Thordarson, Fannar Ørn

Hoved-emnet i denne afhandling er “grey box” modellering af hydrologiskesystemer, samt formulering og vurdering af deres indbyggede usikkerheder.“Grey box” modellen er en kombination af en “white box” model, en fysiskbaseret model, der traditionelt er formuleret som deterministiske ordinære differentialligninger,og en “black box”model, dvs. modeller baseret på statistiskeinput-output relationer. “Grey box” modellen består af en system beskrivelse,defineret ved et endeligt sæt af stokastiske differentialligninger, og enobservationsligning. System- og observtionsligninger repræsenter tilsammenen stokastisk tilstandsrum model. I “grey box” modellen er den samlede støjopdelt i målestøj og processtøj. Processstøjen skyldes model approksimationer, uopdagede input og usikkerheder i input-serien. Estimater af process støjenkan bruges til at fremhæve manglende fit i tilstandsrumformuleringen, ogyderligere støtte beslutninger for en model udvidelse. Ved brug af stokastiskedifferentialligninger til at formulere dynamikken i det hydrologiske system,forøger man enten kompleksiteten af modellen ved at inkludere de nødvendigehydrologiske processer i modellen, eller formulering for proces støjen kan betragtes,således at den opfylder de fysiske grænser for det hydrologiske system oggiver en tilstrækkelig beskrivelse af den integrerede usikkerhed i modelstrukturen.Afhandlingen består af to dele: en sammenfatning og en del som indeholderseks videnskabelige artikler. Sammenfatningen er ydeligere opdelt i tre adskiltedele, der introducerer de vigtigste begreber og metoder, der anvendesi følgende artikler. Den første del indeholder de grundlæggende begreber ihydrologi og relaterede hydrologiske modeller. Den anden del forklarer “greybox”modellen, ved at præsentere stokastiske differentialligninger og vise hvordanligninger kan være knyttet til de tilgængelige målinger. Desuden bliverimpuls respons funktion modeller introduceret, som alternativ til stokastiske differentiallignings baserede modeller, men ved at udnytte kendte hydrologiskemodeller som impuls respons funktion i denne model, bliver den delvistfysisk-baseret. Til at estimere parametrene i “grey box” modellerne anvendesmaximum likelihood metoden. Den tredje vigtige del i sammenfatningener forudsigelser, ogmed fokus på usikkerheder for forudsigelsesintervallerskal de tilsvarende performance mål også omfatte intervallerne. Afhandlingenillustrerer tre performance mål for vurderinger af usikkerhedsintervaller:pålidelighed, skarphed og opløsning. For beslutningstagere er et performancekriterium,der sammenfatter alle dissemål i et enkelt tal at foretrække, et sådantmål er fraktil skill scoren, der er diskuteret i denne afhandling.Anden del af afhandlingen som indeholder artiklerne er opdelt i to forskelligefag. Først er fire artikler, der betragter “grey box” model metoden til enkildeplads med flere driftmæssige pumper. Modelfundamentet er de styrendeligninger for grundvandsstrømning, der kan forenkles og repræsenteres på entilstandsrums form, der ligner de fremgangsmåder, der benyttes i numeriskemetoder til kildeplads modellering. Målet i de første to artikler er at demonstrere,hvordan en simpel “grey box” model er formuleret og, efterfølgende,udvidet med hensyn til parameterestimering ved hjælp af statistiske metoder.De simple modeller i disse artikler betragter kun en del af kildepladsen, mendataanalysen afslører, at boringer på kildepladsen er parvis højt korreleret. Iden tredje artikel, er alle boringer der pumper fra det samme grundvandsmagasininkluderet i tilstandsrum formuleringen af modellen, men i stedet for atudvide den fysiske beskrivelse af systemet er usikkerheden formuleret til athåndtere den spatio-temporale variation i outputet. Usikkerheden i modellener derefter vurderet ved hjælp af fraktil skill score kriteriet. I den fjerde artikeler kildepladsen formuleret ved at betragte impuls respons funktion modellertil at beskrive vandstands variationen i boringerne, som en funktion afalle tilgængelige pumpe rater i kildepladsen. Artiklen illustrerer ved hjælpaf et casestudie, hvordan modellen kan anvendes til at definere og løse kildepladsstyringproblemet.Anden halvdel af del 2 består af to artikler, hvor den stokastiske differentialligningsbaserede model er brugt for kloak afstrømning fra et afløbssystem.En simpel model er anvendt til at beskrive en kompleks nedbør-afstrømningsproces for et opland, men den stokastiske del af systemet er formuleret til atomfatte den stigende usikkerhed, når regnvand løber gennem systemet, samtat beskrive den nedre grænse for usikkerheden, når flow-hastigheden nærmersig nul. Den første artikel viser i detaljer “grey box” modellen og alle relateredeforandringer der er nødvendige for at opnå en realistisk model for kloak afstrømning.I den sidste artikel anvendes denne model til at forudsige afstrømning,desuden er performance af intervallerne evalueret ved hjælp af fraktil skillscore kriteriet.The main topic of the thesis is grey box modelling of hydrologic systems, aswell as formulation and assessment of their embedded uncertainties. Grey boxmodel is a combination of a white box model, a physically-based model thatis traditionally formulated using deterministic ordinary differential equations,and a black box model, which relates to models that are obtained statisticallyfrom input-output relations. Grey box model consists of a system description,defined by a finite set of stochastic differential equations, and an observationequation. Together, system and observation equations represent a stochasticstate space model. In the grey box model the total noise is divided into a measurementnoise and a process noise. The process noise is due to model approximations, undiscovered input and uncertainties in the input series. Estimates ofthe process noise can be used to highlight the lack of fit in state space formulation, and further support decisions for a model expansion. By using stochastic differential equations to formulate the dynamics of the hydrological system,either the complexity of the model can be increased by including the necessary hydrological processes in the model, or formulation of process noise can beconsidered so that it meets the physical limits of the hydrological system andgive an adequate description of the embedded uncertainty in model structure.The thesis consists of two parts: a summary report and a part which containssix scientific papers. The summary report is divided into three distinct partsthat introduce the main concepts and methods used in the following papers. The first part contains the basic concepts in hydrology and related hydrological models. The second part explains the grey box model by presenting stochastic differential equations and show how the equations can be linked to the available measurements. Moreover, impulse response function models are introduced as an alternative to stochastic differential equation based models, but by exploiting known hydrological models as the impulse response function in this model makes this model framework partly physically-based. For estimating the parameters in the grey box models maximum likelihood method is used. The third important part of the summary report is predictions, and with focus on uncertainty of prediction intervals the corresponding performance measures have to include the intervals. The thesis illustrates three performance measures for this performance evaluations: reliability, sharpness and resolution. For decision making, a performance criterion is preferred that quantifies all of these measures in a single number, and for that the quantile skill score criterion is discussed in this thesis. The second part of the thesis, which contains the papers, is divided into two different subjects. First are four papers, which consider the grey box model approach to a well field with several operating pumps. The model foundation is the governing equation for groundwater flow, which can be simplified and represented a state space form that resembles the methods used in numerical methods for well field modelling. The objective in the first two papers is to demonstrate how a simple grey box model is formulated and, subsequently, extended in terms of parameter estimation using statistical methods. The simple modelsin these papers consider only part of the well field, but data analysis reveals that the wells in the well field are highly correlated. In the third paper, all wells pumping from the same aquifer are included in the state space formulation of the model, but instead, but instead of extending the physical description of the system, the uncertainty is formulated to handle the spatio-temporal variation in the output. The uncertainty in the model are then evaluated by using the quantile skill score criterion. In the fourth paper, the well field is formulated by considering the impulse response function models to describe water level variation in the wells, as a function of available pumping rates in the well field. The paper illustrates, through a case study, how the model can be used to define and solve the well field management problem. The second half of part II consists of two papers where the stochastic differential equation based model is used for sewer runoff from a drainage system. Asimple model is used to describe a complex rainfall-runoff process in a catchment, but the stochastic part of the system is formulated to include the increasing uncertainty when rainwater flows through the system, as well as describe the lower limit of the uncertainty when the flow approaches zero. The first paper demonstrates in detail the grey box model and all related transformations required to obtain a feasible model for the sewer runoff. In the last paper this model is used to predict the runoff, and the performances of the prediction intervals are evaluated by the quantile skill score criterion

Online Research Database In Technology

Grey Box Modelling of Hydrological Systems:With Focus on Uncertainties

http://orbit.dtu.dk/ws/files/7946865/phd263_ft.pdf

Grey Box Modelling of Hydrological Systems:With Focus on Uncertainties

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