We investigate the approximation of Rainfall data using  additive   models .  In our model, space and elevation are treated as the   predictor variables.  The multi-dimensional approximation problem is   demonstrated using rainfall data collected by ACTEW    Corporation

Hegland, M.

Roberts, S.

Zoppou, C.

English

Australian Mathematical Society (AustMS): E-Journals

ANZIAM J. 42 (E) ppC1599{C1611, 2000 C1599Spatial and temporal rainfall approximationusing additive modelsC. Zoppou S. Robertsy M. Heglandy(Received 7 August 2000)AbstractWe investigate the approximation of Rainfall data using additivemodels. In our model, space and elevation are treated as the predictorvariables. The multi-dimensional approximation problem is demon-strated using rainfall data collected by actew Corporation.CSIRO Land and Water, Canberra, Water Division, ACTEW Corporation,Canberra, Australia.yResearch School of Information Science and Engineering, Australian NationalUniversity, Canberra, ACT0200, Australia.0See http://anziamj.austms.org.au/V42/CTAC99/Zopp for this article and ancillaryservices, c© Austral. Mathematical Soc. 2000. Published 27 Nov 2000.Contents C1600Contents1 Introduction C16002 Additive Models C16053 Application to ACT Rainfall Data C16074 Conclusions C1608References C16101 IntroductionRainfall is a fundamental component of any water resource assessment strat-egy. Many catchment authorities have implemented expensive and elaboraterainfall monitoring networks to capture the spatial and temporal variabilityin the rainfall. Due to nancial constraints, rainfall monitoring networks aregenerally sparsely distributed within a catchment. To obtain information ata specic location in a catchment, either interpolation or extrapolation ofthe existing data is required.Unfortunately, elementary techniques are often used to estimate the spa-tial and temporal distribution of precipitation from point measurements of1 Introduction C1601rainfall. Many of these approaches only provide a spatial distribution of rain-fall from a set of xed rainfall gauges. The temporal distribution is ignoredby averaging the rainfall over a predetermined period at each rainfall gauge.The crudest method in common use for estimating the precipitation over aregion, is to plot contours of equal precipitation manually or with the as-sistance of a structured grids, called isohyets [2]. The average precipitationis computed between successive isohyets. This is simply linear interpolationand does not utilise all the available data. It is also dicult to produce iso-hyets with sparse data, although with the advent of Geographic InformationSystems, the task of plotting isohyets has been automated. Consider theaverage annual rainfall recorded at the 55 rainfall gauges shown in Figure 1surrounding and within the Australian Capital Territory.A common interpolation approach is to use Theissen polygons or Voronipolygons [10], see Figure 2(a). In this approach, areas closest to a rainfallgauge adopt the rainfall recorded at that gauge. This results in constantrainfall regions with discontinuities between regions. In addition, there is nojustication in assuming that point rainfall measurements provide reliableestimates of precipitation in the surrounding region [4], [9]. The inverse dis-tance weighted method is an alternative approach. The rainfall at any desiredlocation is interpolated from the given data using weights that are based onthe distance from each rainfall gauge and the desired location. This approachproduces a smooth rainfall distribution along with the undesirable troughsand peaks located at the rainfall gauges shown in Figure 2(b). Kriging [1]and thin plate splines [3], which have similar desirable statistical properties,are other examples of interpolation techniques. Kriging requires prior esti-1 Introduction C16026100000mN700000mE0 10 20kmsFigure 1: The location of the rainfall stations operated by actew Corpo-ration over the period 1870{2000.1 Introduction C1603(a)(b)Figure 2: Various methods for interpolation rainfall data: (a) Thiessenpolygons; (b) inverse distance weighted; (c) Kriging; and (d) nite elementthin plate splines (continued next page)1 Introduction C1604(c)(d)Figure 2: (continued) Various methods for interpolation rainfall data: (c)Kriging; and (d) nite element thin plate splines.2 Additive Models C1605mation of the auto-covariance structure, which can be dicult to estimate.Naive implementation of thin plate splines and Kriging do not scale linearlywith the number of data points. This has been recently addressed for thinplate splines for one, two and three-dimensional data sets with the develop-ment of the nite element thin plate spline [7, 6]. However, for moderatelyhigher dimensional data sets (4{5 dimensions) interpolation is dicult. Ad-ditive models [5] provide a class of models that can be used to approximatemulti-dimensional data sets.2 Additive ModelsThe additive model assumes that the data has the form (yi; xi1; xi2; : : : ; xid)where i = 1; 2; : : : ; N are the N data points, y is the response variable andx = (x1;x2; : : : ; xd) is the d-dimensional vector space of predictor variables.The form of the additive model with second-order interaction terms is givenbyf(x) = y +d∑j=1fj(xj) +∑1jkdfj;k(xj ; xk) (1)in which y is the mean response. Interpreting fj(xj) = fj;j(xj ; xj) thenf(x) = y +∑1j<kdfj;k(xj ; xk):2 Additive Models C1606This model assumes that the high dimensional data is well approximated bya sum of low dimensional contributions. The additive model requires theestimation of d one-dimensional functions fj and d(d−1)=2 two-dimensionalfunctions fj;k. The original high-dimensional problem has been reduced to aset of coupled one and two-dimensional problems.The estimation of fj and fj;k becomes a smoothing problem because theprojection of the multi-dimensional data onto lower dimensional space canproduce data with large gradients and perhaps discontinuities. The ttingof an interpolant through this data would be dicult. Therefore, a smooth-ing problem is solved. Since the additive model does not pass through thedata points it is not an interpolant, it does however, approximate the multi-dimensional data set by a smoothing function.Suppose all the functions fl;m except one are known, then the unknownfl;m is a two-dimensional smoothing function of xj ; xk which ts the datay − y −∑l;m6=j;kfj;k:Any suitable smoother can be used, such as thin plate or nite element thinplate splines.3 Application to ACT Rainfall Data C16073 Application to ACT Rainfall DataThe additive model is used to approximate rainfall data collected by actewCorporation. The location of the 55 rainfall gauges are shown in Figure 1.The rainfall gauges are distributed throughout the Cotter River catchment,to the south west of Canberra and Queanbeyan River catchment, which liesto the south east of Canberra. Reservoirs in these two catchments providethe potable water for the residents of Canberra, with a population of 300,000and Queanbeyan, to the west of Canberra, which has a population of 26,000.For the rainfall data, the two spatial dimensions and time are obviouspredictor variables. Hutchinson [8] found that when interpolating annualmean rainfall, an appropriately scaled elevation makes a dominant contribu-tion to the accuracy of the tted surface. Wind direction, which is controlledby topography, can signicantly influence the variability in rainfall [4]. Allthese parameters can be treated as predictor variables when approximatingthe response variable, rainfall. To avoid the generation of negative rainfall,the logarithm of the response variable could also be used. This would re-sult in a generalised additive model [5]. However, taking logarithms of zerorainfall is problematic.Rainfall recorded during a storm that occurred on the 10th of February1996 was approximated. All fty-ve rainfall gauges were operating duringthe storm. For each rainfall gauge, a one-dimensional interpolating functionyi = f i(t) in which yi is the rainfall recorded at the i−th rainfall gauge and4 Conclusions C1608t is the time, is tted to the rainfall data. Interpolation will ensure thatrainfall is conserved. This data is interpolated at a particular time t = Tand an additive model is tted through the values yi(T ) using the locationand elevation of the rainfall stations as the predictor variables.The additive model is used to interpolate the rainfall data on a 32 32 rectangular mesh covering the catchment. Contour plots, illustrated inFigure 3 were produced from the interpolated rainfall for the February storm.The plots represent the interpolated rainfall over an eight minute intervalcommencing at 2:00pm on the 10th of February, 1996. In each plot thecontour interval is 5mm/hr with the minimum contour level arbitrarily setto 5mm/hr. For this event, with the exception of an intense rainfall burst inthe southern ACT catchment, most of the rain fell within Canberra. Theseresults demonstrate that the additive model is capable of providing detailedinformation on the spatial and temporal variability in the rainfall.4 ConclusionsIn this work, the application of an additive model is presented. Its mainadvantage is that it is relatively simple to include any number of predictorvariables that are thought to influence the response variable in the additivemodel. With an ecient additive model, it is possible to obtain real timesimulations of rainfall distributions. These could be used in a rainfall-runomodel to provide more timely and accurate predictions of major flood events.4 Conclusions C16092:00pm 2:02pm 2:03pm 2:04pm2:05pm 2:06pm 2:07pm 2:08pmFigure 3: Contours of interpolated rainfall using an additive model for the10th February 1996 storm.References C1610The ability to produce real time simulations have obvious applications inflood forecasting.Acknowledgement: This project was supported by the ACSYS CRC.References[1] J. P. Delhomme. Kriging in the hydrosciences. Advances in WaterResources, 1(5):251{266, 1978. C1601[2] S. L. Dingman. Physical Hydrology. Prentice-Hall, New York, 1994.C1601[3] J. Duchon. Splines minimizing rotation-invariance semi-morms inSoblev spaces. Lecture Notes in Mathematics, 571:85{100, 1977. C1601[4] D. C. Goodrich, J-M. Faures, D. A. Woolhiser, L. J. Lane, andS. Sorooshian. Measurement and analysis of small-scale convectivestorm rainfall variability. Journal of Hydrology, 173:283{308, 1995.C1601, C1607[5] T. J. Hastie and R. J. Tibshirani. Generalized Additive Models.Monographs Chapman and Hall, 1990. C1605, C1607References C1611[6] M. Hegland, S. Roberts, and J. Atlas. Finite element thin plate splinesfor surface tting. In B.J. Noye, M.D. Teubner, and A.W. Gill, editors,Computational Techniques and Applications: CTAC97, pages 289{296,Singapore, 1998. World Scientic Publishing. C1605[7] M. Hegland, S. Roberts, and I. Altas. Finite element thin plate splinesfor data mining applications. submitted to Mathematical Methods forCurves and Surfaces II, Eds. M. Daehlen, T. Lyche, andL.S. Schumaker, 1998. C1605[8] M.F. Hutchinson. Interpolating mean rainfall using thin platesmoothing splines. International Journal on Geographical InformationSystems, 9(4):385{403, 1995. C1607[9] Z. Sen. Average areal precipitation by percentage weighted polygonmethod. Journal of Hydrologic Engineering, American Society of CivilEngineers, 3(1):69{72, 1998. C1601[10] A.H. Thiessen. Precipitation averages for large areas. MonthlyWeather Review, 39(7):1082{1084, 1911.C1601

Spatial and temporal rainfall approximation using additive models

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Spatial and temporal rainfall approximation using additive models

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