Modeling Ice Growth on Canadian Lakes Using Artilicial Neural Networks

Home , Baker Lake (Nunavut)

WATERRESOURCES RESEARCH, YOL. 42, wl1407, doi:10.1029/2005WR004622.2006 Cllck Here lq Full Article Modeling ice growth on Canadian lakes using artilicial neural networks

o. Seidou,rT. B. M. J, Ouarda,rL. Bilodeau,2M. Hessami.tA. St-Hilaire.rand P. Bruneau2

Received5 October2005: revised30 May 2006; accepted7 July 2006; publishedl4 November2006. lrl This paper presents artiticial neural network (ANN) models designedto predict ice in Canadian lakes and reservoirs during the early winter ice thicknessgrowth period. The rnodels fit ice thickness measurementsat one or more monitored lakes and predict ice thickness during the growth period either at the same locations for dates without measurements(local ANN models) or at any site in the region (regional ANN model), provided that the required meteorologicalinput variables are available. The input variables were selectedafter preliminary assessmentsand were adaptedfrom time series of daily mean air telnperature,rainfall, cloud cover, solar radiation, and averagesnow depth.The resultsof the ANN models comparedwell with thoseof the deterministic physics-drivenCanadian Lake Ice Model (CLIMO) in termsof root-mean-squareerror and in terms of relative root-mean-squareerrors. The ANN modelspredictions were also marginallymore precisethan a revisedversion of Stefan'slaw (RSL), presentedherein. They reproducedsome intrawinter and interannual growth rate fluctuations that were not accountedfor by RSL. The performanceof the models results in good part from a carefulchoice of input variables,inspired from the work on deterministicmodels such as CLIMO. ANN models of ice thicknessshow good potentialfor the use in contexts where ad hoc adjustmentsare desirablebecause of the limited availability of measurementsand where poor data nature,availability, and quality precludesusing deterministicphysics-driven models.

Citation: Seidou, O., T. B. M. J. Ouarda, L. Bilodeau, M. Hesstrmi,A. St-Hilaire, and P. Bruneau (2006), Modeling ice growth on Canadianlakes using artificial neural networks, lYaterResour. Res.,42, Wll407, doi:10.1029/2005WR004622.

1. Introduction to any location in the country, while the others are specific fo the measurementstation to which they [z] This paper investigates the capacity of artificial were adjusted. neural networks to simulate the growth of ice thickness The resultsof the ANN models are comparedto thoseof a on Canadianlakes. Although the growth of ice thickness modified versionof Stelàn'slaw and to publishedresults of can be modeled with nurnericalphysically driven models, the detern.rinisticmodel CLIMO (CanadianLake lce Model) the lack of suffrcientand precise data is often a linritation lMenard et a\.,200?a.2002b; Dugualt et aL.,20031. to the applicability of this kind of models. On the other [+] The remainderof this paper is composedof six parts: hand, extensiverecords of meteorological variables such an overview of ice growth rnodelsbased on thermodynamic as temperature,rainfall and snow on the ground are widely principles (section 2), an introduction to artificial neural available over the Canadian territory, along with ice networks (section3), the methodology for choosing input thickness measurements on a number of lakes. Neural variablesand rating results(section 4), the casestudy ofice network models are often proposed for situations where growth on Canadianlakes (section 5), resr.rltsand discussion (section the physics of the involved processesmay be cornplex or 6), and finally, conclusions(section 7). not fully defined, and where an extensive data sct is available for training a network while being incomplete 2. Ice Growth Models Based on from the point of view of deterministicphysically driven Thermodynamic Principles models [e.g., Olsson et a1,, 2001; Cannon and Whitfield, [s] This section presents an overview of ice growth 2002; Hewett,2003l. models basedon thermodynamicprinciples. Specialatten- Several artificial neural network (ANN) models of [l] tion is given to the formula known as Stefan'slaw, and to a lake ice thicknessgrowth are consideredin this study.One modified formr.rlareferred herein as the revised Stefan's law of theseANN models is regional in scopeand is applicable (RSL), as this formula is later used with the samedata sets as the ANNs in order to compare their performancesin rCentre Eau,'['ene et Ënvironnement,Institut National de la Recherche predictingice thickness. Scientifi que. Quebec, Quebec, Canada. rHydro-Québec, Formation and evolution of river and lake ice are Montreal, Canada. [o] Quebec, govemed by heat fluxes in the water body, heat transfers at

Copyrighl 2006 by the American Geophysical Union. the interfacesof air-watcr,air-ice, water-bed, and water-ice, 0043- | 397i 06 t2005WRO04622$09.00 and by radiationexchanges with ahnosphere.The different

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components of the energy budget are not easy to quantifu cover in any given year, and fr is a constant. In practice É since they are related to hydraulic conditions (turbulence is used as an adjustable parameter with a value that is and velocity distribution), turbidity (which has an effect on lower than the theoretical value to account lbr varying the absorption ofradiation energy), the albedo and the depth conditions of exposure and insulation; Michel ll97ll and compactnessof snow above the ice cover. gives a range of values adapted for a variety of lakes and [r] All numerical ice growth rnodels use a more or less rivers. simplified version of energy budget. Some of thesemodels [rz] The date of the onset of the ice cover is a basic apply to a specificaspect of ice developmentsuch as the ice parameter for using Stefan's law as it determines the date at cover initiation lSchulyakovskii, 19661,border ice formation which the accumulation of freezing degree-days is started fMatousek,1984;Svensson et a\.,19891,frazil ice formation for any given winter. For the majority of sites of interest in fomstedt, 1985a,1985b; Svensson and Omstedt,i9941, and this study, this date is not known so that the normal usageof ice cover growth [e.g., B eltaos, 1995; Schulyaknvskii, 1966; Stefan's law was not feasible. lt was therefore decided to "revised Lock, 19901.Other models are more complete and may use a Stefan'staw" (RSL), which is basedon Dn, sirnulate ice fonnation, transport, growth and decay lShen the accumulation of freezing degree-daysstarting with the and Chiang,1984; Shen and Ho, 1986; Shen et al.,1990, first day of below freezing air temperature in any given 1e951. season.RSL presentsone more adjustable parameter,C, the [t] They all use energybalance to computeice evolution. eflèctive number of degree-daysto be subhacted from Dn in They differ from each other by the level of detailsused to order to obtain D.1. describettre different aspectsof the phenorncnon:hydraulics, heattransfer and radiation.Lake ice modelsare sirnpler ",_[rJD*T ,"f Dr2C t)\ sincewater velocity is low enoughto justify the use of one- Vr l0 { Dr

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rons in the hiddenlayer and a linear neuroniu the output layer lHagan, 19961:

antt : J:n+t(ilr^+t a^ + B'*t) m = 0, | (3) LINEAR NEI}RON

where,a0 is ttrenetwork input, al is the output of the hidden layer, az is the network ouçut, /t andf2'are respectively sigmoid and linear transferfunctions:

.f'(r):#, (4)

HIDOENLAYÊR OUPW LAYER

f,(n) :,, (5) Figure l. Architecture of a tèed forward network with two neurons at the hidden layer and one neuron at the output L/'"*t and,B'''L arenetwork weights and biaseswhich are layer. definedwith the following formulas: 4'T' /iij "black | I [àî*''l It is a box" type of model for which the user has w^rt : ,. ' 8"": no control on the intemal behavior, and it can poorly | "t_:1., ,(-,-lrr.l tu' perform when used for generalizationespecially when the L lru,;] m:0,1 number of neurons is too high for the complexity of the problem (overfitting). In hydrology, ANNs have been used where ,56is the nr.rmberof input variables,Sr is the number for hydrologic data classification [e.g., Liang and Hsu, of neuronsat the first layer and 52 is the number of neurons 19941,river dischargeprediction le.g., Sham,seltlin,19911, at the secondlayer. evaluationand forecastingof \vater quality [e.g., Zhang et al., 1994], inflow forecastingfor irrigation and hydroelec- [ta] One-hiddenJayerneural nefworks with sigmoid activation function on the hidden layer and a linear activation tric darns le.g., Coulibaly et al., 20001, rainfall estimation neuronwere shown to be able to approximateany bounded [e.g., Xiao and Chantlrttsekar, 1997], and correction of continuous function with arbitrarily small enor streamflow under ice lOuarda et a1.,2003). fCybenkn, 1989], provided the number ofneurons in the hidden layer is sufficient.Because of the neuronswith sigmoid activation 4. Methodology functionsin the hiddenlayer, this ANN model is not a linear [ta] The capabilities of RSL and ANNs to model ice nrodel. Other architecturesmay have been chosen,br.rt there growth on individual Canadian lakes are investigated.The are no rules for choosing the number and the sizes of an performancesof the ANN models are investigatedas ANN model. For instance,it is believedthat increasingthe function of its input variables and their intemal structure, numberof layerscan increasethe learningcapabilities of the and a procedure is set up to select the best perfomting model, but it also increasesthe number of parametersand models. thus the length ofthe seriesrequired to properly train it. For [ts] Two kinds of ANN models were consideredin this the onset of this study, the simpler and most popular stndy: (l) site specific ANN models were trained and architecturewas chosen. validated with ice thickness data from a single site and [rs] Figure I showsthe architectureof a one-hidden-layer (2) a regionalANN model was also constructedin order to neural network with three input variables, two neurons in simulate ice growth for sites without field measurementsof the hidden layer and one neuron in the output layer. The ice thickness. Each model is characterizedby the set of its synaptic weights are obtained using a supervisedtraining input variables (referred to as a combination of input algorithm using Bayesian regulation lMackay, 1992, 19951. variables), and the ANN architecnrre (defined by the num- In supervised training, both the inputs and the outputs are ber of neuronson the hidden layer). provided. The network then processes the inputs and [te] Choosingthe best set of input variablesand assess- compares its resulting outputs with the desired outputs. ing the performanceof a model requiressome performance Emorsare thenpropagated back throughthe system,cansing criteria,some model validation procedures,and a methodof the systemto adjustthe weights which control the nefwork. picking the best model among all possibleconfigurations. [zo] An iterative trial and error process,described in more Sinceour modelsare data-driven,two validationprocedures detailsin the restof the paper,will be usedto settlre optimal were used to ensure that the final ANN rnodels are properly number and natureof input variablesas well as the number hained and can be used with confidencefor generalization of neuronsin the hidden layer. purposes. These validation procedures are described at 4.2. Selectionof lnput and Output Variables section4.4. [zr] ANNs are data-ddvenmodels, so their performance 4.1. Architecture of the Artilicial Neural Networks for a given problem relies on the relevanceof the inpuV [n] The artificial neural network used in this resealch output variablesthat are consideredin the training process, was a one-hidden-layerneural network with sigmoid neu- and on the complexity of the relationshipbetween inputs

3 of 15 w11407 SEIDOUET AL.: ICE GROWTHON CANADIAN LAKES wll407 and outputs.As it is the casefor physically driven models, Table l. Tested Combinations of Meteoroloqical Variables problem pararfieterization can dramatically change algorithm efficiency. The purpose of the ANN being the Clombinationsof MeteorologicalVariables Variables' prediction of ice thickness, the formulation of its input and output variables was guided by these practicai consid- I Da; Rad,, erations: (l) inputs should be rneaningful for the ice growth 2 Da; Rad,..+ 0.25 Rad,, + process and available at all ice measurementstations, (2) the 3 Dd ; Rad'," 0.50 Rad, Da; Rad,. +.0.75 Rad" sensitivity ofthe output variable to input variations should 5 D1 ;Rad,,"+ Rad" be as high as possible to enhance training efficiency, and 6 D6;Rad,.;Fg (3) the sensitivity of the output variable to input variations 7 Da; Rad,,.+0.25 Rqd; Fls should have the sane order of magnitude for all input 8 D1 ; Rad* + 0.50 Rad";I{s + g variables. 9 Da ; Rad," 0.75 Rad";È l0 Dd; Rad^" + Rad"; H s 4.2.1. Output Variable ll D7; Rad^";R lzzl At first, it seernedobvious hhatHl the ice thickness, t2 Da; Rad,,"+ 0.25 Rad.;R should serve as tlre output of the ANN. Early runs of the TJ Da; Rad,,.+ 0.50 Rdd.;ll + ANN showed however that the ANN relied almost exclu- l4 D,; Rad," 0.75 Rad";R l5 Da: Rad,,"+ Rodc;R sively on the sum of freezing degree-days and made little t6 Da; Rad,";H"; R use ofthe other variables(presented below). The reasonfor t7 Da ; Rad,. + 0.25 Rad; FIs I R this is that the influenceofthe sun offreezing degree-days l8 Dd ; Rad^.+ 0.50Rad"; F!3; R on the output function was so strong that the training l9 Dd;Rad,,.+ 0.75 Radc;Ê3;-R 70 D6 Rgd," + r1o4;Às; F algorithm (which is basically a numerical optimization 2l Da )_Hs algorithm) was unabie to account for the effect of the otirer 22 DaiR variables.Other variablesclosely relatedto the ice thickness 23 Da;Hs;R were then sought with the intent of improving the fit 'Noteis usedonly for datapreprocessing and posprocessing. betweenobservations and predictionsof ice tlrickness. [z:] After some exploration, the output variable that was retained for this study is the parameter HilDl, which correspondsto the Stefan coefficient. Once a time serics D"("Cdav), as was already describedearlier in this paper, ofthe growth rateofthe squareofthe ice thicknesshas been (2) the sum of solar radiation during the period of ice computed, a time series of ice thickness is easily recon- growth for days with precipitation(W daylm'), divided by structed. The parameter HjlDl is drawn directly from the the sum of degree-days ,Rad" (this quantity is a proxy to classicStefan's law and representsan instantaneousevalu- the quantity of solar radiation attemratedby cloud cover), ation of the square of its constant Ë. Using this output (3) the sum of solar radiation during the period of ice variable led to a reductionof the weight of the degree-days growth for days without precipitation(W daylm-), divided input variablein the ANNs, comparedto that of other input by the sum of degree-days Radn", (4) the average daily variables, and presnmably allowed the ANN to extract rainfall (over time) during the ice growth period R (mm), additional information from the other input variables. It and (5) the average on-ground snow depth (over time) was found that this choice of an output variable increased during the ice growth period Hs{cm). appreciablythe precisionofthe ice thicknessprediction and [zo] The sum of degree-days D, was used to compute led to a better comprehensionof the eftèct of the other HllD, on the calibration data set (preprocessing),and to variablesin rrodeling the ice thickness. reconstruct ice thickness time series from simulated series 4.2.2. Input Variables (postprocessing).It wiil be referred to as an input variable [z+] In the present shrdy, selecting the input variables of in the remainder of the text. the ANNs was a two step process. First, candidate physi- [zz] The sum of solar radiation was split into two parts cally observedvariables were identified. Then, ANN input because only the total amount that reaches the ground variables were formulated using the physically observed should be considered. Consequently,attenuation due to variables. These physically observed variables were cloud cover has to be accounted for by multiplying Rad" retained according to their availability and recognized by a positive factor * smallerthan or equalto l. Five setsof significance for the heat budget involved in ice growth: combinations of Rad," and Rad. were considered: Radn., (1) daily mean air temperatrre,(2) daily total solar radia- Radn,+025 Rad,, Radn. Rad., Radn"+0.75 Rad", Rad,,"+ tion, (3) daily rainfall, and (4) daily snow depth on the Rad., correspondinglespectively to o : 0,0.25,0.50, 0.75 ground (as measured at weâther stations). Longitude and and 1. A total of 23 setsof combinationsof meteorological latitude were also tested as additional parametersfor the variables (listed in Table 1) were tested as inputs for the regional neural network model. artificial neural network models. [zs] The preceding variables were used to construct a [zs] Other important variables for ice growth would have variety of variables,to be used singly or in linear combi- beenvariables based on lake morphology suchas maxlmum nations as input variables for the ANNs. When these depth, mean depth, surface area, perimeter length and other variables are sums, the sumrnation starts on the first day variables that could be derived from these. For most of the of frost basedon the daily mean air temperatureand endson lakes considered,depth information could not be found the day assignedto the variable. The following variables within the scope of this study. Other parameters baseclon were consideredand computedfor eachwinter: (1) the sum the surface area and lake shape ol1 maps were not used in of freezing degree-days derived from the air temperature the eird. It should be noted that this information is never-

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(RMSE),relative root-mean-squal€ error (RRMSE),model explainedvariance (l), Nash criterion (NASH), and bias (BIAS). The criteriaare definedas follows: RMSE: - uïf)' (7) Gtr,

RRMSE:(;e e"!)')' (8)

^'l)2 cov([n].....n'l,lt,..., ',, - . (e) var(fv|,...,nil)var(in j ,. ,nf}"

140 120 100 80 60 1,O 120 lm 80 Longitude(W) B'AS:li - rf) (ro) tr-' fal Figure 2. Location of the 26 lake ice measLrrement stations. - thelessconsidered to be relevant,particularly for the eval- D@f nf)' uation of the date of freezeup. It is felt by the investigators NASH:t-oi (1r) that the use of morphological lake data could provide \ta'; - rt,1' interesting avenues for the use of the ANNs ibr the modeling of lake ice in future studies.They would require a specializedfocus on sources of data that could not be where .Hf,i : 1,..,n and nf,i : 1,...,n are observedand explored within the framework of the present study. simulatedice thicknesses. 4.3. Performance Criteria 4.4. Validation Procedures [zo] The performanceof a given ANN model is evaluated po] Two validation procedures were used: the leave- using five performance criteria: root-mean-sqr.rareerror one-out cross-validationprocedure to find the best combi-

Table2. Ice ThicknessMeasurernent Stations

Longitude, Latitude, Station Code Station Name Water Body

HA1" Quaqtaq Unnamed Lake 68.36 61.03 LI'I Alert Upper Dumbell Lake 61.5 82.46 WFN' Cree Lake Cree Lake 105.t5 ) /.JJ WIQ Primrose Lake Primrose Lake 109.93 54.76 WLH" Lansdowns house Attawapiskat Lake 86.09 52.21 WHO Sainte agathe des monts Lac des sables 73.69 46.03 WTL" Big trout Lâke Big trout Lake 88.11 53.8r YAH" La grande IV Lac la Tanière I L.JO )J. /O YBK, Baker Lake Baker Lake 95.96 64.3 YBT" Brochet Brochet bay of Reindeer Lake 100.3I 57.86 YBX' tslancMsablon Lac à la Truite 56.8I 51.45 YEI" Ennadai Lake Ennadai Lake 99.09 61.11 YGK" Kingston Lake Ontario-Horsey Bay /).J 44.7 YGM" Ginli Lake Winnipeg 95.01 50.61 YGV Havre Saint Piene Patterson Lake 63.88 50.29 YIV" Island Lake TslandLake 93.31 53.84 YKL' Scheiferuille Knob Lake 65.r9 54.78 YNE Noruay house Little Playgreen Lake 96.16 53.98 YNI Nitchequon Nitechequon Lake 69.08 53.18 YPY" l'ort chipewyan Lake Athabasca I I 0.83 58.7 'l'hunder YQl' bay Thunder Bay 88.78 48.43 YVP, Kuujjuaq Stewart Lake 67.53 58.11 YYR" (ioose bay Tenington Basin 59.58 53.33 YZE South baynrouth Huron Lake (South Bay) 8l .9t3 45.54 YZF" Yellorvknife Great Slave Lalie (Back Bay) 114.34 62.45 YFL, Quaqtaq Great Slave Lake (Fort Reliance) I 08.86 62.70 "Stations with enoush data to calibrate local ANN models.

5of15 w11407 SEIDOUET AL.: ICE GROWTHON CANADIAN LAKES wlt407 nation of input variables aud ANN structure, and the standard split sample procedure in which the data is split in two subsets, the first being used to hain the neural network and the latter being used to check the model performance. [lt] The leave-one-out procedure aims to assessthe generalization capability of a given ANN model. In the leave-one-out procedure, the data is divided in subsets.A subsetis representedby all measuremeûtsduring a year (for local ANN modelsand RSL) or all measurementsat a given site (for the regional ANN model). All subsetsbut one are used to train the neural network. Then the trained ANN model is used to simulate the subset that is left out. The same process is repeatedso that every subset ofthe data is 500 100û 150û :û0t 2gD 3u 35û) left out once. The performance indices are then computed Sumûf0e0pe-0ays using the observed and sirnulated values over the whole data set. The pair of input variables and ANN structure which best explain the data is then selected for the rest of the procedure. Unfortunately, the leave-one-out procedure does not give a final model since the neural network is trained severaltimes (one time for eaclrsubset) to compute the perflomranceindices. E i;c pz] The objective of the split sample validation procedure is to have a trained neural network to use for the rest of t0: the study. In this procedure. the data is repeatedlyand c[ randomly divided into two parts containing 80% and 20% of the observations.The first part is used to train the ANN and the other is used to rate it. The operation is performed twenty times and the pârameterset of the run which gives f, the srnallestRMSE is retainedand completesthe assembly 0 5I] 10ûr 150û 2ttû &Ê 3ù+l 35æ 4m i5m of the rrodel. Sumo, ljegre.Days

4.5. Selectionof the Best Combination of Figure 3. Identification of ice growth phase in the data Meteorological Variables and ANN Structure sets at station YZF: (a) all observations and (b) ice growth [r] First, all possiblepairs composedof a set of input phase. rneteorologicalvariables (listed in Table l) and a network structure (defined by the number of neurons in the hidden (2) daily meteorologicaldata obtained from Environment layer) ale formed, given that the number of neurons on the Canada, and (3) incident solar radiation at the top of hidden layer is constrainedto be less than 10 for compu- atmospherewhich have been computedas function of time tationalpurposes. For example,the pair (3, 7) rcpresentsan and geographic location of the studied sites using the ANN model with sevenrleurons on the hidden layerand for formulas presentedin Appendix A. which the inputs variablesarc D4, Rad,,, + 0.50 Rarl.. The pz] The data set containsice tbicknessand snow depth performance criteria of each pair are then rated using the measurements on 53 sites located on Canadian lakes leave-one-outprocedure. fLenormand et a1.,20021.Only 26 ofthese siteshave years [l+] When somevalues of an input variable (suchas surn with enough measurementsto calibrate Stefan's law and the of rain or snow on ground) are missing at a given site, the local ANN. Eight of these stations were not used for combinations of meteorological inputs containing that calibration and validation of the regional ANN because of variable cannot be used. In this case, only a part of the data availability issues that will be further explained. The 23 possible combinations of inputs ale tested when 26 ice measurement stations are listed in Table 2, and searching the best local ANN model. To avoid this their geographic locations are presentedin Figure 2. situation with the regional ANN rnodel, it was trained [:a] The weather data was provided by the national and validated with tlie data of the measurement stations climatic archivesof Environment Canadaand containsthe where the entire input variable were available. daily data of temperature, precipitation (snow and rain) psl The bestpair is then chosen,and the cornbinationof and snow on the ground for more than 10,000 stations meteorologicalvariables in this pair is considercdto be the distributed all over the Canadian territory. one which best explains the data, while the number of neulons in the pair definesthe best ANN structr-rre. 5.2. Proxy Variable for Solar Radiation at Ground Level 5. CaseStudy [:e] Solar radiationwas consideredas a relevantvariable at the onset of tbis study because radiative fluxes were 5.1. Data recognized to be a significant part of the lake energy by pol The data used in this study is of three types: (l) ice most authors[e.g., Lock, 1990;F'ang et a|.,1996; Menard et tlrickness data from the Canadian lce Service [2005], a1.,2002a, 2002b; Duguay et al., 2003). Howeveq solar

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a) temperature on a given date at a given point is interpolated using the observed values at the nearby stations: a data processing program secks the nearest weather station suc- 90- cessively in the northeast,northwest, southwest, and south- east quadrants.the temperature of the ice measurementsite is computed as the weighted average of the temperaturesat 80. the weather stations. The weights are inversely proportional to the distancesfrorn the site of measurementto the weather stations. If historical measurementsare not available at the ten closest stations in a quadrant on a given date, the data is considered missing. When a variable (such as rain, degree- days or snow) has missing values, the combinations of inputs containing this variable cannot be used. This may happen when weather 50' stations are too far frorn the ice measurementstation. In this case only part of the 23 possible combinationsof input listed in Table I can be 40" considered. At eight of the 26 lake ice thickness lneasure- l ment stations, some irregularities were observed in the data (such as a long delay between first freezing air temper- w- 110 124 1m 80 60 140 120 100 80 Longitudo(W) a) b) BAKERLAKE YËK. CALIBRATIOI{ 300

æ0 *- :tt iÀI *+'* {.* 1ù0 + 0 2m 3m0 40m BAKÉRI.AKË Y8K. VALIDATION 250 * I * 3 t5o + jhFiÇF- + Ë too Esu + ObseP6d - Simulaled(revised 0 0 2m0 3000 40to Srmof 0egree-days

KINGSIONYGK. CALIBRATIOI.I 140 120 1oo 80 120 1oo 80 8B -'l1*"1ffi 60

Figure 4. Spatial variability of RSL parameters: 40 a) parameter parameter C; b) k. x

radiationmeasurements at the ground level were not avail- KINGSTONYGK. VALIDATION able in su{Iicient quantity since they are not measuredat al1 BO weather stations. As an altemative, assumed ground level E .-.. solarradiationwas calculatedusing solar radiationon top of !lN the atmospireremultiplied by a factor o; ci is equal to the Ë40 unity on dayswithout precipitation,and lower than the unity on days when rain or snow was observed.The values of ;20 solar radiation on top of the atrnospherewere calculated as 0 functionsof the date and the geographiclocation according 0 10ù 2ao s)l 400 5t0 6tl ?ff 80ù to an algorithm giveir in Appendix A. Sum of Deqree-davc

5.3. Spatial Interpolation of Weather Data Figure 5. Observedand simulated (RSL) ice thicknesses [.lo] As geographicpositions of the ice measuringsites versus sum of degree-days at some ice measurement did not coincide with that of the weather stations.the air stations:(a) YBK (Baker Lake) and (b) YGK (Kingston).

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Table 3. Performance Criteria tbr the Revised Stefan's Law

Calibration Validation

k, L, RMSE, BIAS, RMSE. BI,AS, oç-o.s Station a'n 6-0.s d'c cm I RRMSE cnl cm .2 RRMSE cm

HAI 2.80 14.06 4.97 0.98 0.05 1.02 9.81 0.96 0.09 -J.)ô LTI 2.57 46.16 I 1.94 0.90 0,12 3.16 30.29 0.72 0.l9 47.30 WFN 1.54 90.23 4.17 0.86 0.07 0.10 14.84 0.68 0.l6 -6.68 WIQ 1.84 231.50 6.21 0.84 0.t7 -2.94 8.49 0.74 0.28 2.90 VYLH t.86 33.67 5.87 0.70 0.10 -l-/o 9.34 0.83 0.14 3.61 woH 2.06 60.66 4.94 0.69 0.08 0.73 8.4r 0.82 0.t8 1.20 WTL 1.95 14.72 ),trI 0.91 0.08 0.49 r3.82 0.87 0.l5 -8.09 YAH 1.76 62.73 5.53 0.93 0.10 -t.99 10.65 0.89 0.l6 7.20 YBK 3.05 154.t2 6.07 0.97 0.06 4.27 I 1.69 0.96 0.l0 -4,22 YBT |.64 t2.87 6.50 0.86 0.ll -0.28 7.32 0.92 0.t4 - t.J) YBX 2,91 56.24 7.92 0.72 0.23 1.08 21.86 0.83 0.50 t9.12 YEI 2.70 44.42 7.29 0.81 0.08 4.06 13.64 0.92 0.l3 2.94 YCK 2.70 209.03 4.83 0.71 0.40 3.17 10.51 0.77 0.91 8.31 YGM 2.22 58.67 5.90 0.60 0.07 l -oJ 14.20 0.85 0.15 ,9.81 YGV 2.5) l 08.00 5.t5 0.87 0.l0 1.24 32.87 0.74 l.l3 28.97 YIV 1.88 140.33 5.l4 0.69 0.08 0.l5 |.22 0.76 0.19 1.78 YKL 2.09 59.87 6.29 0.92 0.07 0.83 l 1.38 0.91 0.15 -ô.) I YN-E 2.t2 10.71 8.05 0.80 0.21 2.88 9.26 0.88 0.17 10.46 Ylit 1.75 57.78 5.81 0.92 0.1I 2.17 9.41 0.89 0.13 -5.44 YPY 1.89 95.19 10.30 0.55 0.26 |.25 l 3.83 0.70 0.29 8.65 YQ' 3.13 147.73 4.t9 0.74 0.08 * 1.06 lt.24 0.80 0.92 51.10 YVP 2.53 58.89 6.24 0.86 0.08 2.78 tt.44 0.95 0.12 6.28 't.62 YYR 2.t6 l 6.58 7.80 0.'73 0.16 | 3.68 0.67 0.19 3.05 YZE 2.32 58.27 5.75 0.70 0.l3 -0.92 6.95 0.74 0.16 2.30 Y7F 2.14 52.98 6.57 0.85 0.09 2.21 13.80 0.84 0.l6 0.27 YFL 1.99 423.2 15.24 0.80 0.09 -0.30 2.21 r3.80 0.l6 0.27 Muimum J.I J 423.20 t5.24 0.98 0.40 32,87 r3.80 l.l3 51.10 Minimum 1.54 10.71 4.17 0.55 0.05 --.2.94 2.21 0.67 0.09 -9.81 Mean 2.23 89.18 6.70 0.80 0.tz 0.98 12.78 1.32 0.27 6.16

atures and the ice growth starting period, or nrissing when designingthe procedureand were usedfor subsequent valuesin someof the interpolatedweather variables) so only analyses. the 18remaining stations were used forthe constructionofthe ANN models.These stations are indicatedin Table2. 6, Results and Discussion 5.4. Elimination of the Period of Thinning at the End of the Winter in the Data Set [+z] The data sets describedin section 5 were used to calibratethe RSL, and to train the local and regional ANN The model developedin the present study is appli- [ar] models. ANN training and simulations were performed cable to the growth phase ofthe ice thickness. It is therefore using the Neural Networks toolbox of Matlab fThe important to eliminate the period of ice thinning which Mathvorks, 20051.The training function (trainbr function happens at the end of winter. For this purpose, we use an in the Matlab environment) uses Bayesian regulation empirical iterative procedure: starting from the end the of p4ackay, 1992, 19951 to enhance generalization capabili- data series and moving toward its beginning, all measure- ties. The maximum number of epochs is set to 1000, and ments are successivelyconsidered for eventual rernoval. All the minirnum gradien^t(for minimization of the objective the dates when the measured ice thickness happened to be function) to I x l0-'". The training algorithm also usesan higher than the average ice thickness of the remaining adjustable parameter p-u which controls how far the next subsequentrneasurements. Previously elirninated values values of the parameters will be searched during the are not accounted for when cornputing the average ice optimization process. was set to its default value in thicknessof the remainingsubsequent measurements. lr,max When Matlab.i.e.-1x10'". a given measurementis removed, all subsequent ûleasure- ments are also rernoved.This practical procedure insur.es 6.1. Comparison of RSL ând Local ANN Models tirat the thinning period, and part of the period when the [+:] The spatial variations of RSL parametersare illus- thicknessstagnates are eliminated.The rcsultsof the appli- tratedûr Figure4, and no trend could be found with respect cation of such a procedureto station YZF are illustratedin to geographicallocation. It was found that RSL l'epresents Figure 3. Figure 3a presentsthe wbole set of measurements an excellent ice growth model when it is calibratedfor a including the thinning period and displays several step given site. Observed and sinnrlated data are representedtbr drops on the ice thicknesswhich would have affectedthe sorneof the ice thicknessneasurement stations in Figure 5. training of the ANN models. The retained nleasuremenrs The pararnetersof Stefan's law are obtained by the least after applicationofthe proposedprocedure are illustratedin squaresmethod using 80% ofthe data.There is a very good Figure 3b. These resultscorespond to what was expected agl€ementbefween simulated data and observations.with a

8 oJ'15 wil407 SEIDOUET AL.: ICE GROWTHON CANADIANLAKES \ry11407

Table 4. Performance Criteria for Revised Stefan's Law and Local the Stefan's law except at two stations: the station YBK ANN Models" (Baker Lake: Figure 5a) and station YGK (Kingston: Figure 5b). The data of YBK (Baker Lake) present very Numberof RMSE RRMSE Best Neuronson the (Stefan), R2 (ANN), / little variabilify and the two models had quasi equal Station Combinâtion Hidden Layer cln (Stefan) cm (ANN) performances. The data of the second station come from Lake Ontario, one of the Great Lakes with an average HAI l0 0.95 8.94 8.02 0.96 depth of 91 m and it comes as WFN 5 I 1.59 0.71 I 1.07 0.74 no surprise that the WLH 23 t4.s4 0.59 r I .28 0.'76 growth of the ice starts rather late comparatively to all the WTL o 10.54 0.85 r0.50 0.85 others (degree-daysof about 200). YAH 22 8.96 0.81 7.97 0.85 [as] The number of cornbinations for which ANN YBK l5 I L59 0.96 n.72 0.96 models perform better than the Stefan's law vary from YBT () 8.53 0.86 8.52 0.86 (Brochet YBX t0 13.63 0.66 r3.03 0.69 two at station YBT lake) to more than 120 at YEI l0 I 15.55 0.86 t4.27 0.88 station YAH (La Tertière Lake). The best combinations YGK 7 9 9.t6 0.74 13.43 0.47 (in terms of RMSE) for eacb station are given in Table 4. YGM 9 3 14.70 0.59 10.99 0.77 The best combination is different for each station, but YIV l0 2 12.03 0.69 n.47 0.72 86% contain the proxy for radiation, 550/ocontain YKL I 9 9.72 0.89 9.36 0.90 snow YPY ll 9 19.98 0.54 19.88 0.55 on the ground and 33Yo contain rainfall. The dominating YVP 20 7 t3.22 0.88 12,25 0.90 factor aparl from the degree-daysin the ice growth processis YYR t4 9 12.86 0.69 I l.9l 0.74 then solarradiation, followed by snow and rainfall. YZF t0 2 13.73 0.83 t2.48 0.86 Once the best combination of variablesfor a given YFL 5 t0 17.10 0.74 16.78 0.75 [.to] station selected,the local neural network model for this "Best combination ancl optimal number of neurons on the hidden layer station is obtained following the procedure described in and perfomranceindices obtained during the leave-one-outcross-validation section4.4. The perfornranceindices of the retainedlocal procedure. ANN models are listed in Table 5. [ar] The variation of the ice thicknessessimulated with both the local ANN and RSL at station YGM when rneanRRMSE of 0.12 fbr the calibrationset and 0.27 for the performing the leave-one-outprocednre are presentedin validation set (Table Hence 3)- RSL may be an excellent Figure6. Figures6a and 6b both correspondto a casewhere rnodel for engineering purposes when a few errors of a year of data has been excluded. It can be seen that the ceutimeters not do matter. However, its major drawback is ANN rnodelfollows data variationsmore closely than RSL. that parameter values ltom lake to It are variable lake. is The ANN model adaptswell to the intrawinter and inter- thus difficult to determine values lake which to apply for a annual variations of the ice growth regime while RSL is without observations, the legional ANN unlike model alwaysrepresented by a single curve that tendsto follow an developedin this paper which can readily be used tbr such average interannual regime. lakes. 6.2. Comparison of the Retained Local ANN Models [a+] Table 4 illustrates the performance of the local ANN rnodelsand the revised Stefan'slaw when using the leave- With ANN Models With Sigmoid Oufput Function and one-out cross-validationprocedure. It tumed out that it is Multiple Linear Regression always possible to find a combination of meteorological [+s] In this study, the choice of one-hidden-layerneural variables such that the neural networks perform better than nehvorkswith linear output funct'ionsand signoid neurons

Table 5. Perfonnance Criteria of RSL and the Retained Local ANN Models'

Combinationof RMSE RMSE BIAS BIAS Meteorological (RSL), (RNA), r I RRvsn RRMSE NASH NASH (RSL), (ANIù), Station Variables cm cm (RSL) (A]\rl\i) (RSL) (Ar.rN) (RSL) (AI.IN) cm cm

HAI l0 l0.ll 5.00 0.98 0.99 0.1I 0.05 0.91 0.98 -8.45 -3.65 WF'N 5 7.84 7.76 0.85 0.85 0.13 0.13 0.83 0.84 r.55 2.10 WLH L5 5.30 4.7| 0.93 0.96 0.1| 0.10 0.90 0.92 3.03 -2.82 WTL 9 u.85 7.36 0.93 0.93 0.13 0.1I 0.90 0.93 4.54 1.33 YAH 22 7.35 6.99 0.93 0,93 0.14 0.13 0.87 0.88 4.63 4.17 YBK l5 7.59 7.28 0.98 0.99 0.06 0.06 0.98 0.98 0.28 0.21 YBT 9 8.31 7.38 0.88 0,89 0.14 0.t2 0.85 0.88 - 1.93 0.26 YBX 4 I 1.82 9.63 0.76 0.83 0.24 0.20 0.71 0.81 4.20 3.32 YEI l0 14.5l It.77 0.88 0.93 0.16 0.l3 0.87 0.91 3.72 3.21 YGK 1 1.36 t0.27 0.92 0.91 0.?7 0.31 0.86 0.72 3.76 -3.35 YGM 9 t4.42 6.t4 0.64 0.93 0.18 0.08 0.63 0.93 0.83 0.I8 YIV t0 8.33 0.88 0.90 0.l5 0.13 0.84 0.87 3.79 r.80 YKL I 10.18 9.66 ('t.92 0.92 0.14 0.11 0.90 0.91 3.83 2.47 YPY ll t8.ll t7.57 0.69 0.70 0.31 0.30 0.67 0.69 032 -2.04 YVP 20 I 0.80 7.40 0.94 0.96 0.13 0.09 0.90 0.95 5.45 1.30 YIR l4 9.64 9.l-5 0.77 0.81 0.14 0.14 0.77 0.79 -0.58 2,61 YZF l0 9.50 8.00 0.93 0.94 0.13 0.1I 0.92 0.94 -4.25 -2.19 1 À< YFL 5 IJ.JO 0.93 0.95 0.18 0.10 0.83 0.94 9.45 3.15 "Data setsrandomlv snlit in ti0-20%subsets,

9o115 wl1407 SEIDOUET AL.: ICE GROWTHON CANADIAN LAKES wlt407

a) of a sigmoid activation function resulted in RMSE reduc- tions of up to 10% at some sites,and an increaseof RMSE Ë ofup to 15% at others(Figure 7, top). In general,the use of a sigmoid activation function for the output neuron gives o slightly better results. On the other hand, the use of '; multiple linear regression always resulted in an increased

a RMSE (Figure 7, bottom), confirming the non linear nature of the ice growth process.

Sumof Degree-days 6.3. Influence of the Data Sets Characteristlcs on the Best Performing Solutions E | The retained combinations of explanatory variables , + ++ [+o] ,r' for the local ANN models are very different from site to site. * [r o * -g Consequently, it is important to investigate whether the J ',! c characteristics of the input variables have an effect on the o best perfbrming variables. Figure 8a presents the mean snow depth on the ground for the ice measurementstations listed in Table 4. Figure 8b presents the list of sites where Sumc,f Degree-dayg the best combination of explanatory variables includes b) snow Similar results are presented lbr rainfall (Figure 9a 150 and 9b) and radiations (Figure 10a and lOb). As for the È parametersof the revised Stefàn's law, there seemsto be no Y variable, and Elm clear relationship befween the magnitude of a o nrfrô Ë whether or not it is chosen as an explanatory variable. _g Figure I I presents the maximum ice rneasurements r* 0bseflÉd l.l Srmulaied (ANN) o (Figure 11a) as well as the RMSE of the regional model + SimulslÊd nL-0 at thesesites (Figure I 1b).The RMSE of the regionalmodel 1000 150û seems to be constant all over the territory. The bias of the 3um of Deqree-days regionalmodel is presentedon Figure 12. It can be seenon 1n Ë Figure 12 that the regional rnodel slightly underestimates û1m .r+l' ice thicknesses in the northwestem part of the country .*+ where ice thickness is generally large. Conversely, it slightly 38û ,t nn Ll c part EdJ + overestimatesice thickness in the southwestern of the -'FhnnJ ' Obæryrd Y country, where ice thickness is smaller. 6 .to û Simulated(ANrl) * Srmulaied 6.4. Limited Comparison of Local ANN Models and 5m 1000 15æ ætO Deterministic Model CLIMO Sumof Degree-days [so] The deterministic model CLIMO was used by Mënard et al. to simulate ice growth at. stations Figure 6. Comparisonof local ANN models and RSL at 12002b) station YCM (leave-one-outprocedure) for (a) 1977 and (b) 1980. Lccal Al.lf.l trodel with sigmoid output function

3 Iri E îoi on the hidden layer was due to the demonstrated value of 5i, this kind of architecture. In theory the capability of an Ë 0;- ANN model to leam the hidden logic in a data set is not '-ral-.-l--l----r supposed to heavily rely on the shape of the activation i È.loi. function unless there is a major incompatibility between the range of the output function and the range of the training data sets (e.g., binary data and continuous activation ftrnction for the output neuron). To investigate this point, E previously describedlocal ANN models (with sinusoidal I activation ftinction for hidden neurons and linear activation function for output neuron) are compared to local ANN models with a sigmoid output activation function for all g neurons.The comparisonexercise was also performedfor a multiple linear regression (with an additional intercept parameter). The same procedures for the selection of explanatory variables were applied and the best models Figure 7. Increasein RMSE when switching from local were selectedfor each site. The changeof RMSE (negative ANN with sigmoid activation function for hidden neurons when the new model is better; positive when the new and linear activation function for the output neuron to (top) model is worse) due to the use of these two nerv rnodels ANN model with sigmoid activation function for all 'fhe at the ice measurementsites is plotted in Figure 7. use neuronsand (bottom) multiple linear regressionmodel.

10of 15 wll407 SEIDOUET AL.: ICE GROWTHON CANADIAN LAKES wll407

a) the local ANN rnodel respectivelygave an RMSE of 14.10and 14.31cm, whichis l3oÂand 15% higher than CLIMO. Both models performedbetter than CLIMO when simulatingice growth at stationYFL (Fort Reliance) on thedata of years197 7 - I 990.The RMSE was I 7.I 0 crnfor revisedStefan's law and16.78 cm for thelocal ANN model, slightlylower than the 18cm obtainedwith CLIMO. Despite the much highercomplexity of the deterministic u model,its performanceis comparableto thoseof the local ANNs and ts tr RSL.

I 40.61cm i 31,45cm a 22.æcm r 13.13cm . 3.97cm

140 130 120 110 100 90 80 70 r-orcrruoe(W)

U o ts

ê-.<-+''È J ...^-,Érf .-È;s 11*>:::3q | 1.74mm .t*.*"1e;"H,.1 --À'-.r-i1 { | 1.33mm a 0.92mm ? -,".. r 0.5'1mm u , 0.10mm t F 120 110 100 90 80 70 60 ) rolcruoe (W)

a locrlANtl $ih smwfr il* grflnd âsinplt

il LùralANNwiltro{tr sw m iheEolld âsinpll

110 100 q) 80 uolcrruoe (W)

g Figure 8. Effect of the statisticalcharacteristics of input 3 variables ts on the best performing solution: (a) mean snow F depth on the ground during the ice growth period and J (b) usage of snow depth on the ground as explanatory variable by local ANN models.

YZF (Great Slave Lake-Back Bay) for the years 1960- a tilâIANN$!ith rsidËtr âs iqlt i991, and at station YFL (Great Slave Lake-Fort Reli- (j LedANllwithodfrinlsll6inp0t ance) tbr years 1977-1990. They obtained an RMSE of 9 cnt at YZF and 18 cm at YFL. Since those stationsare included in our database,the results of the two studies 140 130 120 110 100 90 80 7a 60 were compared.It can be seen in Table 4 that RSL gives r-ollcrruoe M) a RMSE of 13.73 cm (12.48 cm tbr the local ANN "leave-one-out" model) using the method on the data of Figure 9. Eftbct of the statisticalcharacteristics of input the years 1958-1996 and 2002-2003 at station YZF. variableson the best perfbrming solution: (a) mean daily When sirnulated on the same period than CLIMO (years rainfall during the ice growth period and (b) usage of 1960 1991) using the leave-one-outprocedure, RSL and rainfàll as explanatoryvariable by local ANN models.

ll of 15 SEIDOUET AL.: ICE GROWTHON CANADIAN LAKES wll407

each ofthe connections ofthe inputs (Rad,"+ Rad" and R) to the hidden neuron, and olle bias and one weight parameter for the connection of the hidden neuron to the output neuron. [sz] The radiation parameter for the regional model is Radn"+ Rad" (a: 1), but the choice to split the radiation value into two parts is justified by the fact that for sorne of

U the local ANN models, the optimal value of o is lower than l. For example, for station WTL, the optimal value ts of o was fotrnd to be 0.75 (see Tables 4 and l). J [s:] It was also noticed when rating the different combinations of input variables for the regional ANN model that | 462.06Wxdav/m2 latitude and longitude do not have an influence on the result. o 411.1BWxdai/m2 However. geographiclocation has a direct influeuce on all | æ0.æwxaay/m1 input variables. In other terms, geographic location has an o 309,ttWxday/mj o 258,53Wxday/mr 130 1n 110 1m 90 80 70 60 a) lorcrruoe(W)

:3ib.*;

u | ffi.fficm E It F | 234.fficm J t | 178.00cm r 12200cm . 66.mcm -rm20t |... .. 1...... l 1ootio 11bioo-'.so-.. 60 70 ô0 LONGITUOEW) b) 90. 140 130 120 "0,-oJfr"oJ?*,* 70 60 xti i Figure 10. Effect of the statistical characteristics i of input IU: variableson the best perfonning solution: (a) mean sum of l solar radiation during the ice growth period and (b) usageof radiation as explanatory variable by local ANN models. u our

3 soi ù\f Regional 6.5. ANN Model i [sr] The perfomance criteria for the five best combina- 40r | 2009cm tions forthe regionalANN model are given in Table 6. The | 19.18cm best combinatiorl(in terms of RMSE) to explain ice growth l ) 18.27cn at tlre regional level is cornbination1-5 (D"; Rad,,"+Rad,;R)

12of 15 \ry11407 SEIDOUET AL.: ICE GROWTHON CANADIANLAKES wlt407

sites to unmonitored sites or periods using only six weight and bias parameters.They also reproduced various ice growth pattems with a t'ew easily obtained meteorological data. [sr] The methodology presented in this paper is of the greatest importance for Nordic hydrologists since no oper- ational model of lake ice growth exists for ungauged sites. This gap needed to be addressed for several reasons: U a practical rnodel of lake ice growth can help assess the F F impact of climate change on lake ecosystems,which is a topic of great concern nowadays. It can also be used to estimate the arnount of water that is lost as ice left on the banks of hydroelectric reservoirs during winter operation, | 1.75cm which may be important for large annual reservoirs in o 0.78cm northem countries. " -0.20 cm [sa] The main limitation of the work presentedin this i2-1.18cm paper is that it applies only to the ice growth period. In {,*-2.16cn other empirical rnodels,a combination of freezing degree- days and n.relting degree-dayshave provided good results 140 130 in modeling the whole duration of the ice cover (w) [e.g., LoNGtruoE Thompson et al., 2005). The extension of the developed rnethod to the whole duration of the ice cover using a Figure 12. Bias the ANN model. of regional single ANN rnodel is a challenging problern becauseof the differences in the two processes to be rnodeled. The leaming perfonnance of the ANN may hence be reduced. influenceon the rangeof input variables,but not directly on A possiblealternative solution is to train an ANN model ice growth relationshipto input variables. for the growth phase and another one for the decay An unexpectedresult in this study is that snow depth [sa] phase. on the ground was not identified as an importantparameter at the regional level for lake ice growth. This is probably due to the greatvariability of this parameterin space,which Appendix A: Solar Radiation at the Top of was estimatedby interpolation from the weather stations Atmosphere locatedat severalkilometers. [:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::r]These relations are obtained from the Solar Radia- To illustrate the ability of the regional ANN model to [s:] tion Monitoring Laboratory [2004]. On average extrater- reproduce ice growth dynamics simulated ice thickness restrialinadiance is 1367 W/m2. This value varies ly t3Yo (using the regional ANN) at station YZF (Great Slave as the earth orbits the sun. Lake-Back Bay) is presentedin Figure 13. It can be seen that the agreementis excellent. 1 : 1 36?(*)'*.o,(Z) KW lm2 (Al) 7. Conclusions [se] It is shown in this paper that artificial neurai where Ào, is the averageSun-Earth distance, R is the actual networks can be valuable altematives to complex thermo- Sun-Earth distance depending on the day of the year, and Z dynamic lake ice growth models, especiallywhen dala is the zenith (the angle of the sun relative to a line not available in sufficient quantity and quality. The perpendicular to the Earth's surface) which depends on performances of tlre proposed ANN models were fairly the declination of the Earth, latitude and solar tirne. If one good when compared to that of the deterministic Cana- notes n the Julian day, Eq, (equation of time) the variation in dian Lake lce CLIMO. They also offered much more minutes of the solar time (f.,1,,.) compared to the standard flexibility than the Stefan's law and were able to trans- time (76,) during the year. w the solar hour angle (in pose information on ice growth dynamics from monitored radians),Longç,.o1the longih,rde of the cenhal meridianline

Table 6. Perl'omranceCriteria of the Resional Model for the Five Best Combinationso

Combination of RMSE RMSE BIAS BIAS Meteorological (RSL), (ANN), r, RRMSË RRMSE NASH NASH (RSL), (ANN) Vari ables cm (RSL) cm (ANN) (RSL) (ANN) (RSL) (ANN) cm cm

l5 25.08 0.65 18.r5 0.82 0,42 0.43 0.65 0.82 -0.15 0.65 20 25.08 0.65 I8.17 0.82 0,42 0.43 0.65 0.82 *0. 15 0.56 l9 25.08 0.65 18.17 0.82 0.47 0.43 0.65 0.82 -0.15 0.12 14 25.08 0.6-5 18.19 0.82 0.12 0.43 0.65 082 -.0.15 t.23 l8 25.08 0.65 18.24 0.82 0.42 0.43 0.65 0.81 0.15 -0.48 uOne neuron on the hidden layer and perfomrance indices obtained during the leave one out cross-validation procedure. Note longitude and latitude are not listed as explanatoryvariables becauseit was fbund that they have no intluence on the output ofthe regional model.

13of l5 \ry11407 SEIDOUET AL.: ICE GROWTHON CANADIAN LAKES 18ol . grl1

1958lfff{ff{ruli${riif$r|$iigsrfi#iis 1964 1970 1976 1982 1988 1994 [-;ô;;--- ; sili;bd iÂNNt: Figure 13. Simulatedand observed ice thicknessat stationYZF. of the tirne zone and Lon5g,n,the longitude of the point of C,, heat transfer coeffrcient from ice to air. calculation,we have the following approximations: Da sum of degree-day from the date of ice cover fomration. f&)'- l.000ll+ 0.034221*.orl'zn'\ + 0.001280 Ds sum of degree-day from the first below zero \R,/ \ l6s/ remperarure. - sin(z'3) + 0 00071e*.o.(+o3) F0.000077 ANN transfer function of the mth laver. \ Jô)/ \ J65/ ice thickness. . L n\ 4t +srn(4;r* (^2) Hr rneanice thickness. I 1- \ JO)/ Hi kth ice thicknessin the data set. Hi estimateof the kth ice thicknessin the data set. on-ground snow depth. cos(Z) : sin(/) sin(d) * cos(/)cos(d) cos(u,) (A3) Us HS lnean on-ground snow depth. k coefficientof Stefan'slaw. Â average rainfall during the ice growth period. tt:23.45."i"(2"?!ff) tool Rad. snm of solar radiation of days with precipitation (snow or rainfall). Rad," sum of solarradiation of days without precipitation (snow or rainfall). (t2 T,,a,.) ...__* (A5) [l''n ANN weight vector of the connectionsform the l2 (m-l)th layer to the (m)th layer.

[oo] lcxnowledgments,The financialsupport provided by the , . Eqt ., Natural Sciencesand EngineeringResearch Council of Canada T,uru,- Tlocal*;0 * (Long,n,- Long1,,,.,1)/15 lAfr\\'--l (NSERC),Hydro-Quebec, the GEOIDE network, and the NodhemStudy Center (CEN) at Laval University is gratefully acknowledged.This researclr was partly carried out while the second author was on sabbaticalleave at the National EngineeringSchool of Tunis (Tunisia). -la.2sin[n/ n+7\ The authorswould like to thmk the associateeditor and three anony- - )t3n

14of 15 wt 1407 SEIDOUET AL.: ICE GROWTHON CANADIAN LAKES wll407

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