynamics, Streeter and Phelps assumed uniform and steady flow conditions and used first-order kinetics to model atmospheric supply of oxygen and oxygen consumption. The advances in computational power during the 70s and 80s allowed several researchers to substantially increase the complexity of the Streeter-Phelps approach. This was achieved by accounting for advection-dispersion phenomena, unsteady two and three dimensional flow conditions, as well as the effect of temperature on various chemical reactions. The QUAL2E model of Brown and Barnwell (1987) is a good example of a moderately complex water quality model where advection-dispersion and temperature effects on several water characteristics and contaminants are considered under one-dimensional steady flow conditions.At present, the state of hydrological modeling and software engineering has reached a point where it is now possible to construct spatial decision support systems (SDDS) capable of simulating the impacts of various management practices (i.e., industrial, municipal and agricultural) on the water quantity and the quality of a watershed's river network. These systems, which idealy should be user-friendly for decision makers, will be both integrated modeling systems (including a database system, hydrologic, soil erosion, agricultural-chemical transport and water quality models) and spatial data analysis systems (including a geographical information system). Currently developed SDDS include PÉGASE (Smitz et al., 1997) and GIBSI (Villeneuve et al., 1996, 1997a,b). In a sustainable water management context, the use of such systems will provide decision makers with a complete tool for exploring a variety of integrated watershed management programs

Évaluation de l'impact de l'assainissement urbain sur la qualité des eaux du bassin versant de la rivière Chaudière à l'aide du système de modélisation intégrée GIBSI

Le programme d'assainissement des eaux de Québec (PAEQ), mis en place à la fin des années 1970, s'est d'abord attaqué au problème de la pollution ponctuelle d'origine urbaine. Plusieurs stations de traitement des eaux usées municipales ont été construites dans le cadre de ce programme réduisant de façon importante les charges de polluants d'origine urbaine. La question demeure toutefois de savoir dans quelle mesure les charges urbaines rejetées avant et l'après l'instauration de ce programme peuvent entraîner des dépassements de différents critères de l'eau. La présente étude a pour objectif d'examiner cette problématique pour le bassin versant de la rivière Chaudière en utilisant le système de modélisation intégrée GIBSI (Gestion Intégrée par Bassin versant à l'aide d'un Système Informatisé). Deux scénarios d'assainissement urbain ont été examinés, l'un représentatif de la période avant la mise en place du programme, le début des années 1980, et un autre représentatif de la période plus récente. Deux chroniques météorologiques ont été utilisées (années 1983 et 1994). L'estimation des probabilités de dépassement de différents critères de l'eau montre une nette réduction de ces probabilités après mise en place du programme. Pour la DBO5 et l'azote total, les charges urbaines rejetées actuellement n'entraînent pas de dépassement des critères de qualité de l'eau pour les deux années retenues. Toutefois, les résultats montrent que les charges en phosphore total d'origine urbaine peuvent à elles seules entraîner des probabilités de dépassement importantes lors d'étiage.In 1978, the Québec Government put in place a provincial municipal clean water program, referred to as the Programme d'assainissement des eaux du Québec (PAEQ), to restore the province's rivers to their natural state. The program focussed primarily on the problem of municipal waste loads and, hence, a large number of wastewater treatment plants (WWTP) were constructed during the 1980's and the 1990's. At the beginning of the 1980's, only a few percent of the population had their waste water treated, but this figure had increased to over 95% by 1997 (Figure 1). In the Chaudière river watershed, this program resulted in the construction of more than 35 WWTP over an 18-year period. Although an impressive effort was devoted to reduce municipal waste loads, a question remains: how does this reduction translate into terms of overall improvement of water quality at the watershed level?The Chaudière river watershed was considered for this application. This watershed covers an area of 6682 km2 and is located south of Quebec City. Land use is dominated by forest (62%) followed by agricultural land (33%), urban area (3.6 %), and water (1.7 %). The total population in 1996 was 173129 and was mainly located in the northern part (i.e., the downstream region) of the watershed. A total of 44 municipal point loads were identified (see Figure 2 and Table 1), most of them corresponding to WWTP. Characteristic data on municipal WWTP for year 1995 were obtained from the Quebec Department of the Environment (MINISTERE DE L'ENVIRONNEMENT ET DE LA FAUNE DU QUEBEC, 1997) and used to estimate average concentrations of total phosphorus and the biological oxygen demand (BOD). Data included affluent and effluent concentrations at various WWTP. Since no data were available for nitrogen, a concentration of 40 mg-N/l was used (TCHOBANOGLOUS et SCHOEDER, 1985; NOVOTNY et CHESTERS, 1981). Nitrogen removal efficiency for different types of treatment was estimated from available data compiled by the Environment Quebec (Table 2). An average, per capita, daily wastewater volume of 0.73 m3 was derived. This large value indicates a large infiltration capacity in the sewer network and, on average, a poor structural state.Results were analysed at four locations distributed along the river. These points corresponded to the locations of the four major water quality monitoring stations (Figure 3). Total annual loads within the sub-watersheds defined by these points were estimated (Table 3). Affluent characteristics were considered for the simulation of the pre-PAEQ period. Figure 4 represents the evolution of total phosphorus loads at station 2340012 (water intake for the town of Charny, the most downstream point). It shows that an important reduction occurred in 1986. For the same sub-watershed, the overall reduction for the 1982-1999 period was 38% for total phosphorus, 37% for total nitrogen and 83% for BOD.Assessment of the impact of the PAEQ was done using the integrated modelling system GIBSI ("Gestion Intégrée par Bassin versant à l'aide d'un Système Informatisé") (ROUSSEAU et al., 2000; MAILHOT et al., 1997 ; VILLENEUVE et al., 1998). Only pollutant loads originating from sewer networks were considered; pollutant loads from industrial plants not connected to a municipal sewer network were not considered in this study. Similarly, diffuse sources of pollution from urban area or agricultural land were not taken into account. Two scenarios were considered: a first scenario associated with the 1983 year corresponding to the pre-PAEQ period; and a second scenario associated with the 1994 year describing the post-PAEQ period (following load reductions). Two hydrological reference years were also selected, namely, years 1983 and 1994. The former corresponded to a significantly drier year than the latter. At station 2340012 (Charny water intake), this difference was even more pronounced when considering that the cumulative summer flow was 6.9 times larger in 1994 than in 1983. Four simulations corresponding to different combinations of representative municipal waste loads and meteorological conditions were performed (Table 5).Simulation results at the four control points were compared with water quality standards (WQS) for total phosphorus (0.03 mg-P/l, aesthetic WQS for prevention of eutrophication in rivers) and BOD (3 mg-BOD5 /l, sanitary WQS for domestic use of water requiring disinfection treatment only). The probabilities of exceeding these WQS were defined (equation 1) as the number of days (daily computational time step) where simulated values exceeded a given value (complementary cumulative distribution function). Table 8 introduces annual probabilities of exceeding WQS at the four control points and for the different simulations whereas Table 9 introduces those for the summer period. At station 2340012 (Charny water intake), for hydrological year 1994, a comparison of pre- and post-PAEQ conditions for total phosphorus shows that post-PAEQ condition lowered the number of days exceeding WQS by 49. Similar reductions were reached at the other control points. For the 1994 summer period, the reduction was not as pronounced since the probability of exceeding WQS was 0.24 for pre-PAEQ situation and it decreased to 0.21 for the post-PAEQ condition. These probabilities worsened under the 1983 summer conditions as concentrations exceeded the WQS 98% of the time at station 2340014

Calcul de probabilités de dépassement d'objectifs environnementaux de rejets de sources ponctuelle et diffuse à l'aide du système de modélisation intégrée GIBSI

Dans un contexte de gestion intégrée de l'eau par bassin versant, on doit, avant d'implanter tout programme d'assainissement, comparer les solutions de rechange, distinguer les contributions des différentes sources de pollution aux problèmes de qualité de l'eau, et définir des objectifs environnementaux de rejet (OER) associés aux activités et aux pressions anthropiques. Dans le cas des sources de pollution d'origine ponctuelle, on détermine un OER et son risque de dépassement en fonction d'un débit d'étiage critique d'une période de retour donnée. Dans le cas de la pollution diffuse d'origine agricole, il n'existe pas de concepts similaires pour définir des OER. L'approche que nous proposons, c'est de simuler, à l'aide de chroniques météorologiques et de divers scénarios de gestion de ces rejets, les concentrations résultantes dans les cours d'eau et de calculer les probabilités de dépassement des critères de qualité de l'eau (CQE). Cette approche permet de caractériser la prise de décision pour la période de contrôle en matière du nombre moyen de jours de dépassement d'un CQE. Ceci représente une quantification plus exacte du risque car ce dernier est interprété en fonction de ce qui est mesuré en rivière, c'est-à-dire les concentrations de polluants. Dans cette étude, on illustre l'application de ce concept à partir d'un cas de figure dans le bassin versant de la rivière Chaudière (Québec, Canada) qui est simulé avec le système de modélisation intégrée GIBSI. Dans le cadre d'une analyse de scénarios de restauration des activités récréatives conduisant à des contacts directs et fréquents avec l'eau, on illustre comment on peut, en réduisant respectivement les pressions agricole et urbaine de 32 et 17%, abaisser globalement les probabilités de dépassement des critères esthétique et bactériologique de 0,32 à 0,19 et de 0,94 à 0, respectivement.In an integrated watershed management context, the implementation of any clean water program requires the evaluation of the contribution of pollutant loads associated with wet (nonpoint or diffuse) and dry weather (point) sources to the studied water quality problem. It is also necessary to set environmental load allocations (ELA) or total maximum daily loads (TMDL) for various anthropogenic activities and to link these loads to water pollutant concentrations. The ELA or TMDL is a numerical quantity determining the maximum load of pollutants from point and nonpoint sources as well as background sources, to receiving water bodies that will meet designated water uses (e.g., swimming or fishing) in terms of water quality standards (WQS). The American Congress included a TMDL program in the Clean Water Act of 1972 (US EPA, 1997; 1999). The outcome of the TMDL program corresponds to the drafting of a watershed management plan (NOVOTNY, 1999).Estimation of ELA from point sources is generally based on simultaneous occurrence of severe low flows and maximum daily loads. The ensuing risk of not meeting WQS is usually linked to the inverse of the return period of the design flow. Meanwhile, estimation of ELA from diffuse sources, and associated environmental risks, is not as well established since diffuse sources of pollution generally occur during important runoff events and strongly depend on land use and management practices. These wet weather loads may be allocated using continuous hydrological modelling. Simulation results can then in return be used to link ELA from point sources and diffuse sources to pollutant concentrations and, for a given time period (e.g., summer), to evaluate the probabilities (namely the risks) of exceeding WQS designating the water use.The objective of this study was to present, using the integrated modelling system GIBSI (VILLENEUVE et al., 1998b; MAILHOT et al.,1997), this new approach to assess the risk associated with the establisment of ELA from point and diffuse sources. A case study on the Chaudière River watershed (Quebec, Canada) was defined and simulations were performed (scenarios A-E). The case study focussed on determining whether WQS defining the designated recreational use of water requiring direct and prolonged contact were attainable. Untreated municipal waste waters from a small town (St. Martin) and nonpoint source pollution were responsible for impairment of the studied river segment. Water contaminants considered were fecal coliforms (FC) and phosphorus (P).Two base case scenarios, A and C, were simulated using four years (1982-1985) of meteorological data to illustrate the degraded bacteriological and aesthetic conditions of the river segment due to dry and wet weather sources, respectively. Dry weather sources were assumed to solely contribute to the bacteriological impairment. Meanwhile, both wet and dry weather sources were assumed to contribute to aesthetic impairment. Scenario B was defined to examine the impact of constructing a waste water treatment plant (WWTP) for the town of St. Martin on the bacteriological conditions. Scenarios D and E were elaborated to quantify the impact of both reducing diffuse and point source loads on aesthetic conditions, respectively. For scenario C, on average 31 kg P/ha were applied on cultivated land according to local fertilization calendars. Similarly, the fertilization rate for scenarios D and E was on average 13 kg P/ha. The design stream flow for ELA from point sources was a30 Q5 (30-day low flow, 5-year return period) and estimated at 6.05 m3 /s for the studied river segment. It is noteworthy to mention that for the 1982-1985 summers (June 21 through September 20), simulated daily stream flows exceeded the design flow 31, 62, 19 and 2 times and that 150, 48, 348, and 270 Mm3 of water flowed through the studied river segment, respectively. This means the 1983 meteorological series was three times as dry as that of 1982 despite the fact it was the latter that produced the30 Q5. The ELA from both point and diffuse sources for the river segment receiving the St. Martin's effluent and the studied river segment were 0.75*1012 UFC and 11.21 kg P and 1.05* 1012 UFC and 15.68 kg P, respectively.For scenario A, simulation results showed that respectively 94% and 14% of the time, the bacteriological (WQSFC < 200 UFC/100 ml) and aesthetic (WQSP < 0,03 P mg/l) WQS were not met over the summer season of the four-year meteorological series. Similarly, for scenario C, the simulation results indicated that 32% of the time, the aesthetic WQS was not achieved. Dry weather sources, namely WWTP and wet weather sources, namely agricultural runoff, accounted for P loads. During the driest summer, dry weather sources accounted for 63% of the P loads. For the other summers, wet weather sources accounted for 71, 88, and 78% of total P loads. For scenario B, simulation results showed that the bacteriological WQS was met 100% of the time. Meanwhile, scenario D was designed to see whether a 27% reduction of the agricultural nonpoint source load could alone restore the aesthetic properties. The simulation results showed an improvement as the probability of exceedence dropped from 0.32 to 0.27. This probability was further lowered to 0.19 when St. Martin's waste waters were treated using aerated lagoons and dephosphotation. This corresponded to an additional 17% abatement of the dry weather sources loads (scenario E).It is noteworthy that this study did not specifically attempt to specify the reasons why WQS were not met, whether it was due to low stream flows or large point or nonpoint sources loads although the integrated modelling system allows for this. Also, this study did not attempt to determine impacts of local management scenarios instead of a systematic reduction though GIBSI also allows for this. However, this study clearly illustrated how the proposed methodology, which is rooted in a risk assessment approach based on evaluation of the probability of exceeding WQS, is well suited to characterise ELA for various anthropogenic activities. Indeed, simulation results clearly demonstrated the benefits of assessing independently the impacts of ELA from point and diffuse sources on the attainability of a designated water use. It is believed that this type of ELA assessment approach will facilitate communication with stakeholders. Ideally, these exceeding probabilities should be evaluated using long meteorological series (e.g., 30 years)

Magnetic Phase Diagram of the Ferromagnetically Stacked Triangular XY Antiferromagnet: A Finite-Size Scaling Study

Histogram Monte-Carlo simulation results are presented for the magnetic-field -- temperature phase diagram of the XY model on a stacked triangular lattice with antiferromagnetic intraplane and ferromagnetic interplane interactions. Finite-size scaling results at the various transition boundaries are consistent with expectations based on symmetry arguments. Although a molecular-field treatment of the Hamiltonian fails to reproduce the correct structure for the phase diagram, it is demonstrated that a phenomenological Landau-type free-energy model contains all the esstential features. These results serve to complement and extend our earlier work [Phys. Rev. B {\bf 48}, 3840 (1993)].Comment: 5 pages (RevTex 3.0), 6 figures available upon request, CRPS 93-

Mean-field phase diagrams of AT2X2AT_2X_2 compounds

Magnetic-field -- temperature phase diagrams of the axial next-nearest-neighbor Ising model are calculated within the framework of a Landau-type expansion of the free energy derived from molecular-field theory. Good qualitative agreement is found with recently reported results on body-centered-tetragonal $UPd_2Si_2$. This work is expected to also be relevant for related compounds.Comment: J1K 2R1 8 pages (RevTex 3.0), 2 figures available upon request, Report# CRPS-94-0

Spin Stiffness of Stacked Triangular Antiferromagnets

We study the spin stiffness of stacked triangular antiferromagnets using both heat bath and broad histogram Monte Carlo methods. Our results are consistent with a continuous transition belonging to the chiral universality class first proposed by Kawamura.Comment: 5 pages, 7 figure

Order induced by dipolar interactions in a geometrically frustrated antiferromagnet

We study the classical Heisenberg model for spins on a pyrochlore lattice interacting via long range dipole-dipole forces and nearest neighbor exchange. Antiferromagnetic exchange alone is known not to induce ordering in this system. We analyze low temperature order resulting from the combined interactions, both by using a mean-field approach and by examining the energy cost of fluctuations about an ordered state. We discuss behavior as a function of the ratio of the dipolar and exchange interaction strengths and find two types of ordered phase. We relate our results to the recent experimental work and reproduce and extend the theoretical calculations on the pyrochlore compound, Gd$_2$Ti$_2$O$_7$, by Raju \textit{et al.}, Phys. Rev. B {\bf 59}, 14489 (1999).Comment: 5 pages, 5 figures, AMSLaTe