Effect of spatial trends on interpolation of river bathymetry Venkatesh Merwade * School of Civil Engineering, 550 Stadium Mall Drive, Purdue University, West Lafayette, IN 47907, United States article info Article history: Received 14 October 2008 Received in revised form 18 March 2009 Accepted 25 March 2009 This manuscript was handled by G. Syme, Editor-in-Chief, with the assistance of Paul Jeffrey, Associate Editor Keywords: River channels Bathymetry Spatial trend Interpolation River modeling summary Continuous surface of river bathymetry (bed topography) is typically produced by spatial interpolation of discrete point or cross-section data. Several interpolation methods that do not account for spatial trend in river bathymetry produce inaccurate surfaces, thus requiring complex interpolation methods such as anisotropic kriging. Although isotropic methods are unsuitable for interpolating river bathymetry, issues that limit their use remain unaddressed. This paper addresses the issue of effect of spatial trend in river bathymetry on isotropic interpolation methods. It is hypothesized that if the trend is removed from the data before interpolation, the results from isotropic methods should be comparable with anisotropic methods. Data from six river reaches in the United States are used to: (i) interpolate bathymetry data using seven spatial interpolation methods; (ii) separate trend from bathymetry; (iii) interpolate residuals (bathymetry minus trend) by using all seven interpolation methods to get residual surfaces, (iv) add the trend back to residual surfaces; and (v) compare resulting surfaces from (iv) with surfaces created in (i). Quantitative and qualitative comparison of results through root mean square error (RMSE), semi-vario- grams, and cross-section profiles show that significant improvement (as much as 60% in RMSE) can be accomplished in spatial interpolation of river bathymetry by separating trend from the data. Although this paper provides a new simple way for interpolating river bathymetry by using (otherwise deemed inappropriate) isotropic methods, the choice of trend function and spatial arrangement of discrete bathymetry data play a vital role in successful implementation of the proposed approach. Ó 2009 Elsevier B.V. All rights reserved. Introduction River bathymetry (bed topography) plays a critical role in numerical modeling of flow hydrodynamics, sediment transport, ecological and geomorphologic assessments. Conventional way of measuring river bathymetry is through cross-sectional surveys where ground profiles are collected at certain locations along the river depending on available resources, river morphology and end use of the data. A recent technological development in bathymetry measurement includes the use of boat-mounted SONAR (Sound Navigation And Ranging) devices such as single or multi-beam echosounder in conjunction with global positioning system (GPS) to give a series of (x,y,z) bathymetry points (Vermeyen, 2006; Rogala, 1999). Although the spatial resolution of bathymetry points collected through echosounding techniques can be much better compared to cross-sections, these data still represent a discrete sample of a continuous bathymetric surface. Continuous mapping of shallow river bathymetry over large areas through air-borne techniques is also an active area of research these days (Hilldale and Raff, 2008; Legleiter et al., 2004; Marcus et al., 2003), but for deeper rivers discrete field data are still the best way for creating accurate bathymetric surfaces (e.g., Lampe and Morlock, 2007; White and Hodges, 2005). Traditional approach to studying flow, sediment, and aquatic habitat in river channels is through one-dimensional models that take bathymetric information in the form of cross-sections (e.g., Gard, 2005; Lee et al., 2006; Martin, 2003; Torizzo and Pitlick, 2004; Yang et al., 2006). Although the normal flow in rivers can be assumed to be one-dimensional in the main channel, this assumption becomes invalid during high floods, thus necessitating the use of 2D/3D hydrodynamic models (e.g., Carrivick, 2006; Dutta et al., 2007; Tayefi et al., 2007). Similarly fish habitat are increasingly being related to three-dimensional nature of river hydraulics, thus requiring 2D/3D models for ecological assessment (e.g., Booker, 2003; de Jalon and Gortazar, 2007; Mouton et al., 2007; Shen and Diplas, 2008). Multi-dimensional models are also becoming popular in sediment transport and geomorphology (e.g., Khosronejad et al., 2007; Mekonnen and Dargahi, 2007; Yue et al., 2008). Although much progress has been made in represen- tation and simulation of river processes in 2D/3D, successful appli- cation of these models is directly linked to accurate bathymetric representation (Buttner, 2007; Crowder and Diplas, 2000; Horritt et al., 2006; Lane et al., 2002). Bathymetry data are incorporated into 2D/3D models by interpolating observed discrete data (points or cross-sections) to get elevations at model nodes (e.g., nodes of a 0022-1694/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2009.03.026 * Tel.: +1 765 494 2176; fax: +1 765 494 0395. E-mail address: [email protected]Journal of Hydrology 371 (2009) 169–181 Contents lists available at ScienceDirect Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol
Continuous surface of river bathymetry (bed topography) is typically produced by spatial interpolation ofdiscrete point or cross-section data. Several interpolation methods that do not account for spatial trend inriver bathymetry produce inaccurate surfaces, thus requiring complex interpolation methods such asanisotropic kriging. Although isotropic methods are unsuitable for interpolating river bathymetry, issuesthat limit their use remain unaddressed. This paper addresses the issue of effect of spatial trend in riverbathymetry on isotropic interpolation methods. It is hypothesized that if the trend is removed from thedata before interpolation, the results from isotropic methods should be comparable with anisotropicmethods. Data from six river reaches in the United States are used to: (i) interpolate bathymetry datausing seven spatial interpolation methods; (ii) separate trend from bathymetry; (iii) interpolate residuals(bathymetry minus trend) by using all seven interpolation methods to get residual surfaces, (iv) add thetrend back to residual surfaces; and (v) compare resulting surfaces from (iv) with surfaces created in (i).Quantitative and qualitative comparison of results through root mean square error (RMSE), semi-vario-grams, and cross-section profiles show that significant improvement (as much as 60% in RMSE) can beaccomplished in spatial interpolation of river bathymetry by separating trend from the data. Althoughthis paper provides a new simple way for interpolating river bathymetry by using (otherwise deemedinappropriate) isotropic methods, the choice of trend function and spatial arrangement of discretebathymetry data play a vital role in successful implementation of the proposed approach.
� 2009 Elsevier B.V. All rights reserved.
Introduction
River bathymetry (bed topography) plays a critical role innumerical modeling of flow hydrodynamics, sediment transport,ecological and geomorphologic assessments. Conventional way ofmeasuring river bathymetry is through cross-sectional surveyswhere ground profiles are collected at certain locations along theriver depending on available resources, river morphology and enduse of the data. A recent technological development in bathymetrymeasurement includes the use of boat-mounted SONAR (SoundNavigation And Ranging) devices such as single or multi-beamechosounder in conjunction with global positioning system (GPS)to give a series of (x,y,z) bathymetry points (Vermeyen, 2006;Rogala, 1999). Although the spatial resolution of bathymetry pointscollected through echosounding techniques can be much bettercompared to cross-sections, these data still represent a discretesample of a continuous bathymetric surface. Continuous mappingof shallow river bathymetry over large areas through air-bornetechniques is also an active area of research these days (Hilldaleand Raff, 2008; Legleiter et al., 2004; Marcus et al., 2003), but fordeeper rivers discrete field data are still the best way for creating
ll rights reserved.
accurate bathymetric surfaces (e.g., Lampe and Morlock, 2007;White and Hodges, 2005).
Traditional approach to studying flow, sediment, and aquatichabitat in river channels is through one-dimensional models thattake bathymetric information in the form of cross-sections (e.g.,Gard, 2005; Lee et al., 2006; Martin, 2003; Torizzo and Pitlick,2004; Yang et al., 2006). Although the normal flow in rivers canbe assumed to be one-dimensional in the main channel, thisassumption becomes invalid during high floods, thus necessitatingthe use of 2D/3D hydrodynamic models (e.g., Carrivick, 2006;Dutta et al., 2007; Tayefi et al., 2007). Similarly fish habitat areincreasingly being related to three-dimensional nature of riverhydraulics, thus requiring 2D/3D models for ecological assessment(e.g., Booker, 2003; de Jalon and Gortazar, 2007; Mouton et al.,2007; Shen and Diplas, 2008). Multi-dimensional models are alsobecoming popular in sediment transport and geomorphology(e.g., Khosronejad et al., 2007; Mekonnen and Dargahi, 2007; Yueet al., 2008). Although much progress has been made in represen-tation and simulation of river processes in 2D/3D, successful appli-cation of these models is directly linked to accurate bathymetricrepresentation (Buttner, 2007; Crowder and Diplas, 2000; Horrittet al., 2006; Lane et al., 2002). Bathymetry data are incorporatedinto 2D/3D models by interpolating observed discrete data (pointsor cross-sections) to get elevations at model nodes (e.g., nodes of a
170 V. Merwade / Journal of Hydrology 371 (2009) 169–181
finite element mesh). Therefore, the accuracy of bathymetric sur-faces represented in 2D/3D models is dependent upon the abilityof interpolation methods in making accurate predictions at unmea-sured locations using discrete data. Recent studies have shown thatcommonly available interpolation methods such as triangulation,inverse distance weighting (IDW), splines or kriging, which assumeisotropy in data, yield inaccurate river bathymetric surface (Goffand Nordfjord, 2004; Merwade et al., 2006). The isotropic assump-tion in most spatial interpolation methods ignores the trend in riv-er bathymetry that is linear (bed slope) in flow direction andnonlinear (cross-sectional shape) across flow direction. As a result,methods that account for river flow direction and topographictrend are recommended for interpolating discrete bathymetrydata.
Additional constraints imposed by river flow direction andtopographic trend limit the choice of methods available for inter-polating river bathymetry to only a few specialized ones such asanisotropic kriging, or custom modifications of existing isotropicmethods (e.g., elliptical IDW by Merwade et al., 2006; anisotropicIDW by Tomczak, 1998). Although anisotropic methods providebetter results compared to isotropic methods, ways to improvethe results from application of simple isotropic methods for inter-polating river bathymetry remain unexplored. Specifically the ef-fect of separating existing trends in river bathymetry beforeinterpolating discrete points is not studied. Such a study can: (a)potentially overcome the limitations of isotropic methods, thusmaking them widely applicable for interpolating river bathymetry;(b) demonstrate the effect of trend on interpolating river bathym-etry; and (c) provide information on accuracy that can be gained by
Fig. 1. Bathymetry dataset for: (a) Brazos (B) River; (b) Ohio (O) River; (c) King Ranch (KR(S) River; and (f) Kentucky River (K).
using complex interpolation methods, such as anisotropic kriging,compared to isotropic methods if the trend is excluded from riverbathymetry. This paper addresses the issue of effect of bathymetrytrend by presenting results from a study that involved interpola-tion of discrete bathymetry point dataset – with and without re-moval of trend – at six river reaches in the United States.Although the topic of spatial trends and anisotropy has receivedconsiderable attention in the field of soil science and groundwaterhydrology (e.g., Petersen et al., 2008; Crawford and Hergert, 1997;Pfannkuch and Winter, 1984), the present study explores this topicfor river channels, thus providing a new way of interpolating riverbathymetry using simple isotropic methods in the absence of spe-cialized complex techniques.
Study areas and datasets
Bathymetry data collected at six river reaches in the UnitedStates are used in this study (Fig. 1a). Of these six reaches, four(Brazos, King Ranch, Rainwater and Sulphur) are located in Texas,and two (Kentucky and Ohio) are located in Kentucky. Except forKing Ranch and Rainwater which are located along the GuadalupeRiver in Texas, names of all other reaches reflect the names of cor-responding rivers. Data for Texas Rivers were provided by TexasWater Development Board (TWDB); whereas data for Kentuckyand Ohio reaches were provided by North Carolina Water ScienceCenter (NCWSC). Data for Texas Rivers were collected for InstreamFlow Program (TWDB, 2008), and data for Ohio and Kentucky werecollected for water quality and sediment transport studies(Wagner and Mueller, 2001, 2002).
) reach of Guadalupe River; (d) Rainwater (R) reach of Guadalupe River; (e) Sulphur
Table 1Summary of data at study reaches (+ obtained by averaging the widths at six equidistant locations along the reach).
River name River characteristics Bathymetry data
Length(km)
Mean width(m)+
Mean depth (m) Slope Substrate type Total points Points/50 �50 m2 area
V. Merwade / Journal of Hydrology 371 (2009) 169–181 171
Data at each reach include bathymetry (x ,y,z) points collectedusing a single-beam echosounder and global positioning system(GPS) mounted on a boat. As the boat moves, the single-beamechosounder measures the water depth (d) by pinging sound sig-nals and the GPS records (x,y) location of each ping. Typically,depth sounders ping multiple signals at one point, and then aver-age the data to give a single recording at that point. Additionally,all GPS recordings are corrected through differential GPS (DGPS)to give more accurate (x,y) locations. Water depth at each pointis then subtracted from water surface elevation to get z, thus cre-ating a series of (x,y,z) bathymetry points. The number of measure-ments and their spatial arrangement depends on the speed andpath taken by the boat as shown in Fig. 1. Data for Sulphur andOhio reach were modified by TWDB and NSWSC, respectively tomatch finite element nodes for a hydrodynamic model, and donot reflect the actual arrangement of field data. It is possible thatsome information was lost during this data transformation forOhio and Sulphur datasets, but for the purpose of this study, thesemodified data are taken as actual measurements. Data at these sixstudy sites provide a good representation of different spatialarrangement ranging from traditional cross-sections (Kentucky)to irregular spacing (Brazos, Rainwater, King Ranch) to regularlyspaced gridded data (Sulphur and Ohio). Morphologic details ateach reach (length, width, slope) including information onbathymetry data are provided in Table 1.
Methodology
The methodology involves the following steps: (i) interpolationof bathymetry data including trend; (ii) separating trend frombathymetry points by using three trend functions; (iii) interpola-tion of residuals (bathymetry minus trend) by using seven meth-ods to get residual surfaces; (iv) adding the trend back toresidual surfaces; and (v) comparison of resulting surfaces from(iv) with surfaces created in (i). Key tasks in the methodology aredescribed below. All geospatial tasks are performed using spatialand geostatistical analyst extensions in ArcGIS from EnvironmentalSystems Research Institute (ESRI).
Testing and validation datasets
Each initial bathymetry dataset is split into two sub-sets: test-ing and validation. Bathymetry points in testing datasets are usedto create interpolated surfaces, which are then compared againstmeasured bathymetry in validation datasets. Different approachesare used in creating testing and validation datasets for each studyarea depending on the configuration of the available data. In thecase of Brazos River, testing and validation datasets are used froma previous study by Merwade et al. (2006), which involved manualseparation of bathymetry points to mimic a cross-sectional config-uration in the testing dataset. In the case of King Ranch, Rainwater,Sulphur, and Ohio River, validation points are extracted throughrandom selection. In the case of Kentucky River, which had points
along cross-sections, alternate cross-sections are selected for vali-dation. Overall, all datasets except Kentucky River are split suchthat 70% of bathymetry points are included in testing dataset and30% in validation dataset. For Kentucky River, the testing and val-idation datasets are split to have 50% points each (alternate cross-sections).
Mapping of data in (s,n) coordinate system
River bathymetry data are collected and stored using Cartesian(x,y) coordinates, but use of these data in this coordinate systemcan introduce issues related to meandering nature of the channel.For example, computing distance between two points along a riverusing (x,y) coordinates will not give the actual flow length betweenthese points for a meandering river. To overcome such issues,channel-fitted coordinate system defined by an s-axis along theflow direction and n-axis across the flow direction (perpendicularto s-axis) is widely used for rivers (e.g., Johannesson and Parker,1989; Nelson and Dungan, 1989; Ye and McCorquodale, 1997).The s-axis can be aligned with either river banks or centerline. Inthis study, the geometric centerline of the channel is treated asthe s-axis with upstream end of the river as its origin (s = 0). Sim-ilarly, looking downstream, all n coordinates are negative on theleft hand side of s-axis, and vice versa. Plotting of river in (s,n)coordinates straightens the river so data can be treated with re-spect to the flow direction as shown in Fig. 2. A GIS proceduredeveloped by (Merwade et al., 2005) is used for mapping bathym-etry in (s,n) coordinates for all six reaches.
Fitting trend to bathymetry points
Mapping of bathymetry in (s,n) coordinates makes the lineartrend (bed slope) a function of s coordinate, and nonlinear trend(cross-sectional shape) a function of n coordinate. Besides usingthe n coordinate, the nonlinear trend is modeled by using otherphysical attributes such as channel width, depth and meanderingcurvature (see e.g., Deutsch and Wang, 1996; James, 1996;Legleiter and Kyriakidis, 2008). It is assumed that the quality oftrend surface, and eventually variance in residuals (measurements–trend) will depend on the selected trend function. In addition,whether a trend function is applied locally (separate functionformulations for individual local areas) or globally (one functionfor the entire reach) can also produce different results. Severaltechniques such as power functions, polynomials, splines andprobability density function (pdf) can be used to fit a trend toriver’s cross-sectional shape. A detailed review of these techniquesincluding their application in fitting river cross-sections can befound in Merwade and Maidment (2004).
To assess the effect of trend function in spatial interpolation,three techniques (two local and one global) are employed in thisstudy. Local techniques include cubic polynomial and a combina-tion of two beta pdf (Eq. (1)), and the global technique includes acubic polynomial function.
(a) (b)
s
n
-n
n
s
x
y
Fig. 2. (s,n) Coordinate transformation for King Ranch dataset. Bathymetry points in: (a) (x,y) coordinates and (b) (s,n) coordinates.
172 V. Merwade / Journal of Hydrology 371 (2009) 169–181
TðzÞ ¼ ff ðxja1;b1Þ þ f ðxja2; b2Þg�k�w; ð1Þ
where, T(z) is the trend function, w is the channel width, k (0 < k < 1)is a scaling factor, and a1, a2, b1, and b2 are the parameters of betapdf given by Eq. (2)
f ðxja; bÞ ¼ 1Bða;bÞ x
a�1ð1� xÞb�1; 0 < x < 1a > 0; b > 0 ð2Þ
Cubic polynomial is selected for its simplicity and easy avail-ability in many software programs for fitting spatial trends. A com-bination of beta pdf is used because comparison of severaltechniques listed earlier for fitting cross-sectional trend has shownthat a combination of two beta pdf produce best results (Merwadeand Maidment, 2004). The objective is not to advocate any onetechnique over others for fitting trends, but to see how differenttechniques affect spatial interpolation results of bathymetrypoints. For local polynomial trend fitting, a circular search neigh-borhood is defined to include 50 points, and for beta pdf, the neigh-borhood is defined to include points covering a channel lengthequal to the average channel width. The neighborhoods for polyno-mial and beta pdf are defined (by trial and error) such that thepoints included in a neighborhood should be able to define a com-plete river cross-section. All parameters in cubic polynomials andbeta functions are optimized to minimize the root mean square er-ror (RMSE, Eq. (5)) between predictions and observations.
Table 2Spatial interpolation methods.
Method Description
IDW Value at an unsampled location (z*) is distance-weighted aobservations (zi). ki = weight at ith point; di = distance betweof points
Tension spline A minimum curvature (smoothest) function [I(f)] is definedof observations. R = 2D euclidian space; T(x, y) = local trend/ = tension weight; K0 = modified Bessel function of zero or
Regularized spline Third and higher order derivatives are added to I(f)
Topogrid A variation of thin plate splines developed by Hutchinson (
Natural neighbor Similar to IDW, but weights are computed based on area. pareas surrounding zi excluding and including z*, respectivel
Ordinary kriging Similar to IDW, but weights are assigned based on distanceThe objective is to minimize r2 to get unbiased estimate w
Anisotropic kriging Ordinary kriging that takes data anisotropy into account
Spatial interpolation
Spatial interpolation is preformed for each training dataset in(s,n) coordinates for two variables: zi and ei where, zi is measuredbathymetry at any point i, and ei ( = zi � T) is corresponding resid-ual after fitting the trend function. Seven spatial interpolationtechniques are used in this study. These are IDW, regularizedspline (RS), spline with tension (TS), topogrid (TG), natural neigh-bor (NN), ordinary kriging (OK) and ordinary kriging with anisot-ropy (AK). A review of these techniques for interpolatingwatershed topography and river bathymetry can be found in Cha-plot et al. (2006) and Merwade et al. (2006), respectively, but abrief description and corresponding equations of each techniqueare presented in Table 2. All these techniques are commonly usedin many disciplines including hydrology through several commer-cial and public domain software programs.
Assessment of trend surfaces
Trend surfaces are assessed for similarity and their ability tobest describe the trend in the data. Such an assessment will helpto understand how much effect a trend surface can have on finalinterpolated bathymetry surface (trend + interpolated residual).Assessment of trend surfaces is performed by using percentage of
Equations
verage of nearbyen zi and z*; N = total no. z� ¼ RN
i¼1kizi; ki ¼1
d2i
RNi¼1
1d2
i
that passes through a set; R(r,rj) = basis function;der; c = 0.577215
Iðf Þ ¼R R
R2 ½/2ðf 2x þ f 2
y Þ þ ðfyxx þ f 2
xy þ f 2yyÞ�dxdy;
f ðx; yÞ ¼ RNi¼1tiRðr; riÞ þ Tðx; yÞ;
1989, 1993)
Rðr; rjÞ ¼ 12p/2 ln dj/
2
� �þ c þ K0ðdj/Þ
h i
i and qi are the Thiesseny
ki ¼ ðpi � qjÞ=pi
and spatial correlation.ith minimum variance
r2 ¼ E ðz� � z�Þ2h i
¼ E½ z� � Rni¼1kizi
� �2� subject to
RNi¼1kizi ¼ 1
Table 3Students t-test results (a = 0.05) between beta (BT), local polynomial (LT) and globalpolynomial (GT) trend surfaces for all six reaches (B = Brazos; KR = King Ranch;RW = Rainwater; S = Sulphur; K = Kentucky; and O = Ohio).
V. Merwade / Journal of Hydrology 371 (2009) 169–181 173
total sum of squares (SST, Eq. (3)), and by conducting t-test(a = 0.05). Although these techniques are commonly used for com-paring the accuracy of surfaces, they can produce spurious resultsfor spatially correlated bathymetry data. Therefore, quantitativeassessments are augmented with visual inspection of surfaces,and comparison of experimental semi-variogram plots.
SST ¼ 1� ðPn
i¼1ðzi � ziÞ2ÞðPn
i¼1ðzi � �zÞ2Þ; ð3Þ
where �z is the mean. A semi-variogram plot is a plot of semi-variance (Eq. (4)) as a function of distance between observationpoints as shown in Fig. 5. In spatially correlated data such as riverbathymetry, semi-variance is smaller for nearer points, and viceversa. Therefore, semi-variance increases with distance betweenmeasurement points, until a threshold is reached in the distanceof separation. This threshold is called the range as shown inFig. 5b. The semi-variance corresponding to the range is called thesill, and semi-variance corresponding to zero separation distanceon a semi-variogram plot is called the nugget. Semi-variogram canbe used to compare the spatial distribution of semi-variance cap-tured by different trend and interpolated surfaces. Moreover, fittingof semi-variogram in different directions will show the extent towhich the trend in the data is captured by a trend-fitting function.For example, if the trend in the bathymetry is captured, the semi-variograms of residuals in all direction should be identical.
cijðhÞ ¼Eððzi � zjÞ2Þ
2; ð4Þ
where cij is the semi-variance between two bathymetry points(zi,zj) separated by distance h.
Cross-validation of interpolated surfaces
After spatial interpolation of points (zi) in testing dataset, theresulting surfaces are validated against observed bathymetry invalidation datasets. Several methods are available to assess thequality of interpolated surfaces compared to measured bathymetryin validation datasets. Commonly used quantitative methods in-clude precision indices such as mean error, mean absolute erroror the root mean square error (RMSE). In this study, RMSE (Eq.(5)) which provides an overall measure of how close or accuratethe estimated elevations are compared to measurements is usedfor quantitative assessment.
RMSE ¼ 1n
Xn
i¼1
ðzi � ziÞ2" #1=2
; ð5Þ
where zi is measurement at ith location, zi is estimate of zi, and n isthe total number of points in the validation dataset. When ei isinterpolated, zi is obtained by adding T(i) to ei.
Besides RMSE, surfaces are also compared by using SST, and t-test (a = 0.05). Similar to trend surfaces, quantitative assessmentsof interpolated surfaces are augmented with visual inspection,and comparison of experimental semi-variogram plots.
Results
Assessment of trend surfaces
To keep the terminology simple, beta trend surface will be re-ferred as BT, local polynomial trend as LT and global polynomialtrend as GT. Application of traditional techniques such as Studentst-test and SST for comparing trend surfaces show some interestingresults (Tables 3 and 4). For example, the mean of BT, LT and GT arenot significantly different for Kentucky, the mean for BT and LT are
significantly different for Brazos, Ohio and Sulphur datasets(Table 3). Overall, the mean is not significantly different for allthree trend surfaces for many study areas (except for Rainwater)at 95% confidence interval. The results of SST (Table 4) show thatthe BT and LT capture more than 80% of variance in the measuredbathymetry for all datasets; whereas GT captures only around 50%of variance in measured bathymetry for most datasets. Lower SSTfor global polynomial trend for all datasets can be attributed toits inability to capture local variations that exist in river bathyme-try. Overall, Table 4 seems to show that beta and local polynomialtrends show similar performance in capturing the variations inmeasured bathymetry. Does this mean that these two techniqueswill have the same effect on bathymetry interpolation? This willbecome clear when results from spatial interpolations are pre-sented in a subsequent sub-section.
A visual comparison of trend surfaces with measured bathyme-try (Fig. 3) show that the BT surface is better in capturing localvariations compared to LT. Only results for King Ranch datasetare presented in Fig. 3 because the overall interpretation of this fig-ure is applicable to all other datasets. Fig. 3a represents the basesurface that is created by interpolating all bathymetry points (test-ing + validation) using anisotropic kriging for comparison. Thesmoothness of surface increases from BT to LT to GT, with GT un-able to capture any local variations (Fig. 3b–d). Plots of ground pro-files at two locations (Fig. 4) show how BT, LT and GT perform indescribing individual cross-sections. Cross-sections produced byGT at both locations are almost identical; whereas BT and LT adjustto fit the trend at individual locations. It is also important to notethat a better trend surface that captures local variations requiresmore parameters and computing power. A visual comparison ofoverall surfaces and cross-section profiles at individual locationsshow that BT, LT and GT produce distinct surfaces that can havedifferent effects on interpolation of residuals, and subsequentlybathymetry surfaces.
Experimental semi-variogram plots of BT, LT, GT, and their cor-responding residuals show how the spatial correlation in thebathymetry is affected (Fig. 5). Results from King Ranch experi-mental semi-variogram and trend surfaces are consistent withearlier statistical analysis. Both BT and LT produce similar semi-variograms suggesting that these two surfaces have similar spatialdistribution of semi-variance. On the other hand, GT is unable tocapture most of the variance in the bathymetry. Semi-variogramplots of bathymetry and trend surfaces in s-direction have higherrange compared to other directions due to smaller variations(or higher correlation) in bathymetry along the flow direction.
Fig. 3. Trend surfaces for King Ranch Reach: (a) bathymetry surface for the entire reach; (b) close-up view of the surface in (a); (c) trend surface using beta function; (d) trendsurface using local polynomial; and (e) trend surface using global polynomial.
Elev
atio
n (m
)
142.5
143.0
143.5
144.0
144.5
0 10 20 30 40 50
Z BTLT GT
142.0
142.5
143.0
143.5
144.0
144.5
145.0
0 10 20 30 40 50
Z BTLT GT
Distance (m)a) b)
Fig. 4. Cross-sections for King Ranch Reach at: (a) x location in Fig. 3b; (b) y location in Fig. 3b (Z represents the base surface; BT = beta trend; LT = local polynomial; andGT = global polynomial).
174 V. Merwade / Journal of Hydrology 371 (2009) 169–181
Similarly semi-variograms of residuals show that BT and LT cap-ture about 75% of semi-variance in the bathymetry (residual sillis approximately 0.1 compared to 0.4 for bathymetry); whereasresiduals from GT produce a semi-variogram that looks similar toobserved bathymetry in terms of range and sill, suggesting thatno or very little variance is captured. Similarity of residual semi-
variograms from BT and LT in omni-direction and s-direction sug-gest that most of the trend is captured by these functions. Althoughoverall results from King Ranch semi-variograms are applicable toall datasets, semi-variograms from Brazos that had relatively bet-ter GT surfaces is also presented for comparison (Fig. 6). Unlikeother datasets that had similar semi-variograms for BT and LT,
0.0
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200
Z BT LT GT
a) b)
0.0
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200
BT LT GT
0.0
0.1
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0 50 100 150 200
BT LT GT
c) d)
0.0
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200
Z BT LT GT
Range
Sill
Nugget
Fig. 5. Semi-variograms of bathymetry, trend and residual surfaces for King Ranch: (a) bathymetry and trend surfaces in all directions; (b) bathymetry and trend surfaces ins-direction; (c) residuals in all directions; and (d) residuals in s-direction (Z represents the base surface; BT = beta trend; LT = local polynomial; and GT = global polynomial).
0.0
0.5
1.0
1.5
2.0
2.5
0 100 200 300 400 500
Z BT LT GT0.0
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1.5
2.0
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Z BTLT GT
a) b)
0.0
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BT LT GT0.0
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Fig. 6. Semi-variograms of bathymetry, trend and residual surfaces for Brazos River: (a) bathymetry and trend surfaces in all directions; (b) bathymetry and trend surfaces ins-direction; (c) residuals in all directions; and (d) residuals in s-direction (Z represents the base surface; BT = beta trend; LT = local polynomial; and GT = global polynomial).
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semi-variograms for Brazos BT and LT show visible difference.Again, the question is: are these differences any significant to affectspatial interpolation? This question is answered in the next sub-section.
Assessment of spatial interpolation techniques
Bathymetry points in all testing datasets are interpolated usingelevations to get first set of bathymetry surfaces (interpolationincluding trend). Next, residuals are interpolated, and added tocorresponding trend surfaces to get second set of bathymetry sur-faces (interpolation excluding trend). As a result four bathymetry
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Interpolation including trend
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Fig. 7. RMSE (m) results for spatial interpolation techniques (BT = b
surfaces are created for each interpolation technique. These are:(a) interpolation including trend; (b) BT + interpolation of corre-sponding residuals; (c) LT + interpolation of corresponding residu-als; and (d) GT + interpolation of corresponding residuals. RMSE foreach surface is then computed by using elevations in validationdatasets (Fig. 7). Mean RMSE values for all techniques for each val-idation dataset are presented in Table 5. The differences betweenmeasured and estimated bathymetry, and between RMSE fromeach technique are found to be statistically significant through Stu-dents t-test at 95% confidence level.
All techniques for all datasets show improvement in RMSEvalues when the trend is excluded from spatial interpolation. The
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LT + interpolated residuals
GT + interpolated residuals
eta trend; LT = local polynomial; and GT = global polynomial).
Table 5Mean RMSE in meter for all reaches by creating surfaces through: (a) interpolationincluding trend; (b) BT + interpolation of corresponding residuals; (c) LT + interpola-tion of corresponding residuals; and (d) GT + interpolation of corresponding residuals.The numbers in parenthesis represent the percentage improvement in RMSE from (b),(c) and (d) with respect to (a).
Table 6Mean percentage reduction in RMSE for each technique compared to interpolationincluding trend: (b) BT + interpolation of corresponding residuals; (c) LT + interpola-tion of corresponding residuals; and (d) GT + interpolation of corresponding residuals.
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percentage change in RMSE range from as little as 6% (with BT forSulphur) to as much as 60% (with BT for Kentucky). With theexception of the Sulphur River, bathymetry estimates after model-ing the trend with BT show the most reduction in RMSE followedby LT and GT. Table 5 show that removing trends in bathymetryeven through a simple function such as global polynomial that re-quire only three parameters can have impact on spatial interpola-
Fig. 8. Interpolated surfaces for area surrounding Y in Fig. 3b for King Ranch: (a) Baspolynomial + IDW residuals; (e) global polynomial + IDW residuals; and (f) natural neig
tion. Other functions that require more parameters such as betapdf and local polynomials can provide better results than globalpolynomial, but at additional computational cost. Fig. 7 also showthat RMSE results obtained through exclusive treatment of trendsare not consistent among all techniques and datasets. For example,accounting for trend though beta pdf has negative impact on spa-tial interpolation (slight increase in RMSE) for Ohio dataset. Thesame is true for local and global polynomial trends for Rainwaterdataset.
Quantitative estimates of how each interpolation techniques isaffected by removing trend in bathymetry is presented in Table 6for all datasets. Table 6 is derived by computing the percentagechange in RMSE for (b), (c) and (d) interpolation categories com-pared to (a) for all reaches, and then taking a mean of these per-centages. Mean values of percentage change (or improvement) inRMSE considering all datasets show that IDW is most benefittedamong all interpolation techniques; whereas regularized splineshow the least improvement in RMSE. In addition, interpolationtechniques that account for anisotropy (e.g., anisotropic kriging),and the ones that are more robust for anisotropic data (e.g., naturalneighbor) do not show much change in RMSE for different trendsurfaces compared to other isotropic techniques (e.g., IDW andtopogrid). Overall, excluding trends from bathymetry points beforespatial interpolation is found to produce better results in terms ofRMSE. A more qualitative assessment through visual inspectionand cross-section plots is presented in the next sub-section.
Assessment of interpolated surfaces
Although RMSE provides information on the accuracy of abathymetric surface, it is necessary to understand how thisnumber affects the spatial representation. First, bathymetry plots(Fig. 8) created using IDW for King Ranch dataset show how thesurface representation is affected by including/excluding bathym-etry trend during spatial interpolation. One advantage (that is
e surface; (b) IDW of z including trend; (c) beta trend + IDW residuals; (d) localhbor interpolation of z including trend.
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Fig. 9. Cross-section at location X for surfaces in Fig 8: (a) Base surface; (b) IDW of zincluding trend; (c) beta + IDW residuals; (d) local polynomial + IDW residuals; and(e) global polynomial + IDW residuals.
Fig. 10. Interpolated bathymetry surfaces with natural neighbor for Brazos and Kentuckyof residuals + trend.
178 V. Merwade / Journal of Hydrology 371 (2009) 169–181
evident from Fig. 8) of interpolating residuals, and then addingtrend to residual surface is to minimize the local influence ofsparse points on interpolation results. For example, the dark circu-lar spots in Fig. 8b are caused due to presence of single bathymetrypoints that have insufficient neighboring points to create a reason-able surface. As a result, a spike (near banks) or depression (nearthalweg) is created surrounding such points. On the other hand,when residuals (bathymetry – trend) are interpolated, the magni-tude associated with each interpolating point is found to be rela-tively small for all study reaches to create this local effect, andwhen the trend is added back, the overall effect is negligible asdemonstrated in Fig. 8c surface. However, the quality of interpolatesurface is dependent on the effectiveness of the trend surface. Forexample, global polynomial which is unable to capture local trendseffectively also produces spikes/depressions in the final surface,but these spikes/depressions are still not as prominent as they ap-pear in Fig. 8a. Interestingly, natural neighbor which is robust (not
: (a)/(d) Base surface; (b)/(c) interpolation including trend; and (c)/(f) interpolation
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Fig. 11. Cross-section profiles at location P and Q for Brazos and Kentucky, respectively for surfaces in Fig. 10: (a)/(d) Base surface; (b)/(c) interpolation including trend; and(c)/(f) interpolation of residuals + trend.
V. Merwade / Journal of Hydrology 371 (2009) 169–181 179
as sensitive to bathymetric trend) compared to other methods pro-duced identical surfaces for all techniques (including/excludingtrends) for King Ranch. Although this may suggest that a robusttechnique is unaffected by how the interpolation is performed(including/excluding trends), this is not true for all datasets as de-scribed in the next paragraph. Only one surface created by naturalneighbor interpolation that includes trend is presented in Fig. 8f forKing Ranch whose IDW counterpart is 8b. The difference in sur-faces is directly reflected in cross-section profiles presented inFig. 9.
The robustness of natural neighbor as seen for the King Ranchdataset is not applicable for all datasets. For example, cross-sec-tions from interpolated surfaces including and excluding trendsusing natural neighbor interpolation for Brazos and Kentucky showsignificant differences in results (Figs. 10 and 11). The surfaces cre-ated by excluding trend are smoother compared to surfaces cre-ated by interpolation including trend. The overall representationof bathymetric surface created by excluding trend is more repre-sentative of measured bathymetry compared to surfaces createdby interpolation of points including trend. The difference in finalresults is more distinct and considerably better for the Kentuckydataset that includes sparsely located bathymetry points alongcross-sections.
Discussion and conclusions
There is a growing need to obtain accurate bathymetry data foruse in multi-dimensional river models (e.g., Dutta et al., 2007; Shenand Diplas, 2008; Yue et al., 2008). Typically, river bathymetry dataare collected as cross-sections or discrete points, which are inter-polated to get elevation at computational nodes in river models.Because of the presence of spatial trends in river bathymetry, com-monly used isotropic interpolation methods that do not accountfor spatial trends yield inaccurate results. The objective of thisstudy was to use discrete bathymetry data from six study areasto investigate the effect of trend on spatial interpolation methodsfor creating continuous surfaces. Although recent studies haveadvocated the use of anisotropic interpolation techniques (e.g.,Merwade et al., 2006; Tomczak, 1998), the present study showsthat isotropic methods can yield improved results if the trend inthe data is excluded during spatial interpolation. The trend fromthe bathymetry data should be excluded at two levels. First, thedata should be mapped using flow oriented (s,n) coordinates to re-move the spatial heterogeneity in the orientation of river slopewith respect to Cartesian coordinates. Second, the cross-sectionaltrend must be removed from the bathymetry by using a nonlinearfunction. After removing the trend, the residuals can be interpo-lated to create a surface to which the trend is added back to get
the final bathymetric surface. Results show that the bathymetricsurface created by excluding trends not only gives better RMSEfor validation datasets, but the final surface is more representativeof the actual bathymetry.
When field data are limited (sparse cross-sections or bathyme-try points), it becomes necessary to make the best use of these datato make accurate predictions in unmeasured locations. Excludingthe trend from the data and then adding it back after interpolatingresiduals insure that the trend is restored in the final interpolatedsurface, thus providing more confidence at unmeasured locations.The idea of separating bathymetry into trend and its residuals isnot new, but it is mainly utilized in the context of kriging (a rela-tively sophisticated approach compared to other simple methodssuch as IDW), and thus not implemented in any existing isotropicinterpolation methods. Because most isotropic methods are not de-signed to handle the trend in the data explicitly, this trend can getlost during interpolation of sparse data (e.g., cross-sections) thusaffecting the quality of the final interpolated surface, and conse-quently affect model results that rely on bathymetry input. Thisstudy shows that results from IDW can improve by as much as40% with regard to RMSE when the trend is excluded from interpo-lation. Improvement in other methods is not as significant as IDW,but still better compared to interpolation including trend. Resultsfrom Kentucky dataset that had typical cross-sectional data showthat spatial interpolation using any method can be significantlyimproved through explicit treatment of bathymetry trend.
Although it is difficult to argue against collection of more fielddata, results from this study show that more bathymetry pointsare not always useful if channel morphology or trend is ignoredwhile taking measurements. For example, Rainwater dataset hasmuch higher density of bathymetry points compared to Brazosand Ohio, but the measurements are taken along flow lines thatare unable to capture the entire cross-section of the channel atany location. Inadequate description of cross-sections results inpoor trend surface, which in turn affects the final interpolation re-sults. Consequently, RMSE of interpolated surface for Rainwater isnearly equal to that of Brazos and Ohio which are relatively larger/deeper rivers with much lower point density. Therefore, spatialarrangement of bathymetric points is equally important in collect-ing field data to get a satisfactory interpolated surface from dis-crete bathymetry points.
Channel trend can be modeled in several ways: process-basedpredictive model, a model based on relationship between channelplanform and cross-sectional asymmetry, and a least-squareregression model that fits a user-defined function to observations.In this study, three regression models (two using polynomial andone using beta pdf) are used to fit bathymetric trends. Overall, itis found that selection of a trend model has effect on final
180 V. Merwade / Journal of Hydrology 371 (2009) 169–181
interpolated surface. Although any model can be used to define thetrend, a model that makes the most use of the measured discretedata to fit the trend is recommended to get accurate resultsthrough spatial interpolation. Trend function defined using betapdf is found to be the best among all three options considered inthis study followed by local polynomial. Considering the heteroge-neity in channel bathymetry, fitting of local trends is recom-mended compared to a global trend that uses a single functionfor the whole data. If use of local trend model is impractical dueto computational demands or any other reasons, modeling thetrend using a global function and interpolating the residuals canstill provide better results compared to traditional interpolationincluding trend. A trend model that can capture even as little as20% of variance in the data (global polynomial trend for KinghRanch) can have effect on final interpolation results. Quality oftrend surfaces should be checked by considering spatial properties(e.g., semi-variogram) rather than using only pure statistical tech-niques, which can give spurious results.
Although the general conclusion that exclusive treatment ofspatial trend while interpolating river bathymetry provides betterresults is applicable to many other rivers, it should be noted thatmore research is needed to look at other factors related to this to-pic. Findings related to spatial arrangement of bathymetry andnumber of data points may be different if channel morphology,geology and substrate type are also taken into account. Comparedto a sandy river, a gravel bed river may need more measurementsto accurately capture the small scale variations in the bathymetry.Another enhancement to the present study would be to look at therole of river flow on the bathymetry, and how flow conditionsshould be incorporated during data collection, and consequentlyspatial interpolation. Similarly, the effect of different techniquesused in this study also needs to be investigated. For example,how would the results change if the test and validation data aresplit 50/50, or all datasets are sampled randomly (Brazos is sam-pled manually for creating validation dataset). Investigation ofthese factors in detail may shed more light into results obtainedin Fig. 7 that show that not all datasets respond equally to differenttrend models and spatial interpolation techniques.
The idea of separating bathymetry into trend and residualsopens a new area of research in bathymetric modeling: stochasticsimulation of river bathymetry. Although the parameters of trendsfunctions used in this study are developed by using field bathym-etry data, approximate trend for river bathymetry can also bedeveloped by using channel planform, width and depth, whichcan be obtained from other datasets without actual field measure-ments. An aerial photograph can be used to get channel width atany location, and drainage area can be used to get average channelwidth and depth through hydraulic geometry relationships. Cross-sectional shape can be modeled by relating channel planform tocross-sectional asymmetry, and this shape (represented as a func-tion of curvature, channel width and depth) can then be used tocreate an approximate trend surface at any location along a river.This trend must then be combined with a surface of residualswhich can be one of the realizations of several random fields.The author is pursuing this idea by using sequential Gaussian sim-ulation for some of the datasets used in this study.
Acknowledgments
The author is grateful to Barney Austin (Texas Water Develop-ment Board), Tim Osting (previously at Texas Water DevelopmentBoard) and Chad Wagner (USGS) for sharing the bathymetry data-sets used in this study. Constructive comments from two reviewersand the editor led to significant improvement of this manuscript,and their input is greatly appreciated.
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