In this work it is constructed a hydro-meteorological factor to improve the adjustment of statistical time series models, such as state space models, of water quality variables by observing hydrological series (recorded in time and space) in a River basin. The hydro-meteorological factor is incorporated as a covariate in multivariate state space models fitted to homogeneous groups of monitoring sites. Additionally, in the modelling process it is considered a latent variable that allows incorporating a structural component, such as seasonality, in a dynamic way

Costa, Marco

Gonçalves, A. Manuela

English

In this work it is constructed a hydro-meteorological factor to improve
the adjustment of statistical time series models, such as state space models, of
water quality variables by observing hydrological series (recorded in time and
space) in a River basin. The hydro-meteorological factor is incorporated as a
covariate in multivariate state space models fitted to homogeneous groups of
monitoring sites. Additionally, in the modelling process it is considered a latent
variable that allows incorporating a structural component, such as seasonality, in
a dynamic way.Fundação para a Ciência e a Tecnologia (FCT

Universidade do Minho: RepositoriUM

Improvement of surface water quality variables modelling that incorporates a hydro-meteorological factor : a state-space approach

Repositório Institucional da Universidade de Aveiro

Poster PresentationImprovement of surface water qualityvariables modelling that incorporates ahydro-meteorological factor: a state-spaceapproachA. Manuela Gonc¸alves1, Marco Costa21 Department of Mathematics and Applications, University of MinhoCMAT-Center of Mathematics, Portugalmneves@math.uminho.pt2 Higher School of Technology and Management of A´gueda-University of AveiroCMAF-UL, Portugalmarco@ua.ptAbstract: In this work it is constructed a hydro-meteorological factor to improvethe adjustment of statistical time series models, such as state space models, ofwater quality variables by observing hydrological series (recorded in time andspace) in a River basin. The hydro-meteorological factor is incorporated as acovariate in multivariate state space models fitted to homogeneous groups ofmonitoring sites. Additionally, in the modelling process it is considered a latentvariable that allows incorporating a structural component, such as seasonality, ina dynamic way.Keywords: hydrological basin; water quality, state-space modelling; Kalmanfilter; hydro-meteorological factor.1 IntrodutionWater quality monitoring is an important tool in the management andassessment of surface water quality. This study focuses on a rather extendeddata set relative to the River Ave basin (Portugal) and consists mainly ofmonthly measurements of biochemical variables in a network of monitoringwater quality stations.A hydro-meteorological factor is constructed for eachmonitoring station based on monthly estimates of precipitation obtained bymeans of a rain gauge network. Through stochastic interpolation (Kriging)it is estimated the mean area rainfall during each month in the area ofinfluence of each water quality monitoring site. These estimates are basedon rain gauges located in the respective area of influence. In a recent work,Costa and Gonc¸alves (2010) show that a set of water quality monitoringsites can be modelled applying cluster techniques that minimize the numberof models.2 Surface water quality variables modelling2 Data Set DescriptionThe Northern Regional Directory for the Environment and Natural Re-sources (DRARN) and the National Institute of Water (INAG) has beencollecting various water quality variables (monthly physical-chemical andmicrobiological analyses) from 16 quality monitoring sites. The data setof the 16 water quality monitoring sites, comprising 11 water quality vari-ables, have been monthly measured between 1988 and 2006. At this time,this work focuses on Dissolved Oxygen (DO) (mg/l) in water because it isone of the most important variables in the evaluation on river water qual-ity. For instance, it is shown the data and the results of one cluster withfive water monitoring sites identified in Costa and Gonc¸alves (2010) as theless polluted cluster.3 MethodsAs starting point, it is constructed a hydro-meteorological factor used ascovariate in the modelling process. This covariate will integrate a hydro-meteorological component that is recognized as crucial in any water qualitymodelling process. This factor is constructed through stochastic interpola-tion (Kriging) based on an udometric network (Figure 1) with 19 meteo-rological stations. The model of spatial continuity, which is inferred frommonthly precipitation estimates, assumes hypothesis of homogeneity of theprocess: the process is stationary of 2nd, i.e., intrinsically stationary andisotropic. Under this hypothesis, two observations in the same locationbut in different times are independent and the spatial variability patternremains the same (Kyriakidis and Journel, 1999). The empirical semivari-ogram is given byγˆZ(h | l) = 12T |N(h|l)|T∑t=1∑(i,j)∈N(h|l)[(Zt(si)− Zt(sj)]2with N(h|l) = {(i, j) : ‖si − sj‖ − ‖h‖ ≤ l; 1 ≤ i ≤ j ≤ n} and |N(h|l)| =#N(h|l).The river basin is discretized in 368 points with 2Km x 2Km (Figure 1)and at each point s0 the estimate of the monthly mean area precipitationis given by the Kriging estimator, i.e., by a linear combination of the 19known points sj , j = 1, ..., 19 and Zt(s0) =19∑j=1λjZt(sj).3.1 Hydro-meteorological factorIt is constructed one covariate for each water monitoring site based on theestimate of the monthly mean precipitation of its influence region. In thisA. Manuela Gonc¸alves and Marco Costa 3FIGURE 1. Spatial distribution of 19 meteorological monitoring sites in the RiverAve basin and discretization of River Ave basin in 368 points.context, the influence regions of each water monitoring site were defined bytechnicians of the INAG and they are supported on the regions topographyand the lands drainage dynamics.Firstly, for each water monitoring site, it was computed the monthly meanarea precipitation in its influence region based on the average of point pre-diction. Naturally, a large influence region tends to have a greater precipi-tation amount. Indeed, it is clear that the precipitation amount influencesoxygen concentration in water. However, if the goal of this work is to founda prediction model to DO in a month t, the covariate should not incor-porate the precipitation amount of the current month, but only the pastinformation.Let P(i)t be the estimate of the precipitation amount in the influence area ofa water monitoring site i at month t. We considered a covariateH(i)t computedas a weighted average of precipitation amount at months t− 1 and t− 2.3.2 State space modelFor each cluster i with homogenous water monitoring site it is fitted a statespace model to Dissolved Oxygen concentration incorporating two struc-tural components: the hydro-meteorological factor and a seasonality. In or-der to simplify, it is considered monthly seasonality assuming 12 known co-efficients (for each month it is taken the month mean; Costa and Gonc¸alves,2010):Y(i)1,tY(i)2,t...Y(i)m,t =St H(i)1,tSt H(i)2,t......St H(i)m,t(X(i)1,tX(i)2,t)+µ(i)1,tµ(i)2,t...µ(i)m,t,where4 Surface water quality variables modelling(X(i)1,tX(i)2,t)=(1µ(i)X2)+(φ(i)11 φ(i)12φ(i)21 φ(i)22)[(X(i)1,t−1X(i)2,t−1)−(1µ(i)X2)]++(V(i)1,tV(i)2,t).Since normal distribution is not always the best distribution to fit mete-orological variables in this work, we adopted consistent distribution-freeestimators developed from the original work by Costa and Alpuim (2010).TABLE 1. Parameters estimates.µˆX φˆ ΣˆV Σˆµ Sites1 0.277 −1.045 0.016 −0.005 0.597 0.000 0.000 0.000 0.000 CANT−0.0003 0.038 0.738 −0.005 0.003 0.000 0.265 0.000 0.000 0.000 GOL0.000 0.000 0.417 0.000 0.000 FER0.000 0.000 0.000 0.383 0.000 V SA0.000 0.000 0.000 0.000 0.737 TAIThe state space model with these parameters estimates associated to theKalman filter produces monthly one-step predictions for Dissolved Oxy-gen concentration at each water monitoring site (Table 1). Figure 2 showsobserved data and predictions in Vizela Santo Adria˜o (VSA) and Gola˜es(GOL) monitoring sites.FIGURE 2. Observed and one-step predictions of Dissolved Oxygen concentrationin Vizela Santo Adria˜o (VSA) and in Gola˜es (GOL).4 ConclusionsIt is possible to conclude that the hydro-meteorological factor is an impor-tant component adding information beyond the usual seasonality. More-over, the adoption of the consistent distribution-free estimators for thestate space models requires a future comparison with gaussian likelihoodestimation, assessing its relative efficiency, and possibly comparing its fore-casts mean square error. However, distribution-free estimators are an easyA. Manuela Gonc¸alves and Marco Costa 5solution without computed problems, nor iterative procedures and neitherrequires initial values. The next step is to analyse the filtered estimates ofstates X(i)t|t given by the Kalman filter, which allows an interesting analysisof these latent variables as calibrate factors of the two structural compo-nents.Acknowledgments: A. Manuela Gonc¸alves acknowledges the financialsupport provided by the Research Centre of Mathematics of the Universityof Minho through the FCT Pluriannual Funding Program.ReferencesBengtsson, T., Cavanaugh, J. (2008). State-space discrimination and clus-tering of atmospheric time series data based on Kullback informationmeasures. Environmetrics, 10, 377-394.Costa, M., Alpuim, T. (2010). Parameter estimation of state space modelsfor univariate observations. J Stat Plan and Inference, 140(7), 1889-1902.Costa M., Gonc¸alves, A. M. (2010). Clustering and forecasting of dissolvedoxygen concentration on a river basin. Stoch Environ Res Risk Assess,DOI: 10.1007/s00477-010-0429-5.Kyriakidis, P.C., Journel, A.G. (1999). Geostatistical space-time models:a review. Mathematical Geology, 31(6), 651-684.

Improvement of surface water quality variables modelling that incorporates a hydro-meteorological factor: a state-space approach

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