Testing Different Interpolation Methods Based on Single Beam

International Journal of Geo-Information

Case Report Testing Diﬀerent Interpolation Methods Based on Single Beam Echosounder River Surveying. Case Study: Siret River

Maxim Arseni *, Mirela Voiculescu , Lucian Puiu Georgescu, Catalina Iticescu and Adrian Rosu

Faculty of Science and Environment, European Center of Excellence for the Environment, “Dunarea de Jos” University of Galati, 111, Domneasca Street, 800201 Galati, Romania; [email protected] (M.V.); [email protected] (L.P.G.); [email protected] (C.I.); [email protected] (A.R.) * Correspondence: [email protected]; Tel.: +40-746407084

Received: 12 September 2019; Accepted: 6 November 2019; Published: 10 November 2019

Abstract: Bathymetric measurements play an important role in assessing the sedimentation rate, deposition of pollutants, erosion rate, or monitoring of morphological changes in a river, lake, or accumulation basin. In order to create a coherent and continuous digital elevation model (DEM) of a river bed, various data interpolation methods are used, especially when single-beam bathymetric measurements do not cover the entire area and when there are areas which are not measured. Interpolation methods are based on numerical models applied to natural landscapes (e.g., meandering river) by taking into account various morphometric and morphologies and a wide range of scales. Obviously, each interpolation method, used in standard or customised form, yields different results. This study aims at testing four interpolation methods in order to determine the most appropriate method which will give an accurate description of the riverbed, based on single-beam bathymetric measurements. The four interpolation methods selected in the present research are: inverse distance weighting (IDW), radial basis function (RBF) with completely regularized spline (CRS) which uses deterministic interpolation, simple kriging (KRG) which is a geo-statistical method, and Topo to Raster (TopoR), a particular method specifically designed for creating continuous surfaces from various elevation points, contour, or polygon data, suitable for creating surfaces for hydrologic analysis. Digital elevation models (DEM’s) were statistically analyzed and precision and errors were evaluated. The single-beam bathymetric measurements were made on the Siret River, between 0 and 35 km. To check and validate the methods, the experiment was repeated for five randomly selected cross-sections in a 1500 m section of the river. The results were then compared with the data extracted from each elevation model generated with each of the four interpolation methods. Our results show that: 1) TopoR is the most accurate technique, and 2) the two deterministic methods give large errors in bank areas, for the entire river channel and for the particular cross-sections.

Keywords: interpolation; river bathymetry; single-beam echosounder; river cross-section; Siret River

1. Introduction The assessment of water bodies (lakes, rivers, etc.) properties makes use of knowledge regarding water quality, temperature, salinity, volumetric and depth measurements. The last contributes to the accurate maintenance of dredging, and the bathymetric mapping of supraglacial lakes, rivers, and streams, etc. The bathymetric models are the best solution to deﬁne the bottom surface of any type of lakes or rivers, and these models are obtained by means of depth measurements. The bathymetric light detection and ranging (LIDAR) scanning technology is currently the most applied surveying method for data collection, and it is applied to capture land and seaﬂoor simultaneously, to create a detailed

ISPRS Int. J. Geo-Inf. 2019, 8, 507; doi:10.3390/ijgi8110507 www.mdpi.com/journal/ijgi ISPRS Int. J. Geo-Inf. 2019, 8, 507 2 of 21

3D elevation model along the coastline [1]. Although this is an effective and cost-efficient method, it cannot provide enough detail about the stream or river bathymetry, due to its inability to measure depth accurately where the water is cloudy or turbid [2]. In some cases, additional independent measurements are required to confirm LIDAR bathymetry performance, especially in deep, turbid waters or river cloudy waters, as mentioned e.g., by Kinzel [3] and Saylam [4]. Thus, if no bathymetric LIDAR is available, an accurate DEM (digital elevation model) for river beds can be obtained, when real-time kinematic (RTK) global positioning system (GPS) technologies are combined with multibeam or single-beam echosounders [5,6]. River bathymetry or the so-called “bed topography” plays a critical role in flow dynamics numerical modeling, in sediment transport modeling, geomorphologic modeling or in ecological assessments [7]. The best way to collect precise data for a river is by surveying it on a specific direction, namely on transversal cross-sectional paths [8,9]. The bathymetry measurement method is based on a boat-mounted single-beam or multibeam echosounder device combined with a global positioning system (GPS) with real-time kinematic (RTK) correction. Using this combination of measurements, a series of three-dimensional data (x, y, z) can be collected. The spatial resolution of collected data can be improved either by using a very expensive echosounder or by applying different methods of spatial interpolation in a post-processing step. Spatial interpolation of scattered data points has been an active research topic during the last decade since it has been applied in various fields. For example, Li and Heap [10] present a review of 53 comparative studies which assess the performance of 72 interpolation submethods and combined methods, used in different domains. Special attention is paid to the use of spatial interpolation techniques in hydrology, for interpolating scattered elevation data in order to create digital elevation models (DEM), which include the minor bed of river channel. According to Liffner [11], and Merwade [12] and Merwade [13], the DEM is affected directly by the type of interpolation methods used. The present study aims at demonstrating the importance of choosing the appropriate spatial interpolation methods for interpolating river channel bathymetry data. The performance of four different interpolation techniques were analyzed by applying different statistical methods of performance assessment, such as: Minim value (MinV), maxim value (MaxV), mean error (ME), mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE), root mean square standardized error (RMSSE) and standard deviation (SD) [13–18].

Study Area The Siret hydrographic basin is located in the eastern part of Romania and it is the drainage basin of River Siret, which is the largest and most important tributary of the Danube [19]. The Siret River basin covers an area of 42,890 km2 in Romania, 28.116 km2 of which are managed by the Siret Water Directorate [20] under the name Siret Hydrographic Space (SHS). The study area covers a 35 km section upstream River Siret, along Galati–Sendreni–Independenta, i.e., from its confluence with the Danube, up to Independenta village. River Siret forms a natural border between Galati and Braila counties. Figure1 shows an aerial map of the area under scrutiny. The Siret riverbed material is composed of a combination of fine sand and silt [21]. The river bankfull along the studied area is approximately 100 m wide. Since no accurate bathymetric measurements have been made in this area [22], analyzing the bathymetry of this watercourse is scientifically relevant. Moreover, the measurements and three-dimensional information corresponding to this river section are of utmost contribution for enlarging the existing national databases. The single-beam echosounder (SBES) is the most common instrument used in ports and in lake and river surveys, because it is an efficient, low cost and accessible instrument. The SBES uses acoustic depth measurements, which are based on measuring the elapsed time that an acoustic pulse takes in order to travel from a transducer to the waterway bottom and back. The general formula of the corrected depth is given by [23]: 1 d = (v t) + k + dr. (1) 2 ∗ ISPRS Int. J. Geo‐Inf. 2019, 8, 507 3 of 21 ISPRS Int. J. Geo‐Inf. 2019, 8, 507 3 of 21 1 ISPRS Int. J. Geo-Inf. 2019, 8, 507 𝑑1 𝑣∗𝑡 𝑘𝑑 3 of(1) 21 𝑑2 𝑣∗𝑡 𝑘𝑑 (1) 2 where: d = corrected depth; v = average velocity of sound in the water; t = time from transducer to where:bottom dand = correctedcorrected back; k = system depth; vindex = averageaverage constant; velocity velocity dr = distance of of sound sound from in the reference water; watert = time surface from to transducer transducer. to bottom and back; k = systemsystem index index constant; constant; drr == distancedistance from from reference reference water water surface surface to to transducer. transducer.

Figure 1. The study area located at the confluence of River Siret with the Danube. The 35 km area Figuremeasured 1. The is positioned study area between located the at theyellow conﬂuenceconfluence lines. of River Siret withwith thethe Danube.Danube. The 35 km area measured is positioned between the yellow lines.lines. 2. Materials and Methods 2. Materials and Methods 2. Materials and Methods 2.1. Materials 2.1. Materials The bathymetry data were collectedcollected by usingusing aa boat-mountedboat‐mounted single-beamsingle‐beam acoustic depth The bathymetry data were collected by using a boat‐mounted single‐beam acoustic depth sounder (SBES) linked to a real-timereal‐time kinematickinematic (RTK) globalglobal positioningpositioning system (GPS), which can sounder (SBES) linked to a real‐time kinematic (RTK) global positioning system (GPS), which can provide sub-decimetersub‐decimeter accuracyaccuracy forfor thethe surveyedsurveyed points.points. Figure2 2 shows shows several several photos photos taken taken during during provide sub‐decimeter accuracy for the surveyed points. Figure 2 shows several photos taken during data collection. data collection.

Figure 2.2. Left: Bathymetric depth measurements with SonarMite single-beamsingle‐beam echosounder (SBES) (mountingFigure(mounting 2. Left: method Bathymetric on on a a fiberglass ﬁberglass depth measurementsrigid rigid boat) boat) linked linked with to toSonarMiteSouth South S82 S82-V‐ Vsingle RTK RTK‐ beamGNSS; GNSS; echosounder middle: middle: GNSS (SBES) GNSS and and(mountingSBES SBES instruments instrumentsmethod onand a and fiberglassaccessories; accessories; rigid right: boat) right: total linked total station to station South and and S82South South‐V RTK S82 S82-V GNSS;‐V RTK RTK middle: GNSS GNSS GNSS used used and for topographicSBES instruments measurements. and accessories; right: total station and South S82‐V RTK GNSS used for topographic measurements. Bathymetric depths were measured using an Ohmex SonarMite/BTX SonarMite/BTX single single-beam‐beam echosounder, echosounder, with Bathymetric aa 235 235 kHz kHz frequency. depthsfrequency. were The accuracyThe measured accuracy of using the of echo anthe soundingOhmex echo soundingSonarMite/BTX equipment equipment used single to collect ‐usedbeam the toechosounder, bathymetric collect the depth data is 0.025 cm, with a 4-degree beam spread, able to operate between 0.30 m to 75.00 m withbathymetric a 235 kHzdepth± frequency. data is ±0.025 The± cm, accuracy with a ±4of ‐degreethe echo beam sounding spread, equipment able to operate used between to collect 0.30 the m depthbathymetricto 75.00 range m depth depth (software range data limited). (softwareis ±0.025 The cm,limited). soundwith aThe velocity±4 ‐sounddegree in velocity beam water spread, ranges in water able between ranges to operate 1400between to between 1600 1400 m to/0.30s, 1600 but, m whentom/s, 75.00 but, a soundm when depth velocity a range sound proﬁler (software velocity (SVP) limited).profiler is not (SVP)The available, sound is not the velocity available, echosounder in water the echosounder uses ranges a 1500 between m /usess average 1400 a 1500 to value. 1600 m/s Dependingm/s, but, when on the a watersound depth, velocity the profiler echosounder (SVP) applies is not available, 3 to 6 Hz ultrasonic the echosounder ping rate, uses with a an1500 output m/s data range of 2 Hz [24].

ISPRS Int. J. Geo‐Inf. 2019, 8, 507 4 of 21 ISPRS Int. J. Geo‐Inf. 2019, 8, 507 4 of 21 average value. Depending on the water depth, the echosounder applies 3 to 6 Hz ultrasonic ping average value. Depending on the water depth, the echosounder applies 3 to 6 Hz ultrasonic ping rate,ISPRS with Int. J. an Geo-Inf. output2019 ,data8, 507 range of 2 Hz [24]. 4 of 21 rate, with an output data range of 2 Hz [24]. The measurements were made in 5 days, within the interval 22.03.2017–01.04.2017. Each The measurements were made in 5 days, within the interval 22.03.2017–01.04.2017. Each bathymetry point is characterized by the longitude and latitude obtained from the RTK GPS, and by bathymetryThe measurements point is characterized were made by in 5 the days, longitude within the and interval latitude 22.03.2017–01.04.2017. obtained from the RTK Each GPS, bathymetry and by the elevation (z), which is obtained by subtracting the water depth from the water surface elevation pointthe elevation is characterized (z), which by is the obtained longitude by andsubtracting latitude the obtained water fromdepth the from RTK the GPS, water and surface by the elevation recorded at gauge station. The variation of the daily water level was obtained from the Sendreni (z),recorded which at is gauge obtained station. by subtracting The variation the waterof the depthdaily water from the level water was surface obtained elevation from the recorded Sendreni at gauge station. As Figure 3 shows, the water level during the measurement campaign varied between gauge station. As The Figure variation 3 shows, of the the daily water water level level during was the obtained measurement from the campaign Sendreni varied gauge between station. 435 and 460 cm. As435 Figure and 4603 shows, cm. the water level during the measurement campaign varied between 435 and 460 cm.

460 460 450 450 440 440 430 430 420 420 Water level (cm) 22/03/2017 23/03/2017 29/03/2017 30/03/2017 1/4/2017

Water level (cm) 22/03/2017 23/03/2017 29/03/2017 30/03/2017 1/4/2017 Date (dd/mm/year) Date (dd/mm/year)

Figure 3. Water level at Sendreni gauge station recorded during the measurements campaign. Flood Figure 3. Water level at Sendreni gauge station recorded duringduring the measurementsmeasurements campaign.campaign. Flood levels: C.A. (attention level) = 550; C.I. (flood level) = 600; C.P. (risk level) = 650. levels: C.A. (attention (attention level) = 550;550; C.I. C.I. (flood (ﬂood level) level) == 600;600; C.P. C.P. (risk (risk level) level) == 650.650. The bathymetry data for the surveyed area (Figures 4 and 5) comprise a set of 216816 (x, y, z) The bathymetry data for the surveyed area area (Figures (Figures 44 andand5 5)) comprise comprise a a set set of of 216816 216816 (x, (x, y, y, z) z) points or approximately 45,000 points/km22 measured along the river channel bed. points or approximately 45,000 pointspoints/km/km 2 measured along the river channel bed.

Figure 4. Top and middle: Ortopthomap views of sections S1 (downstream)–S4 (upstream); the blue Figure 4. Top and middle: Ortopthomap views of sections S1 (downstream)–S4 (upstream); the blue Figureline shows 4. Top the and trajectory middle: of Ortopthomap the boat; the river views banks of sections are marked S1 (downstream)–S4 in yellow. Bottom: (upstream); ortopthomap the viewblue line shows the trajectory of the boat; the river banks are marked in yellow. Bottom: ortopthomap lineof the shows whole the bathymetric trajectory surveyof the boat; area, withthe river the ﬁrst banks four are sections. marked in yellow. Bottom: ortopthomap view of the whole bathymetric survey area, with the first four sections. view of the whole bathymetric survey area, with the first four sections. Figures4 and5 present the 9 sections of the whole surveyed area. The total length of the surveyed Figures 4 and 5 present the 9 sections of the whole surveyed area. The total length of the pathFigures reached 4 103.22 and 5 km present with anthe average 9 sections boat of speed the whole of 1.75 surveyed knots (0.9 area. m/s). The The total distance length between of the surveyed path reached 103.22 km with an average boat speed of 1.75 knots (0.9 m/s). The distance thesurveyed cross-section path reached ranges 103.22 from 25km to with 100 an m. average This distance boat speed depends of 1.75 on knots the linearity (0.9 m/s). or The sinuosity distance of between the cross‐section ranges from 25 to 100 m. This distance depends on the linearity or thebetween main the river cross channel‐section being ranges larger from in areas25 to where 100 m. the This river distance track is depends straight, on and the smaller linearity where or sinuosity of the main river channel being larger in areas where the river track is straight, and smaller thesinuosity river of is the meandered. main river Since channel the being water larger discharge in areas rate where is high the river in the track case is of straight, River Siretand smaller (about where the river is meandered. Since the water discharge rate is high in the case of River Siret (about 1000where m 3the/s for river a 4 is m meandered. water level) Since [25], itthe was water almost discharge impossible rate tois maintainhigh in the equal case distances of River betweenSiret (about the 1000 m3/s for a 4 m water level) [25], it was almost impossible to maintain equal distances between cross-sections1000 m3/s for a measured. 4 m water level) [25], it was almost impossible to maintain equal distances between the cross‐sections measured. the crossAn ortopthomap‐sections measured. with a 0.5 0.5 m planimetric resolution, produced in 2016, was used in order to × represent the data in terms of geospatial location. This map was provided by the National Agency ISPRS Int. J. Geo‐Inf. 2019, 8, 507 5 of 21

ISPRS Int. J. Geo-Inf. 2019, 8, 507 5 of 21 An ortopthomap with a 0.5 × 0.5 m planimetric resolution, produced in 2016, was used in order to represent the data in terms of geospatial location. This map was provided by the National Agency for Cadastre and Land Registration of Romania (ANCPI).(ANCPI). The bathymetric measurements and data processing observedobserved the the International International Hydrographic Hydrographic Organization Organization (IHO) S-44(IHO) regulation S‐44 regulation since Romania since doesRomania not have does a not clearly have deﬁned a clearly national defined regulation national for regulation river bathymetric for river measurements.bathymetric measurements. According to thisAccording regulation, to thethis accuracy regulation, of bathymetric the accuracy depth of needs bathymetric to comply depth with theneeds requirements to comply of thewith Special the Orderrequirements [26] which of the is usedSpecial only Order for those[26] which areas is where used under-keelonly for those clearance areas where is critical. under The‐keel accuracy clearance of depthis critical. data The by accuracy IHO-S44 of [26 depth] regulation data by for IHO this‐S44 type [26] of regulation area is 0.25 for m,this by type applying of area the is ±0.25 maximum m, by ± allowableapplying the total maximum vertical uncertainty allowable total (TVU), vertical thus theuncertainty results given (TVU), by thethus SBES the results used for given the present by the researchSBES used article for the complies present with research the IHO article S-44 complies regulations. with the IHO S‐44 regulations.

Figure 5. The same as in Figure 44,, butbut for for the the last last ﬁve five sections sections of of the the bathymetric bathymetric surveying: surveying: S5S5 (downstream)–S9 (upstream).(upstream).

The Lat-LongLat‐Long coordinates and elevation (Z) (Z) of of each each point point were were measured measured with with an an RTK RTK GPS GPS – – South S82-VS82‐V equipment. equipment. The The South South S82-V S82 is‐V an is RTK an GNSS RTK receiver,GNSS receiver, designed designed for precision for acquisition precision data.acquisition This surveying data. This instrument surveying can instrument receive GPS can and receive satellite GPS signals and satellite from GLONASS signals from and GLONASS GALILEO. Theand planimetricGALILEO. pointsThe planimetric were measured points in were the WGS84 measured coordinate in the system WGS84 and coordinate the altimetric system point and in the Blackaltimetric Sea 1975point Constanta in the Black height Sea Datum. 1975 ByConstanta additionally height using Datum. the RTK By method additionally the accuracy using increasesthe RTK up to 8 mm + 1 ppm RMS (Root Mean Square) for horizontal data surveying, and up to 15 mm + method± the accuracy increases up to ±8 mm + 1 ppm RMS (Root Mean Square) for horizontal± data 1surveying, ppm RMS and for up vertical to ±15 data mm surveying + 1 ppm RMS [27]. for vertical data surveying [27]. The bank points were measured by using a Trimble M5M5 surveyingsurveying totaltotal station.station. The maximum distance between the points which mark the left and right river banks waswas 3030 m.m. A combination of bathymetric and topographic measurements was used in order to achieve higher accuracy when the four interpolationinterpolation methodsmethods were were applied. applied. The The accuracy accuracy of of the the DEM DEM depends depends directly directly on on the the number number of measuredof measured points points and and it is it crucialis crucial for for ﬂood flood mapping mapping and and hydraulic hydraulic modeling. modeling. To performperform various various functions, functions, such such as automatedas automated mapping mapping functions, functions, data managementdata management or spatial or analysisspatial analysis function, function, the ArcGIS the versionArcGIS 10.2.,version provided 10.2., provided by ESRI (Environmentalby ESRI (Environmental Systems ResearchSystems Institute),Research Institute), was used. was A GIS used. software A GIS populated software withpopulated the correct with datathe correct can help data in thecan organization, help in the interpretation,organization, interpretation, and communication and communication of environmental of environmental data. For example, data. For this example, particular this GIS particular software hasGIS beensoftware used has to createbeen used digital to soil create mapping, digital generatedsoil mapping, by kriging generated interpolation, by kriging in interpolation, diﬀerent urban in areasdifferent [28]. urban Also, areas the GIS [28]. software Also, the was GIS used software [29] for was comparing used [29] spatial for comparing interpolation spatial methods interpolation in order tomethods estimate in theorder precipitation to estimate distribution.the precipitation In our distribution. study, the In GIS our software study, the was GIS used software for spatial was used data

ISPRS Int. J. Geo-Inf. 2019, 8, 507 6 of 21 exploration and surface generation, selecting diﬀerent interpolation methods and statistical analyses which allow developing a DEM for riverbed channel from various data measured at discrete points.

2.2. Methods The digital elevation model (DEM) is a widely used product and provides a three-dimensional representation (X, Y, Z) of the studied areas. According to Burrough [30], the term DEM can be defined as “a regular matrix representation of continuous variations of space relief units”. This section describes the four different methods which are commonly used for interpolating scattered point data in order to obtain an accurate DEM for this section of River Siret. Calculations were performed by using the z values resulting from the bathymetric field measurements.

2.2.1. Inverse Distance Weighting (IDW) IDW represents a deterministic spatial interpolation model, based on an assumption that the value at an unsampled point can be approximated by a weighted average of observed values within a circular search neighborhood [31]. The radius of the search circle is deﬁned by the range of a ﬁxed number of closest points. In most cases, the weights used for averaging the data value are a decreasing function of the distance between the sampled and unsampled points [32–34]. The general equation used by IDW in order to estimate a value at an unsampled point is given by:

XN z∗ = λiZi . (2) i=1 where Zi (i = 1, 2, ... , N) are measured values at N points, and the λi is given by:

N 1 X 1 λi = ( p )/( p ) (3) di i=1 di

* where di is the distance between ith sampled point (zi) and the unsampled point (z ), and p is the exponent variable. A higher power exponent results in less inﬂuence from distant points, and vice versa. The standard power value is two.

2.2.2. Simple Kriging (KRG) Kriging (KRG) is a method developed in the 1960s by the French mathematician Matheron [35] and exemplified in detail by Coburn [36]. This is a geostatistical method of interpolation which is very useful in different fields. Maps generated by this method have a very well-structured visual appearance, all based on irregular spatial data [35]. The general model of this method is based on a constant µ(s) for the data set and random errors ε(s) which have spatial dependence [37–40]:

Z(s) = µ(s) + ε(s). (4)

The prediction of value in a new cell position is done by summing the values of the weight of the multiplied points with the Z value of the data used in the interpolation [41]:

XN XN Zˆ (s0) = λiZ(si), λi = 1 (5) i=1 i=1

Being a very ﬂexible method, it can be used by default or customized to match the data by specifying the nearest semi-variogram model. The semi-variogram consists of two components: The experimental semi-variogram (empirical) and the theoretical model of the semi-variogram [42]. ISPRS Int. J. Geo-Inf. 2019, 8, 507 7 of 21

The experimental semi-variogram results from the calculation of the variance of each measured point versus the other points used in spatialization [43]:

Nj 1 X 2 λˆ hj = [Z(si) Z(si + h)] (6) 2Nj − i=1 where: Nj is a set of pairs of locations separated by the distance h; h—represents the average of the distances between distinct pairs Nj. Data analysis for KRG interpolation was performed using two particular cases of this method, namely: Simple kriging (SKRG) and universal kriging (UKRG). The SKRG method involves determining the unknown value by estimating the weighted average values of the neighboring points [44]:

XN Z(s ) = λ Z(s ) ε. (7) 0 i i ± i=1 where Z(s0) represents the estimated value, Z(si) is the value of the neighboring points, λi is the PN coeﬃcients of weight, which satisfy the condition i=1 λi = 1, and ε is the default estimation error. The UKRG method uses the assumption that the spatial variation of z is dependent on three components: a data set structure, a correlated random component, and a residual error. This is a hybrid method, where the spatial trend is measured by polynomial surfaces of diﬀerent orders, derived globally or locally [45]: XN λi fk(si) = fk(s0). (8) i=1 where Z(s0) represents the estimated value of the unknown variability, µ(s0) is the deterministic function, and ε(s0) is random variation.

2.2.3. Radial Basis Function (RBF) The radial basis function (RBF) is a interpolation method similar to the kriging family, but it does not beneﬁt from the contribution of space spatial data analysis through the variogram [46]:

XN Zˆ (s ) = ω ϕ( s s ) + ω + . (9) 0 i k i − 0 k n 1 i=1 where ϕ(r) is the radial base function, r = s s is the radial distance between the point for which a k i − 0 k new s0 value is calculated and the points with measured values si, and ω symbolizes the weights to be estimated. Basic radial functions are often used to create neural networks. This method is used to calculate surfaces composed of a very large number of points [47]. According to Dumitrescu [48], the most applied RBF functions are:

1. Multiquadric function (MQ): 1/2 ϕ(r) = r2 + σ2 . (10)

2. Thin-plate spline (TPS): ϕ(r) = (σ + r)2 ln(σ + r). (11)

3. Spline with tension (ST): r ϕ(r) = ln σ + K σ r + Ce (12) ∗ 2 0 ∗ ISPRS Int. J. Geo-Inf. 2019, 8, 507 8 of 21

where K0(x) is the modiﬁed Bessel function. 4. Completely regularized spline (CRS):

n 2n X∞ ( 1) (σ r) r 2 r 2 θ(r) = − ∗ = ln σ + E σ + Ce. (13) − n!n ∗ 2 1 ∗ 2 n=1

where E1(x) is the exponential integration function, and Ce is the Euler constant.

2.2.4. Topo to Raster Interpolation (TopoR) The Topo to Raster interpolation method (TopoR) is specifically designed for creating hydrologically correct digital elevation models (DEMs) [49]. It is based on the ANUDEM program developed by Michael Hutchinson [50–53]. This method was designed to take advantage of all types of possible input data which typically characterize the landforms (elevation points, contour lines, stream centerline, sink, boundary, lake, cliff, coast) [54]. The Topo to Raster interpolation was developed by the ANU (Australian National University), but this method is often used in many countries and research papers [55,56]. It uses a finite differential interpolation technique and it is optimized to streamline the local interpolation computational method such as IDW interpolation, without losing the continuity of surfaces obtained by the kriging or RBF Spline interpolation methods. The difference is defined as the first and second degree of partial derivation f of the interpolation method described by the following Equations [56]: Z 2 2 J1( f ) = fx + fy dxdy. (14)

Z 2 2 2 J2( f ) = fxx + fxy + fyy dxdy. (15)

J1 and J2 must be minimized in order to eliminate the maximum and minimum peak eﬀect (excessively smooth or peaked surface) and to obtain a realistic surface terrain. If only J2 is minimized, a very smooth surface is obtained, and vice versa, if only J1 is minimized, maximum and minimum peaks occur [57]. Hutchinson [50] suggests applying a roughness penalty by taking into account the cell resolution so as to reduce this eﬀect:

2 J( f ) = 0.5h− J1( f ) + J2( f ). (16)

Essentially, TopoR is a combined interpolation method which uses a discrete technique based on spline polynomial functions of degree m and smoothness k, where the roughness can be modified to allow the generation of DEM with steep changes in the field, such as areas influenced by depths of currents, ridges or cliffs. These landforms are also called local maximums or local minimums. In the present research, point elevation features were used.

3. Results and Discussion

3.1. Block Data Performance Analysis After collecting and validating topo-bathymetric data, digital models were generated (Figure6), by using the four interpolation methods described in the previous section. Figure6a–d presents the four digital elevation models: (a) inverse distance weighting (IDW) interpolation; (b) kriging (KRG) interpolation; (c) radial basis function (RBF) interpolation with completely regularized spline (CRS) method; (d) Topo to Raster interpolation method, for a selected river section of 1500 m. The spatial resolution of 5 5 m was used for all DEMs. × ISPRS Int. J. Geo‐Inf. 2019, 8, 507 9 of 21 ISPRS Int. J. Geo-Inf. 2019, 8, 507 9 of 21 ISPRS Int. J. Geo‐Inf. 2019, 8, 507 9 of 21 Legend Elevation [m] 6 Legend5 Elevation [m] 46 35 24 13 (a) (b) 02 -11 -20 (a) (b) -3-1 -4-2 -5-3 -6-4 -7-5 -8-6 -9-7 (c) (d) -8 -9 Figure 6. Digital elevation(c) model (DEM) of a 1500 m river section(d) of the main channel obtained by using the 4 interpolation methods: (a) inverse distance weighting (IDW); (b) kriging (KRG); (c) radial Figure 6.6. Digital elevation model (DEM) of a 1500 m river section of the main channel obtained by basis function with completely regularized spline (RBF); (d) Topo to Raster. using the 4 interpolation methods: (a) inverse distance weighting (IDW); (b) kriging (KRG); ((c)) radial basis function with completely regularizedregularized splinespline (RBF);(RBF); ((dd)) TopoTopo to to Raster. Raster. All interpolation methods were run by using Spatial Analyst Tools from ArcGIS 10.2. For the simple kriging method, a single spherical variogram model was fitted to the sample variogram of all All interpolation methods were run by usingusing SpatialSpatial AnalystAnalyst Tools fromfrom ArcGISArcGIS 10.2.10.2. For the data by varying different nugget, sill, and range to fit the variogram model. Special attention was simple kriging method, a single spherical variogram model was fittedfitted to the sample variogram of all paid to matching the slope for the first several reliable lags. Due to this fact a spherical data byby varyingvarying di differentfferent nugget, nugget, sill, sill, and and range range to fitto thefit the variogram variogram model. model. Special Special attention attention was paidwas semivariogram, with Nugget = 0.012, Sill = 1.15, Range = 106.44, and Lag Size = 13.30 was adopted. topaid matching to matching the slope the for theslope first for several the reliablefirst several lags. Duereliable to this lags. fact aDue spherical to this semivariogram, fact a spherical with In the case of the TopoR interpolation method, the differences in elevation ranges were least Nuggetsemivariogram,= 0.012, Sillwith= Nugget1.15, Range = 0.012,= 106.44, Sill = 1.15, and LagRange Size = 106.44,= 13.30 and was Lag adopted. Size = 13.30 was adopted. obvious than those resulting from the IDW and RBF interpolation methods. The TopoR DEM has a In thethe case of thethe TopoRTopoR interpolationinterpolation method,method, thethe didifferencesfferences inin elevationelevation rangesranges were leastleast much smoother and flatter graphic representation, with lower sinuosity elements. obvious than those resulting from the IDWIDW andand RBFRBF interpolationinterpolation methods. The TopoR DEM has a The basic statistical value which defines the dispersion of the frequency distribution of much smoother and flatterflatter graphicgraphic representation,representation, withwith lowerlower sinuositysinuosity elements.elements. deviations between the measured and unmeasured interpolated values is the mean‐standard The basicbasic statistical statistical value value which which defines defines the dispersion the dispersion of the frequencyof the frequency distribution distribution of deviations of deviation (mean SD). Figure 7 shows the Gauss curve representation (a) and boxplot chart (b) for betweendeviations the between measured the and measured unmeasured and interpolatedunmeasured values interpolated is the mean-standard values is the deviation mean‐standard (mean each model, which illustrate how closely the created model predicts the measured values. The Gauss SD).deviation Figure (mean7 shows SD). the Figure Gauss 7 curve shows representation the Gauss curve (a) and representation boxplot chart (a) (b) and for boxplot each model, chart which(b) for distribution shown in Figure 7a proves that the narrowest line is associated to the TopoR method. A illustrateeach model, how which closely illustrate the created how model closely predicts the created the measuredmodel predicts values. the The measured Gauss distribution values. The shown Gauss good model will be described by a narrow error distribution and a small, close to zero, SD. Both indistribution Figure7a provesshown thatin Figure the narrowest 7a proves line that is the associated narrowest to line the is TopoR associated method. to the A TopoR good model method. will A plots in Figure 7a,b suggest that TopoR is the best model, since the associated error histogram is the begood described model will by a be narrow described error by distribution a narrow anderror a small,distribution close toand zero, a small, SD. Both close plots to zero, in Figure SD. 7Botha,b narrowest and the mean SD is the smallest. suggestplots in Figure that TopoR 7a,b suggest is the best that model, TopoR since is the the best associated model, since error the histogram associated is the error narrowest histogram and is thethe meannarrowest SD is and the the smallest. mean SD is the smallest.

Histogram of error distribution Boxplot Graph 7000 0.7 IDW 0.38 0.6 0.33 KRG Histogram of error distribution Boxplot Graph 6000 0.27 7000 RBF 0.50.7 0.21 TopoR IDW 0.38 5000 0.40.6 0.33 KRG 6000 0.27 RBF 0.30.5 4000 TopoR 0.21 5000 0.20.4 3000

Frequency 0.10.3 4000 2000 0.00.2 3000

Frequency -0.10.1 Mean 1000 Mean±SE 2000 -0.20.0 Mean±SD 0 -0.3-0.1 Mean 1000-1.69 -1.13 -0.58 -0.03 0.53 1.08 1.63 IDW KRG RBF TopoR -0.2 Mean±SE Mean±SD 0 (a) -0.3 (b) -1.69 -1.13 -0.58 -0.03 0.53 1.08 1.63 IDW KRG RBF TopoR Figure 7. Statistical results of model(a) ﬁt: (a) The frequency of error distribution(b) for each of the four

models: IDW (blue), KRG (red), RBF (green), TopoR (magenta); (b) Mean standard deviation between Figuremeasured 7. Statistical and interpolated results of z-values model fit: for (a each) The of frequency the four models. of error IDW,distribution inverse for distance each of weighting; the four models:KRG, simple IDW kriging;(blue), RBF,KRG radial (red), basis RBF function. (green), TopoR (magenta); (b) Mean standard deviation Figure 7. Statistical results of model fit: (a) The frequency of error distribution for each of the four between measured and interpolated z‐values for each of the four models. IDW, inverse distance Formodels: a fair IDW comparison (blue), KRG of the (red), interpolated RBF (green), points TopoR on the (magenta); chosen section, (b) Mean the samestandard number deviation of points weighting; KRG, simple kriging; RBF, radial basis function. and thebetween 5 m measured5 m spatial and resolution interpolated of z the‐values raster for were each used.of the Infour order models. to assess IDW, the inverse accuracy distance of each × interpolationweighting; methods KRG, simple with kriging; the SBES RBF, measurements, radial basis function. some statistical tests whose results are shown in

ISPRS Int. J. Geo‐Inf. 2019, 8, 507 10 of 21

For a fair comparison of the interpolated points on the chosen section, the same number of points and the 5 m × 5 m spatial resolution of the raster were used. In order to assess the accuracy of each interpolation methods with the SBES measurements, some statistical tests whose results are shown in Tables 1 and 2, were performed. Table 1 describes the results of obtained mean elevation and variance for all points along the 1500 m river section, and the differences between the depth measurements (SBES) with the four interpolators by using a two‐tailed t‐test. Values of correlation ISPRS Int. J. Geo-Inf. 2019, 8, 507 10 of 21 coefficients and results of the regression analysis are shown in Table 2.

Tables1 andTable2, were 1. Report performed. on the Table results1 describesfrom T‐test the to test results the significance of obtained of mean the compared elevation method. and variance for all points along the 1500 m river section, and the differences between the depth measurements Results T‐test (SBES) with the four interpolators by using a two-tailed t-test. Values of correlation coefficients and Method Mean Variance Count T‐stat P(T < = t) Two‐Tail results of the regression analysis are shown in Table2. SBES −1.060 4.170 8709 ‐ ‐ IDW −1.087 3.724 8709 0.917 0.358 Table 1. Report on the results from T-test to test the significance of the compared method. KRG −1.065 3.963 8709 0.174 0.861 RBFResults −1.091 3.708 8709 T-Test1.062 0.288 TopoR −0.983 4.119 8709 −2.271 0.023 Method Mean Variance Count T-stat P(T < = t) Two-Tail SBES 1.060 4.170 8709 - - Table 2. Results of correlation between− SBES and the DEMs obtained by using interpolation methods IDW 1.087 3.724 8709 0.917 0.358 − for the 1500mKRG river section (left column)1.065 and 3.963 regression 8709 (right column) 0.174 for each 0.861of the four DEMs. − RBF 1.091 3.708 8709 1.062 0.288 Correlation− Test Regression Analysis Method TopoR 0.983 4.119 8709 2.271 0.023 Pearson −Coefficient R2− SD P‐Value IDW 0.979 0.959 0.414 1.53x10‐38 Table 2. Results of correlation between SBES and the DEMs obtained by using interpolation methods ‐4 forKRG the 1500m river section (left column)0.984 and regression (right0.969 column) for each0.356 of the four DEMs.4.15x10 RBF 0.973 0.947 0.468 1.54x10‐30 Correlation Test Regression Analysis ‐109 TopoRMethod 0.988 0.973 0.306 6.54x10 Scatterplots of interpolatedPearson Coe (Vi)ffi andcient measured R2values (Vm) can SD be seen in Figure P-Value 8a–d, together with the correspondingIDW correlation 0.979 coefficient. The 0.959 four methods 0.414 give reliable1.53 results10 38 for most × − elevation values,KRG except for the 0.984 interval 4–6 m. Correlation 0.969 coefficients 0.356 are close4.15 to 1 10for4 all methods × − RBF 0.973 0.947 0.468 1.54 10 30 (see Table 2); however, not all methods give similar results. Table 2 shows that the× RBF− method has TopoR 0.988 0.973 0.306 6.54 10 109 the lowest performance, with the smallest regression coefficient, R2 = 0.947, and× −a correlation coefficient of 0.973. This is in line with Arseni [18] and Rosu [58], who found a similar result. The Page:Scatterplots of interpolated (Vi) and measured values (Vm) can be seen in Figure8a–d, together10 withFigure the correspondingshould be aligned correlation center with coeffi nocient. first The line four indent. methods The current give reliable indent results is—5.95 for most elevation values,The except TopoR for the interpolation interval 4–6 m.method Correlation gives coetheffi bestcients results, are close with to 1 the for allhighest methods values (see Tablefor both2); however,regression not and all correlation methods give coefficients similar results. (R2 = 0.973 Table and2 shows 0.988). that Also, the the RBF DEM method produced has the with lowest TopoR performance,has the lowest with number the smallest of points regression within coetheffi problematiccient, R2 = 0.947,sector and of 4–6 a correlation m. This is coe in ffilinecient with of 0.973.Arseni This[22], is who in line found with a Arseni similar [ 18result.] and Rosu [58], who found a similar result.

Scatterplot Vi / Vm Scatterplot Vi / Vm Scatterplot Vi / Vm Scatterplot Vi / Vm 6 6 6 6

4 4 4 4

2 2 2 2

0 0 0 0

-2 -2 -2 -2 SBES SBES SBES SBES -4 -4 -4 -4

-6 -6 -6 -6

-8 -8 -8 -8

-10 -10 -10 -10 -10-8-6-4-20246 -10-8-6-4-20 2 4 6 -10-8-6-4-20 2 4 6 -10-8-6-4-20246 IDW KRG RBF TopoR Vi:Vm: r2 = 0.9589 Vi:Vm: r2 = 0.9694 Vi:Vm: r2 = 0.9469 Vi:Vm: r2 = 0.9728 (a) (b) (c) (d)

FigureFigure 8. 8.The The linear linear regression regression for thefor four the interpolationfour interpolation methods: methods: (a) IDW; (a) ( bIDW;) KRG; (b ()c )KRG; RBF; ((dc)) TopoR.RBF; (d) TopoR. The TopoR interpolation method gives the best results, with the highest values for both regression 2 and3.2. correlation Local Cross coe‐Sectionﬃcients Performance (R = 0.973 Analysis and 0.988). Also, the DEM produced with TopoR has the lowest number of points within the problematic sector of 4–6 m. This is in line with Arseni [22], who found a similar result.

3.2. Local Cross-Section Performance Analysis In order understand better the qualities of each of the four interpolation methods, the individual behaviour of the models for ﬁve cross-sections chosen for validation was also analyzed. Thus, ﬁve ISPRS Int. J. Geo‐Inf. 2019, 8, 507 11 of 21

Thus, five random cross‐sections were chosen for repeated surveying and data validation from the 1500 m studied area (Figure 9). ISPRS Int. J. Geo‐Inf. 2019, 8, 507 11 of 21 ISPRS Int. J. Geo-Inf. 2019, 8, 507 11 of 21 In order understand better the qualities of each of the four interpolation methods, the individual behaviour of the models for five cross‐sections chosen for validation was also analyzed. randomThus, five cross-sections random cross were‐sections chosen were for repeated chosen surveyingfor repeated and surveying data validation and data from validation the 1500 m from studied the area1500 (Figurem studied9). area (Figure 9).

Figure 9. Locations of the five‐cross‐section measured with SBES technique.

The SBES profile was obtained from measurements by using the linear method (cross‐sectional survey). The four modeled profiles were obtained by extracting each point value from the generated DEMs. Each cross‐section surveyed by SBES bathymetric measurements was compared with the values extracted from each DEM characteristic of each interpolation method. The differences Figure 9. Locations of the five five-cross-section‐cross‐section measured with SBES technique. between each model and measurements for each cross‐section is shown in Figure 10. Detailed informationThe SBES about profileprofile the was five obtained statistical from parameters measurements (variables) by using for the models linear and method measurements (cross-sectional(cross‐sectional are presentedsurvey). The in Tablefour modeled A1 (Appendix profiles profiles B): were minimum obtained value by extracting (MIN), maximum each point value value (MAX), from the mean generated value (MEAN),DEMs. Each Each median cross cross-section (MEDIAN),‐section surveyed surveyed and standard by by SBES SBES deviation bathymetric (SD), measurements calculated by was using compared the ArcGIS with 10.4 the software.values extractedextracted These parameters from from each each DEM were DEM characteristic used characteristic to assess of the each ofsuccess interpolationeach ofinterpolation the data method. analysis. method. The diff erencesThe differences between eachbetweenThe model differenceseach and model measurements between and measurements SBES for measurements each cross-section for each and cross isinterpolated shown‐section in is Figure elevationshown 10 .in value, Detailed Figure for 10. information each Detailed of the fiveaboutinformation section, the five areabout statisticalshown the infive parametersFigure statistical 10 in (variables) orderparameters to assess, for (variables) models in a direct and for manner, measurementsmodels theand performance measurements are presented of each are in model.Tablepresented A1 Figure (Appendixin Table 10 shows A1B ):(Appendix that minimum all models B): value minimum reproduce (MIN), value maximumvery (MIN), well the value maximum river (MAX), bed value except mean (MAX), for value two mean regions (MEAN), value of aboutmedian(MEAN), 20–30 (MEDIAN), median m from (MEDIAN), the and river standard banks, and deviationwhichstandard are deviationbasically (SD), calculated regions(SD), calculated of by small using depths by the using ArcGIS or positive the 10.4 ArcGIS elevation software. 10.4 (asThesesoftware. seen parameters in These Figure parameters A1/Appendix were used were to assess A). used the to assess success the of success the data of analysis. the data analysis. The differences between SBES measurements and interpolated elevation value, for each of the Cross-section 1 five section,5 are shown in Figure 10 in order to assess, in a direct manner, the performance of each model.4 Figure 10 shows that all models IDW reproduce KRG very RBF well the TopoR river bed except for two regions of about3 20–30 m from the river banks, which are basically regions of small depths or positive elevation 2 (as seen1 in Figure A1/Appendix A). Error [m] 0 -1 Cross-section 1 5 0 102030405060708090100110 4 IDW KRG RBF TopoR Distance [m] 3 2 Cross-section 2 1

Error [m] 5 40 IDW KRG RBF TopoR -13 2 0 102030405060708090100110 1 Distance [m] Error [m] 0 -1 0 102030405060708090100110120130 Distance [m] Figure 10. Cont.

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Cross-section 3 5 4 IDW KRG RBF TopoR 3 2 1 Error [m] 0 -1 0 102030405060708090100110 Distance [m] Cross-section 4 5 4 IDW KRG RBF TopoR 3 2 1 Error [m] Error 0 -1 0 102030405060708090100110120 Distance [m] Cross-section 5 5 4 IDW KRG RBF TopoR 3 2 1 Error [m] 0 -1 0 102030405060708090100110120130140 Distance [m] Figure 10. DifferenceDifference between measurements (by SBES) and interpolated elevation values from each DEM model (SBES–DEM): IDW—red line, KRG—green line, RBF—magenta line, TopoR—black line; DEM model (SBES–DEM): IDW—red line, KRG—green line, RBF—magenta line, TopoR—black line; negative values correspond to overestimating the elevation. negative values correspond to overestimating the elevation. The differences between SBES measurements and interpolated elevation value, for each of the The deviation increases when approaching both banks for all models. TopoR slightly five section, are shown in Figure 10 in order to assess, in a direct manner, the performance of each overestimates the elevation for all sections, while IDW and RBF underestimate the elevation at the model. Figure 10 shows that all models reproduce very well the river bed except for two regions of shores. The KRG model has an irregular behavior, with values which are either larger or smaller about 20–30 m from the river banks, which are basically regions of small depths or positive elevation than measurements. Obviously the TopoR model is the best, since the deviations from (as seen in Figure A1/AppendixA). measurements are the smallest (1 m). On the other hand, IDW has the worst performance, with The deviation increases when approaching both banks for all models. TopoR slightly overestimates differences between the model and the field measurements of more than 5 m (5.35 m for the elevation for all sections, while IDW and RBF underestimate the elevation at the shores. The KRG cross‐Section 4, close to right bank of the river flow). It seems that the IDW, KRG and RBF methods model has an irregular behavior, with values which are either larger or smaller than measurements. are not reliable as long as the river banks have steep slopes. Obviously the TopoR model is the best, since the deviations from measurements are the smallest Further, all data have been normalized by the value of the elevation (Z), which represents the (1 m). On the other hand, IDW has the worst performance, with differences between the model and elevation of the river bed channel. Normalization was performed by recalculating the Z values the field measurements of more than 5 m (5.35 m for cross-Section 4, close to right bank of the river between 0 and 1, using the following formula: flow). It seems that the IDW, KRG and RBF methods are not reliable as long as the river banks have steep slopes. 𝑋 min 𝑋 𝑍 (17) Further, all data have been normalized bymax the𝑋 valuemin of 𝑋 the elevation (Z), which represents the whereelevation X = of (X the1, X river2, …, bed Xn) channel. and Zi represent Normalization the ith was normalized performed data. by recalculating the Z values between 0 andThis 1, using type the of followingnormalization formula: is also called “feature scaling”, or “unity‐based normalization”. Another type of normalization (like normalization by dividing to maximum value) cannot provide Xi min(X) scaled data between 0 and 1, consideringZi =the negative− elevation values. (17) max(X) min(X) The normalized profiles obtained from SBES measurements− and the four interpolation methods wherefor eachX cross= (X1‐,Xsection2, ... are,X nshown) and Z ini represent Figure 11. the ith normalized data.

ISPRS Int. J. Geo-Inf. 2019, 8, 507 13 of 21 ISPRS Int. J. Geo‐Inf. 2019, 8, 507 13 of 21

ThisThis typetype ofof normalizationnormalization isis alsoalso calledcalled “feature“feature scaling”,scaling”, oror “unity-based“unity‐based normalization”.normalization”. AnotherAnother typetype ofof normalizationnormalization (like(like normalizationnormalization byby dividingdividing toto maximummaximum value)value) cannotcannot provideprovide scaledscaled datadata betweenbetween 00 and and 1, 1, considering considering the the negative negative elevation elevation values. values. TheThe normalizednormalized proﬁlesprofiles obtainedobtained fromfrom SBESSBES measurementsmeasurements andand thethe fourfour interpolationinterpolation methodsmethods forfor eacheach cross-sectioncross‐section are are shown shown in in Figure Figure 11 11..

1.1 SBES IDW 1.1 SBES IDW 1 KRG RBF 1 KRG RBF 0.9 TopoR 0.9 TopoR 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2

Normalized Z value 0.1 Normalized Z value 0.1 0 0 0 102030405060708090100110 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Distance (m) Distance (m) (a) (b)

1.1 SBES IDW 1.1 SBES IDW 1 KRG RBF 1 KRG RBF 0.9 TopoR 0.9 TopoR 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2

Normalized Z value 0.1 Normalized Z value 0.1 0 0 0 102030405060708090100110 0 102030405060708090100110120 Distance (m) Distance (m) (c) (d)

1.1 SBES IDW 1 KRG RBF 0.9 TopoR 0.8 0.7 0.6 0.5 0.4 0.3 0.2

Normalized Z value 0.1 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 Distance (m) (e)

Figure 11. Comparison between the SBES normalized profile (red) and normalized profiles extracted fromFigure DEM 11. Comparison (orange: KRG, between black: the TopoR, SBES green: normalized IDW and profile blue: (red) RBF), and for normalized five cross-sections profiles forextracted River Siret:from (DEMa) Cross-Section (orange: KRG, 1; (b )black: Cross-Section TopoR, green: 2; (c) Cross-Section IDW and blue: 3; ( dRBF),) Cross-Section for five cross 4; (e‐sections) Cross-Section for River 5. Siret: (a) Cross‐Section 1; (b) Cross‐Section 2; (c) Cross‐Section 3; (d) Cross‐Section 4; (e) FigureCross‐Section 11 shows 5. that each transversal profile obtained by interpolation methods reasonably describes the measured profile of the selected cross-sections of the river bed channel. The good performanceFigure 11 of theshows TopoR that DEM each (black transversal line) is profile illustrated obtained in Figure by 11interpolation, since the corresponding methods reasonably profile isdescribes very close the (sometimes measured even profile superimposed) of the selected to the cross profile‐sections given of by the SBES river bathymetric bed channel. measurements The good (redperformance line). The of other the threeTopoR models DEM give (black sometimes line) is erroneousillustrated results, in Figure especially 11, since in the the area corresponding close to the rightprofile and is leftvery banks, close as (sometimes shown in Figure even 10superimposed). This can be alsoto the seen profile in Appendix given byA, FigureSBES bathymetric A1, where plotsmeasurements of normalized (red line). profiles The are other compared three models to real give profiles. sometimes This is erroneous in accordance results, with especially the observed in the higherarea close dispersion to the right seen forand low left depths banks, (0–5as shown m) in Figurein Figure8. 10. This can be also seen in Appendix A, FigureFigure A1, where12 shows plots results of normalized of the regression profiles are analysis compared for to each real of profiles. the four This methods is in accordance on SBES measurementswith the observed (normalized higher dispersion values). seen for low depths (0–5 m) in Figure 8. Figure 12 shows results of the regression analysis for each of the four methods on SBES measurements (normalized values).

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SBES vs IDW SBES vs KRG SBES vs RBF SBES vs TopoR SBES SBES SBES SBES

1 1.2 1.2 1.2 1.2

1.0 1.0 1.0 1.0 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 section IDW RBF KRG ‐ 0.4 0.4 0.4 0.4 TopoR 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0

Cross -0.2 -0.2 -0.2 -0.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.20.00.20.40.60.81.01.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

1.2 1.2 1.2 1.2

2 1.0 1.0 1.0 1.0

0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 IDW RBF KRG TopoR section 0.4 0.4 0.4 0.4 ‐ 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0

Cross -0.2 -0.2 -0.2 -0.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

1.2 1.2 1.2 1.2

3 1.0 1.0 1.0 1.0

0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 IDW RBF KRG section 0.4 0.4 0.4 TopoR 0.4 ‐ 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0

Cross -0.2 -0.2 -0.2 -0.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

1.2 1.2 1.2 1.2

4 1.0 1.0 1.0 1.0

0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 IDW KRG RBF TopoR section 0.4 0.4 0.4 0.4 ‐ 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0

Cross -0.2 -0.2 -0.2 -0.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

1.2 1.2 1.2 1.2

5 1.0 1.0 1.0 1.0

0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 IDW KRG RBF TopoR section 0.4 0.4 0.4 0.4 ‐ 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0

Cross -0.2 -0.2 -0.2 -0.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

FigureFigure 12. 12.Simple Simple linear linear regression regression of theof the four four DEM-based DEM‐based normalized normalized proﬁle profile on the on normalized the normalized SBES proﬁleSBES profile (left to (left right: to IDW, right: KRG, IDW, RBF, KRG, TopoR) RBF, TopoR) for each for selected each selected cross-section. cross‐section.

The regression analysis shows that the IDW method has a diffuse data dispersion in all 5 The regression analysis shows that the IDW method has a diffuse data dispersion in all 5 cross-sections. Thus, this is the most inaccurate method. The other two methods, i.e., KRG and RBF, cross‐sections. Thus, this is the most inaccurate method. The other two methods, i.e., KRG and RBF, have less regular behavior, with a higher data dispersion for some sections. Again, TopoR is the have less regular behavior, with a higher data dispersion for some sections. Again, TopoR is the least least dispersive method, whose results are almost perfectly linearly dependent on measurements for dispersive method, whose results are almost perfectly linearly dependent on measurements for all allcross cross-sections.‐sections. TheThe statistical statistical analysis analysis was was performed performed by by using using the the Pearson Pearson correlation correlation factor factor (R) (R) and and root root mean mean squaresquare error error (RMSE). (RMSE). The The results results obtained obtained are shownare shown in Figure in Figure 13. The 13. root The mean root square mean errorsquare (RMSE) error or(RMSE) so-called or so root‐called mean root square mean deviation square deviation (RMSD) measures (RMSD) themeasures deviation the ofdeviation output data of output from thedata initial from inputthe initial data [input59,60 ].data A small [59,60]. RMSE A small together RMSE with together a high with R indicate a high a R good indicate fit of a the good model. fit of Obviously,the model. TopoRObviously, has the TopoR best has performance, the best performance, closely followed closely by followed the KRG by method, the KRG while method, IDW while is the IDW poorest is the interpolationpoorest interpolation method in method all cases. in all cases. OurOur results results are are in in line line with with Curebal’s Curebal’s [ 61[61]] conclusions conclusions for for the the Keçidere Keçidere basin, basin, or or for for models models of of thethe Aswan Aswan high high dam dam reservoir reservoir storage storage capacity capacity and and morphology morphology development development [62], which[62], which both show both thatshow TopoR that isTopoR the most is the accurate most accurate method formethod obtaining for obtaining hydrologically hydrologically correct DEM. correct The DEM. IDW andThe KRG IDW methodsand KRG are methods reported are as reported being inaccurate as being inaccurate when analyzing when analyzing river bed topographyriver bed topography by Goff [63 by] andGoff Merwade[63] and [Merwade13]. However, [13]. sometimesHowever, sometimes the kriging the (KRG) kriging method (KRG) estimates method the estimates elevation the better elevation than IDWbetter [64 than]. Also, IDW Šiljeg [64]. [Also,57] shows Šiljeg that [57] that shows the that results that based the results on TopoR based has on no TopoR significant has no di significantfferences (smalldifferences errors) (small in comparison errors) in with comparison the aero-photogrammetric with the aero‐photogrammetric measurements (which measurements made up his (which base model).made up However, his base Šiljeg model). [57] However, reports that Šiljeg TopoR [57] is notreports recommended that TopoR for is thenot micro-level recommended analysis for of,the e.g.,micro rocks‐level or specificanalysis micro-landforms. of, e.g., rocks Aor complex specific analysis micro‐landforms. of many interpolation A complex methods analysis for of creating many DEMsinterpolation in the Belgorod methods regionfor creating of Podsadneea DEMs in [the65], Belgorod shows that region TopoR of isPodsadneea the most accurate [65], shows method. that TopoR is the most accurate method. However, despite the accuracy of interpolation techniques for

ISPRS Int. J. Geo-Inf. 2019, 8, 507 15 of 21

ISPRS Int. J. Geo‐Inf. 2019, 8, 507 15 of 21 However, despite the accuracy of interpolation techniques for generating accurate DEM assessed by severalgenerating studies, accurate there DEM are still assessed no consistent by several ﬁndings studies, about there the are performances still no consistent of the spatial findings interpolators about the forperformances river bathymetry of the spatial [66]. interpolators for river bathymetry [66]

RMSE_IDW RMSE_KRG RMSE_RBF RMSE_TopoR R_IDW R_KRG R_RBF R_TopoR

2 1 0.98 0.96 1.5 0.94 0.92 1 0.9 0.88 0.86 0.5 0.84

0.82 Pearson coefficient (R) 0 0.8

Root mean square error (RMSE) 12345 Cross‐section Figure 13. Root mean square error (RMSE) bar, left axis, and the Pearson correlation coefficient coeﬃcient (R) diamonds, rightright axis, axis, for for modeled modeled values values for for the the selected selected cross-sections: cross‐sections: blue: blue: IDW, IDW, orange: orange: KRG, KRG, grey: RPF,grey: yellow: RPF, yellow: TopoR. TopoR. Please Please note that note the that best the ﬁt best is given fit is given by high by R high and R small and RMSE.small RMSE.

4. Conclusions

The creation of the digital elevation model of an area depends on the interpolation method. A good DEM helps calculating the plan and volumetric areas of the river more precisely; thus, DEM can be very useful for evaluating any morphometric changes. This study aimed at identifying the best interpolation method which can give the most accurate representation of a river bed from single-beam river bathymetry measurements. The bathymetric survey was carried out the on section of River Siret and was based on single- beam echosounder combined with an RTK GPS system. The digital elevation bathymetric models created by single-beam measurement technique depend on the interpolation methods used. In order to assess the accuracy of the interpolated surface, commonly available interpolation methods such as IDW, KRG, RBF and TopoR [61] were used. The Mean SD was one of the statistical indicators which was used to assess the interpolation accuracy. There were no differences between the measured and unmeasured interpolated values in the case of accurate interpolation (e.g., IDW). One can only guess that these differences were determined by cross-validation. The cross-validation of block data showed that the TopoR interpolation has the smallest standard deviation (0.306 m). The coefficients of the 2 2 2 determination R between the predicted and the measured values were RIDW = 0.959, RRBF = 0.947, 2 2 RKRG = 0.969 and RTopoR = 0.973, confirming that TopoR is the most accurate method. Differences were also observed in the case of the digital elevation model representation. The TopoR interpolation method described an attenuated and smoothed DEM (AppendixA, Figures A2–A5). The DEM obtained by using the four interpolation methods were compared with the SBES measurements for the five cross-sections of the river in order to check the accuracy of each method. The TopoR interpolation profile is almost similar to the SBES measurements profile. The profiles obtained by using the IDW, KRG, and RBF methods had sizeable differences as compared to the measured profile, especially in the areas close to the river banks. The statistical results show that all DEMs are statistically significant, regardless of the method; however, the TopoR interpolation method is the best, with the smallest RMSE and the highest Pearson correlation coefficient, when comparing the modeled and measured depth data. The statistical parameters are: RMSETopoR = 0.276 m and RTopoR = 0.997 m, against the IDW method, which has a mean RMSEIDW = 1.532 m and RIDW = 0.897 m.

ISPRS Int. J. Geo-Inf. 2019, 8, 507 16 of 21

Thus, all results show that the TopoR method is the most accurate interpolation method to be used when creating a DEM, for a given number of points and grid. Ideas of a future study have emerged from the analysis of these tests, and one refers to adding bathymetric longitudinal measurements (along the river channel or perpendicular to cross-sections), thus creating a more regular measurement grid. This may improve the analysis of interpolation methods and will also lead to a higher accuracy. The number of points and pixel size (spatial resolution) is another important parameter worth taking into account. The investigation of the eﬀect of these on digital elevation models is presently on the way.

Author Contributions: Maxim Arseni and Adrian Rosu performed the measurements, establish the methodology, validated the data and analyzed the data; Adrian Rosu designed the figures; Lucian Puiu Georgescu and Catalina Iticescu supervised the progress of teamwork; Maxim Arseni did the statistical analysis and, together with Mirela Voiculescu, prepared the manuscript. All authors commented on the original manuscript. Funding: This work was partially co-financed by the European Social Fund, through The Human Capital Operational Programme 2014–2020 (POCU), Romania, POCU/380/6/13. Acknowledgments: Arseni Maxim’s and Rosu Adrian’s research work was supported by the project “ANTREPRENORDOC”, Contract no. 36355/23.05.2019, financed by The Human Capital Operational Programme 2014–2020 (POCU), Romania, and the work of Voiculescu Mirela, Iticescu Catalina and Georgescu Puiu Lucian was supported by the project “EXPERT”, financed by the Romanian Ministry of Research and Innovation, Contract no. 14PFE/17.10.2018. Conflicts of Interest: The authors declare no conflict of interest. ISPRS Int. J. Geo‐Inf. 2019, 8, 507 17 of 21 Appendix A Appendix A

5 SBES IDW 5 SBES IDW 4 KRG RBF 4 KRG RBF 3 3 2 2 1 1 0 0 ‐1 0 102030405060708090100110 ‐2 ‐1 0 102030405060708090100110120130

Elevation Z [m] ‐3 Elevation Z [m] ‐2 ‐4 ‐3 ‐5 ‐4 Distance [m] Distance [m] (a) (b) 5 SBES IDW 5 SBES IDW 3 KRG RBF 4 KRG RBF 3 1 2 ‐1 1 0 102030405060708090100110 0 ‐3 ‐1 0 102030405060708090100110120 ‐5 ‐2

Elevation Z [m] Elevation Z [m] ‐3 ‐7 ‐4 ‐9 ‐5 Distance [m] Distance [m] (c) (d) 5 SBES IDW 4 KRG RBF 3 2 1 0 0

‐1 10 20 30 40 50 60 70 80 90 Elevation Z [m] ‐2 100 110 120 130 140 ‐3 Distance [m] (e)

Figure A1. Proﬁle shape from right to left bank of SBES and extracted from DEM proﬁles for the Siret River:Figure ( aA1.) Cross-Section Profile shape 1; from (b) Cross-Section right to left bank 2; (c) Cross-Sectionof SBES and extracted 3; (d) Cross-Section from DEM 4; profiles (e) Cross-Section for the Siret 5. River: (a) Cross‐Section 1; (b) Cross‐Section 2; (c) Cross‐Section 3; (d) Cross‐Section 4; (e) Cross‐Section 5.

Figure A2. River Siret depth map obtained by using the IDW interpolation method.

ISPRS Int. J. Geo‐Inf. 2019, 8, 507 17 of 21 ISPRS Int. J. Geo‐Inf. 2019, 8, 507 17 of 21 Appendix A Appendix A 5 SBES IDW 5 SBES IDW 54 SBESKRG IDWRBF 54 SBESKRG IDWRBF 43 3 2 KRG RBF 4 KRG RBF 3 32 1 1 20 2 1 0 ‐1 0 102030405060708090100110 1 0 ‐1 0 102030405060708090100110120130 ‐1‐2 0 Elevation Z [m] 0 102030405060708090100110 Elevation Z [m] ‐2 ‐2‐3 ‐1 0 102030405060708090100110120130 ‐4 ‐3 Elevation Z [m] ‐3 Elevation Z [m] ‐2 ‐4‐5 Distance [m] ‐3‐4 Distance [m] ‐5 ‐4 Distance [m](a) Distance [m](b) 5 (SBESa) IDW 5 SBES(b) IDW 3 SBESKRG IDWRBF 4 SBESKRG IDWRBF 5 53 31 KRG RBF 4 KRG RBF 32 ‐11 1 0 102030405060708090100110 20 ‐1‐3 1 0 102030405060708090100110 ‐10 0 102030405060708090100110120 ‐3‐5 ‐1‐2 Elevation Z [m] Elevation Z [m] ‐3 0 102030405060708090100110120 ‐5‐7 ‐2‐4 Elevation Z [m] Elevation Z [m] ‐3 ‐7‐9 Distance [m] ‐4‐5 Distance [m] ‐9 ‐5 Distance [m](c) Distance [m](d) (c) 5 SBES IDW (d) 54 SBESKRG IDWRBF 43 KRG RBF 32 21 10 0

‐1 10 20 30 40 50 60 70 80 90

Elevation Z [m] 0 100 110 120 130 140

‐2 0

‐1 10 20 30 40 50 60 70 80 90 Elevation Z [m] 100 110 120 130 140 ‐2‐3 Distance [m] ‐3 Distance [m](e) (e )

Figure A1. Profile shape from right to left bank of SBES and extracted from DEM profiles for the Siret FigureRiver: A1.(a) ProfileCross‐ Sectionshape from 1; (rightb) Cross to left‐Section bank of 2;SBES (c) andCross extracted‐Section from 3; DEM(d) Cross profiles‐Section for the 4; Siret (e) ISPRSRiver:Cross Int. J.‐ Geo-Inf. Section(a) Cross2019 5. ,‐Section8, 507 1; (b) Cross‐Section 2; (c) Cross‐Section 3; (d) Cross‐Section 4; (e17) of 21 Cross‐Section 5.

Figure A2. River Siret depth map obtained by using the IDW interpolation method. Figure A2. River Siret depth map obtained by using the IDW interpolation method.

ISPRS Int. J. Geo‐Inf. 2019, 8, 507 18 of 21 ISPRS Int. J. Geo‐Inf. 2019, 8, 507 18 of 21 Figure A3. River Siret depth map obtained by using the KRG interpolation method. Figure A3. River Siret depth map obtained by using the KRG interpolation method. Figure A3. River Siret depth map obtained by using the KRG interpolation method.

Figure A4. River Siret depth map obtained by using the RBF interpolation method. Figure A4. River Siret depth map obtained by using the RBF interpolation method. Figure A4. River Siret depth map obtained by using the RBF interpolation method.

Figure A5. River Siret depth map obtained by using the TopoR interpolation method. Figure A5. River Siret depth map obtained by using the TopoR interpolation method. Figure A5. River Siret depth map obtained by using the TopoR interpolation method. Appendix B Appendix B Table A1. Statistics of model fits as given by the ArcGIS report. Table A1. Statistics of model fits as given by the ArcGIS report. Results Number of Measured Width MIN MAX MEAN MEDIAN SD Method Results NumberPoints of Measured Width[m] MIN MAX MEAN MEDIAN SD MethodSBES −4.636 4.272 −0.420 −0.396 2.629 Points [m] SBESIDW − −4.5464.636 1.5124.272 −1.1280.420 −1.1290.396 1.8582.629 Cross‐Section 1 KRGIDW −4.6534.546 3.5831.512 −0.5731.128 −0.5621.129 2.4451.858 237 109 Cross‐Section 1 KRGRBF − −4.6684.653 2.0523.583 −0.8800.573 −0.7580.562 2.0632.445 237 109 TopoRRBF − −4.5304.668 4.5282.052 −0.3300.880 −0.2870.758 2.7022.063 TopoR −4.530 4.528 −0.330 −0.287 2.702 SBES −3.661 5.567 0.156 −0.778 2.928 SBESIDW − −3.6053.661 2.1635.567 − 0.1560.824 − −0.7060.778 1.7452.928 Cross‐Section 2 KRGIDW −3.6523.605 4.5962.163 −0.2550.824 −0.8530.706 2.5851.745 190 128 Cross‐Section 2 KRGRBF − −3.6493.652 2.9964.596 −0.6110.255 −0.8450.853 2.0562.585 190 128 TopoRRBF − −3.6553.649 5.5252.996 −0.1110.611 − −0.8440.845 2.8612.056 TopoR −3.655 5.525 0.111 −0.844 2.861 SBES −7.973 4.569 −1.700 −1.989 4.172 SBESIDW − −7.9277.973 2.3044.569 −2.5681.700 −2.3661.989 3.2774.172 Cross‐Section 3 KRGIDW −8.0387.927 4.2082.304 −2.1192.568 −2.3892.366 3.8333.277 174 109 Cross‐Section 3 KRGRBF − −8.0008.038 3.0704.208 −2.3472.119 −2.3642.389 3.5133.833 174 109 TopoRRBF − −7.8068.000 4.8683.070 −1.5452.347 −2.0852.364 4.2673.513 TopoR −7.806 4.868 −1.545 −2.085 4.267 SBES −4.748 4.713 −0.624 −0.879 2.755 SBESIDW − −4.5264.748 2.0404.713 −1.4230.624 −1.5240.879 1.8942.755 Cross‐Section 4 KRGIDW −4.7514.526 4.4442.040 −0.7971.423 −1.2541.524 2.6531.894 189 118 Cross‐Section 4 KRGRBF − −4.7194.751 2.9014.444 −1.1180.797 −1.2521.254 2.1842.653 189 118 TopoRRBF − −4.6884.719 5.2232.901 −0.4811.118 −0.9821.252 2.9222.184 TopoR −4.688 5.223 −0.481 −0.982 2.922 SBES −2.394 4.224 −0.326 −0.599 1.756 Cross‐Section 5 197 137 SBESIDW − −2.2942.394 2.3794.224 −0.8900.326 −0.9010.599 1.2021.756 Cross‐Section 5 197 137 IDW −2.294 2.379 −0.890 −0.901 1.202

ISPRS Int. J. Geo-Inf. 2019, 8, 507 18 of 21

Appendix B

Table A1. Statistics of model ﬁts as given by the ArcGIS report.

Results Number of MIN MAX MEAN MEDIAN SD Width [m] Method Measured Points SBES 4.636 4.272 0.420 0.396 2.629 − − − IDW 4.546 1.512 1.128 1.129 1.858 − − − Cross-Section 1 KRG 4.653 3.583 0.573 0.562 2.445 237 109 − − − RBF 4.668 2.052 0.880 0.758 2.063 − − − TopoR 4.530 4.528 0.330 0.287 2.702 − − − SBES 3.661 5.567 0.156 0.778 2.928 − − IDW 3.605 2.163 0.824 0.706 1.745 − − − Cross-Section 2 KRG 3.652 4.596 0.255 0.853 2.585 190 128 − − − RBF 3.649 2.996 0.611 0.845 2.056 − − − TopoR 3.655 5.525 0.111 0.844 2.861 − − SBES 7.973 4.569 1.700 1.989 4.172 − − − IDW 7.927 2.304 2.568 2.366 3.277 − − − Cross-Section 3 KRG 8.038 4.208 2.119 2.389 3.833 174 109 − − − RBF 8.000 3.070 2.347 2.364 3.513 − − − TopoR 7.806 4.868 1.545 2.085 4.267 − − − SBES 4.748 4.713 0.624 0.879 2.755 − − − IDW 4.526 2.040 1.423 1.524 1.894 − − − Cross-Section 4 KRG 4.751 4.444 0.797 1.254 2.653 189 118 − − − RBF 4.719 2.901 1.118 1.252 2.184 − − − TopoR 4.688 5.223 0.481 0.982 2.922 − − − SBES 2.394 4.224 0.326 0.599 1.756 − − − IDW 2.294 2.379 0.890 0.901 1.202 − − − Cross-Section 5 KRG 2.325 4.180 0.448 0.737 1.821 197 137 − − − RBF 2.328 2.260 0.816 0.769 1.209 − − − TopoR 2.238 4.962 0.195 0.713 1.993 − − −

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