�����������������

Citation: Alcaras, E.; Amoroso, P.P.;

Parente, C. The Influence of

Interpolated Point Location and

Density on 3D Bathymetric Models

Generated by Kriging Methods:

An Application on the Giglio Island

Seabed (Italy). Geosciences 2022, 12,

62. https://doi.org/10.3390/

geosciences12020062

Academic Editors: Vincent Lecours

and Jesus Martinez-Frias

Received: 28 December 2021

Accepted: 27 January 2022

Published: 29 January 2022

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional affil-

iations.

Copyright: © 2022 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

geosciences

Article

The Influence of Interpolated Point Location and Densityon 3D Bathymetric Models Generated by Kriging Methods:An Application on the Giglio Island Seabed (Italy)Emanuele Alcaras 1,* , Pier Paolo Amoroso 1 and Claudio Parente 2

1 International PhD Programme “Environment, Resources and Sustainable Development”,Department of Science and Technology, Parthenope University of Naples, Centro Direzionale,Isola C4, 80143 Naples, Italy; pierpaolo.amoroso@studenti.uniparthenope.it

2 DIST—Department of Science and Technology, Parthenope University of Naples, Centro Direzionale,Isola C4, 80143 Naples, Italy; claudio.parente@uniparthenope.it

* Correspondence: emanuele.alcaras@uniparthenope.it

Abstract: In relation to 3D bathymetric modelling, this article aims to analyze the performance ofKriging approaches in dependence of the location and density of the measured depth points. Theexperiments were carried out on a multi-beam sonar (MBS) dataset that includes 240,000 soundingscovering a sea-bottom area near Giglio Island (Italy). Seven subsets were derived in random wayfrom the initial regular MBS dataset, selecting an increasing number of points uniformly spaced.Seven models were generated for both Ordinary Kriging and Universal Kriging. Each model wassubmitted to leave-one-out cross-validation to define the exactness of the predictive values andcompared with the initial grid to better evaluate the accuracy in dependence of the point number anddissemination. To investigate this relationship, a new index called MVI (Morphological VariationIndex) was introduced as a measurement of the level of variation of seabed morphology. The resultsvalidate the efficiency of the Kriging methods and remark the influence of the dataset distributionon the 3D model, highlighting MVI as a useful index to represent the seabed variation as a uniquevalue. Finally, in no rugged areas using 1 point every 1000 m2, the RMSE of the differences betweenmeasured and interpolated values falls below 1 m, while a further increment of soundings is requiredin the presence of a high level of variation of seabed morphology.

Keywords: interpolation; topography; 3D model; DEM; multi-beam; point density; bathymetry; Kriging

1. Introduction

Nowadays, 3D modeling is applied in many fields, such as cultural heritage preser-vation [1], underground survey [2], coastal monitoring [3], indoor mapping [4], quarry’sstudies [5], and landfill monitoring [6].

Three-dimensional models include digital elevation models (DEMs) that are typicallyused to visualize the variation of the Earth’s surface [7] and produced through manyprocedures such as remote sensing, photogrammetry, and land surveying [8]. A continuousrepresentation of the examined part of the Earth surface is typically generated usingappropriate interpolators that can be applied to topographic data as well as to hydrographicdata [9].

Interpolation is founded on the principles of spatial autocorrelation, which assumesthat closer points are more similar compared to farther ones [10], in accordance with thefirst law of geography formulated by Tobler [11]. In other terms, we can define spatial in-terpolation as the procedure to predict the value of attributes at unobserved points within astudy area covered by existing observations [12]. Using points with known values (samplepoints), interpolation techniques allow for predicting unknown values for any geographic

Geosciences 2022, 12, 62. https://doi.org/10.3390/geosciences12020062 https://www.mdpi.com/journal/geosciences

Geosciences 2022, 12, 62 2 of 23

point data, such as elevation, rainfall, chemical concentrations, noise levels, etc. [13]. Typi-cally applied to grids with estimates made for all cells, interpolation methods generate asurface from the points [14]. They have been classified as global or local [15]. The differencebetween the two groups lies in the use of points with known values for assessing unknownvalues: a global method uses all the data available in the initial dataset while a local methoduses only a selection of them, typically the points closest to the single grid node in whichthe value must be calculated [16]. Interpolation methods are also classified as either exactor approximate methods because of the characteristic of preserving or not preserving theoriginal sample point values on the inferred surface [17]. Another distinction is madebetween deterministic and stochastic methods. The former use point values directly andcan be applied when there is sufficient knowledge about the geographical surface beingmodeled so as to describe its character as a mathematical function [18]. The latter utilizethe statistical properties of the measured points: they quantify the spatial autocorrelationamong sample points and consider the spatial configuration of the measured points aroundthe prediction location [19].

Some of the most used deterministic methods are: Inverse Distance Weighting (IDW),local polynomial functions (first, second, third, . . . , nth order), and radial basis func-tions [20]. Particularly, IDW estimates interpolated points considering their distance fromknown values: each measured point is weighted by the inverse of its distance from theinterpolation point [21]. Consequently, as the distance increases, the weight decreasesrapidly [22].

Polynomial interpolation fits a smooth surface defined by a mathematical function(a polynomial) to the input sample points [23]. The order of the polynomial ranges from first-order to higher-order: in every case, the values of the coefficients included in the polynomialequation must be determined using the measured values in the sample points [24]. Globalpolynomial functions use the entire dataset to define the equation of a single surfacecapable of representing the entire area by itself. Local polynomial functions use a localsubset defined by a window rather than all the measured points: the window is movedacross the analyzed region, positioned on each grid node, and the surface value at thecenter of this moving box is assessed. The window is so large that it contains a sufficientnumber of data points for defining the surface equation [25], e.g., no less than 10 points fora third-order polynomial function [26].

Radial basis functions are conceptually similar to fitting a rubber membrane throughthe measured sample points while minimizing the total curvature of the surface [27].They include different basis functions (e.g., completely regularized spline, spline withtension thin-plate spline, multiquadric function, inverse multiquadric function, etc.) anddepending on which you select, it will determine how the rubber membrane will fit betweenthe values [15]. Radial basis functions are exact interpolators, thus they require the surfaceto pass through the measured points; however, they can predict values above the maximumand below the minimum measured values.

Geostatistical methods include the concept of randomness, whereby the interpolatedsurface is hypothesized as one of many that might have been observed and all of whichcould generate the known data points [28]. Some of the most used stochastic methodsare Ordinary Kriging and Universal Kriging [29]. Particularly, since they were used forthe experiments carried out in our research, these methods are described in detail in thenext section.

Several interpolation algorithms are available in literature and a large part of them isalready implemented in GIS software, but the choice of the most suitable of them cannotbe made a priori since it requires them to be evaluated each time [26]. Their use concernsterrain as well as sea-bottom representation.

Three-dimensional bathymetric models, which are investigated in this article, arefundamental for many purposes: the spatial representation of the seabed morphologyallows for the development of studies such as those on navigation safety [30], coastal

Geosciences 2022, 12, 62 3 of 23

dynamics [31], sea ecosystems [32], seagrass meadows mapping [33], port dredging [34],infrastructure projects [35], etc.

Depth data that can be interpolated for 3D bathymetric model creation are usuallyacquired by the hydrographic authority through bathymetric survey and used for nauticalchart production. The acquisition of bathymetric data of the sea surrounding the Italianpeninsula and the construction of the relative nautical charts is the prerogative of the“Istituto Idrografico della Marina Militare” (I.I.M.M.), which is the hydrographic authorityin Italy [36].

Bathymetric surveys can be carried out using different types of instrumentation basedon acoustic signal transmission through the water; single-beam sonar (SBS) [37], multi-beam sonar (MBS) [38], and side-scan sonar (SSS) [39] are widely diffused techniquesfor hydrographic survey that supply datasets for seabed representation. Furthermore,bathymetric data can be extracted from multispectral satellite images (Satellite DerivedBathymetry, SDB) [40] or from the Electronic Navigational Chart (ENC) [41]. SBS, MBS,and SSS convert into a range the measurement of the time lag between transmitting andreceiving an acoustic signal that travels through the water, springs back the seabed, andreturns to the sounder [42,43]. SDB is carried out by processing the remotely sensed imagesusing specific techniques, such as the band ratio method [44], which takes into account therelationships between bands [45].

Single-beam data as well as depth data derived from a nautical chart are point clouddatasets that are submitted to spatial interpolation processes that allow for generating3D models [46]. Those irregularly spaced measured points are used as input to calculatethe depths in the nodes of appropriate grids that are spaced in relation to the accuracyof the input data [26,47]. In other terms, established by the extension of the study area,the accuracy of the generated model is proportional to the number of nodes that composethe grid covering this area: the greater the number of nodes, the better will be the outputmodel. Obviously, particular attention must be paid to choosing this number because it isof no use to create a 1000 × 1000 grid if the map only contains three sounding values [48].

The accuracy of the resulting 3D models is affected by the points numerosity anddistribution [49]. In literature, mathematical models are proposed to take into accountseveral independent variables, such as the surface slope, the density of points, the spatialdistribution of these points, and the interpolation method, in order to explain the accuracyof a given DEM [50]. Aguilar et al. (2002) [51] recognized three factors that influence the ac-curacy of a DEM: the morphology of the terrain, the density of the measured points, and theinterpolation method used. Several studies have been conducted on the influence of pointsdensity for DEMs generation and a large part of them considers laser scanner techniques,particularly LiDAR (Light Detection and Ranging) applications [52–54]. Regardless of theacquisition technique used to capture measured points, some studies focus on the relation-ship between the DEM accuracy and the sampling density. Aguilar et al. (2005) analyzedthis relationship and analytically adjusted it to a decreasing potential function [55]. Chaplotet al. (2005) evaluated the performance of interpolation techniques for the generation of theDEM of natural landscapes of differing morphologies and over a large range of scales [56]:five interpolation techniques (inverse-distance weighting, Ordinary Kriging, UniversalKriging, the multiquadratic radial basis function, and regularized spline with tension)were applied and their performance evaluated using data for both a mountainous areaof northern Laos and a gentle landscape of western France at nested spatial scales, withsampling densities from 4 to 109 points/km2.

This article aims to analyze the performance of Kriging approaches for bathymetricdata interpolation in dependence of the location and density of the sample points relatedto the morphological situations. Each algorithm available in GIS software to interpolateheight/depth values has its own advantages and disadvantages [57] but Kriging interpola-tors are included among the most performing ones [58], thus we preferred to use two ofthem in this study, namely Universal Kriging and Ordinary Kriging. Experiments werecarried out on MBS datasets covering a sea-bottom area near Giglio Island (Italy) and

Geosciences 2022, 12, 62 4 of 23

results analyzed in reference to the sounding density. A new index named the Morpho-logical Variation Index (MVI) is proposed to numerically express the level of variationof the seabed morphology to establish the useful number of points to be interpolated independence of the bottom conformation.

2. Study Area and Datasets

The experiments are carried out on a MBD that includes 240,000 measured points(Figure 1) and covers an area of the sea bottom near the coasts of the Giglio Island (Italy) inthe Tuscan archipelago (Figure 2), with an extension of 240,000 m2 (one point per squaremeter). The original dataset counts a total of 825,602 points and is provided by I.I.M.M.,who also carried out the bathymetric survey in 2012.

Geosciences 2022, 12, x FOR PEER REVIEW 4 of 24

height/depth values has its own advantages and disadvantages [57] but Kriging interpo-lators are included among the most performing ones [58], thus we preferred to use two of them in this study, namely Universal Kriging and Ordinary Kriging. Experiments were carried out on MBS datasets covering a sea-bottom area near Giglio Island (Italy) and re-sults analyzed in reference to the sounding density. A new index named the Morpholog-ical Variation Index (MVI) is proposed to numerically express the level of variation of the seabed morphology to establish the useful number of points to be interpolated in depend-ence of the bottom conformation.

2. Study Area and Datasets The experiments are carried out on a MBD that includes 240,000 measured points

(Figure 1) and covers an area of the sea bottom near the coasts of the Giglio Island (Italy) in the Tuscan archipelago (Figure 2), with an extension of 240,000 m2 (one point per square meter). The original dataset counts a total of 825,602 points and is provided by I.I.M.M., who also carried out the bathymetric survey in 2012.

Figure 1. Three-dimensional representation of the seafloor obtained from the entire multi-beam da-taset (MBD): the upper image represents the point cloud of the initial MBD; the lower image repre-sents the continuous 3D model of the seafloor.

Figure 1. Three-dimensional representation of the seafloor obtained from the entire multi-beamdataset (MBD): the upper image represents the point cloud of the initial MBD; the lower imagerepresents the continuous 3D model of the seafloor.

The study area was chosen because it is of great interest; in fact, several hydrographiccampaigns have been carried out on it over time. Furthermore, this area is characterized bya high level of variability of the seabed; in fact, it has both particularly steep and jaggedareas, and some almost flat areas [59]. The sea bottom, beneath 50 m of depth, consists ofmore than 60% clay [60].

The area extends 400 m × 600 m within the following UTM/WGS84 plane coordinates−32T zone: E1 = 658,640 m, E2 = 659,240 m, N1 = 4,690,639 m, and N2 = 4,691,039 m.

MBS data were organized as grid points containing X, Y, and Z values. Depth valuesrange between −5.45 m and −108 m in the selected area, with a precision of 1 cm ondepths. Different subsets were derived in random way from the initial MBS, selectingan increasing number of points (24, 48, 120, 240, 480, 1200, and 2400). Particularly, thearea was subdivided in six sectors extending 200 m × 200 m (Figure 3) and the selectionwas carried out to ensure an equal number of points included in each sector. As shown

Geosciences 2022, 12, 62 5 of 23

in Sections 3 and 4, this data organization permits us to identify different morphologicalsituations so that each sector can represent a local typical variability of the seabed.

Geosciences 2022, 12, x FOR PEER REVIEW 5 of 24

Figure 2. The study area: on the left, localization of Giglio Island into Tyrrhenian Sea, in equirectan-gular projection and WGS84 geographic coordinates (EPSG: 4326); on the right, Giglio Island visu-alization in RGB composition of Sentinel-2B images in UTM/WGS 84 plane coordinates expressed in meters (EPSG: 32632), with the study area highlighted in the red rectangle.

The study area was chosen because it is of great interest; in fact, several hydrographic campaigns have been carried out on it over time. Furthermore, this area is characterized by a high level of variability of the seabed; in fact, it has both particularly steep and jagged areas, and some almost flat areas [59]. The sea bottom, beneath 50 m of depth, consists of more than 60% clay [60].

The area extends 400 m x 600 m within the following UTM/WGS84 plane coordinates −32T zone: E1 = 658,640 m, E2 = 659,240 m, N1 = 4,690,639 m, and N2 = 4,691,039 m.

MBS data were organized as grid points containing X, Y, and Z values. Depth values range between −5.45 m and −108 m in the selected area, with a precision of 1 cm on depths. Different subsets were derived in random way from the initial MBS, selecting an increas-ing number of points (24, 48, 120, 240, 480, 1200, and 2400). Particularly, the area was sub-divided in six sectors extending 200 m × 200 m (Figure 3) and the selection was carried out to ensure an equal number of points included in each sector. As shown in Sections 3 and 4, this data organization permits us to identify different morphological situations so that each sector can represent a local typical variability of the seabed.

Figure 2. The study area: on the left, localization of Giglio Island into Tyrrhenian Sea, in equirect-angular projection and WGS84 geographic coordinates (EPSG: 4326); on the right, Giglio Islandvisualization in RGB composition of Sentinel-2B images in UTM/WGS 84 plane coordinates ex-pressed in meters (EPSG: 32632), with the study area highlighted in the red rectangle.

Geosciences 2022, 12, x FOR PEER REVIEW 6 of 24

Figure 3. The study area with 2400 chosen points (in red) in UTM/WSG 84 plane coordinates ex-pressed in meters (EPSG: 32632).

3. Methods Each of the seven subsets described above was submitted to Kriging interpolation to

obtain 3D bathymetric models of the study area. Using the extension of ArcGIS [61], named Geostatistical Analyst [62], seven models were generated for both Ordinary and Universal Kriging. Each model was submitted to leave-one-out cross-validation [63] to define the exactness of predictive values; in addition, each product was compared with the initial grid to better evaluate the accuracy of the model in dependence of the point number and dissemination. The experiments were carried out to investigate the efficiency of the Kriging methods and analyze the impact of both the dataset location and density on the 3D model exactness. Particularly, we explored the influence of the variability of the seabed morphology on the measured point number and distribution in order to achieve accurate results.

In the following sections, the proposed approach to produce 3D bathymetric models and to investigate their accuracy in dependence of sample point location and dissemina-tion is described in detail. First of all, main characteristics of Kriging interpolation (Section 3.1) are resumed; then, the steps for model construction and global accuracy evaluation are illustrated (Section 3.2.1); and finally, the proposed index for analyzing the morpho-logical variation of the seabed (MVI) based on the inversions of the slope and the extent of the differences in depth between adjacent pixels is illustrated, remarking its rule for evaluating the local accuracy of the 3D bathymetric model produced (Section 3.2.2).

3.1. Kriging Interpolation The Kriging interpolation method is based on the application of a geostatistical

model which takes into account the spatial correlation between the sampled points and adapts it to estimate the value at an unsampled point [64].

The fundamental principle of Kriging is to assume that the spatial variation of any attribute is too irregular to be modeled by a simple mathematical function [65]. This irreg-ular variation can be modelled, often achieving better results, by applying a stochastic surface [29]. Indeed, Kriging is one of the stochastic interpolation methods and for this reason, it is based on the principle according to which the link between neighboring points

Figure 3. The study area with 2400 chosen points (in red) in UTM/WSG 84 plane coordinatesexpressed in meters (EPSG: 32632).

Geosciences 2022, 12, 62 6 of 23

3. Methods

Each of the seven subsets described above was submitted to Kriging interpolationto obtain 3D bathymetric models of the study area. Using the extension of ArcGIS [61],named Geostatistical Analyst [62], seven models were generated for both Ordinary andUniversal Kriging. Each model was submitted to leave-one-out cross-validation [63] todefine the exactness of predictive values; in addition, each product was compared withthe initial grid to better evaluate the accuracy of the model in dependence of the pointnumber and dissemination. The experiments were carried out to investigate the efficiencyof the Kriging methods and analyze the impact of both the dataset location and densityon the 3D model exactness. Particularly, we explored the influence of the variability of theseabed morphology on the measured point number and distribution in order to achieveaccurate results.

In the following sections, the proposed approach to produce 3D bathymetric modelsand to investigate their accuracy in dependence of sample point location and disseminationis described in detail. First of all, main characteristics of Kriging interpolation (Section 3.1)are resumed; then, the steps for model construction and global accuracy evaluation areillustrated (Section 3.2.1); and finally, the proposed index for analyzing the morphologicalvariation of the seabed (MVI) based on the inversions of the slope and the extent of thedifferences in depth between adjacent pixels is illustrated, remarking its rule for evaluatingthe local accuracy of the 3D bathymetric model produced (Section 3.2.2).

3.1. Kriging Interpolation

The Kriging interpolation method is based on the application of a geostatistical modelwhich takes into account the spatial correlation between the sampled points and adapts itto estimate the value at an unsampled point [64].

The fundamental principle of Kriging is to assume that the spatial variation of anyattribute is too irregular to be modeled by a simple mathematical function [65]. Thisirregular variation can be modelled, often achieving better results, by applying a stochasticsurface [29]. Indeed, Kriging is one of the stochastic interpolation methods and for thisreason, it is based on the principle according to which the link between neighboring pointscan be expressed by a statistical correlation which does not necessarily express physicalmeaning [66]. For Kriging, this turns out to be a very efficient technique and one of themost used in many fields of application, such as: data reconstruction [67], rainfall intensityestimation [68], temperature spatial modelling [69], soil composition variability [70], andDEM generation [71].

In order to evaluate the variability of the attributes of the points as the distanceincreases, Kriging makes use of the semi-variogram [72]. The semi-variogram is a diagramin which the semi-variance is represented as a function of the distance between twopoints [73]. The semi-variance function is given by the following formula [74]:

γ(h) =1

2n

n

∑i=1

(z(xi)− z(xi + h))2 (1)

where γ(h) is the value of the semi-variance at the distance d; n is the number of couples ofpoints separated by d; z is the value of the depth; and xi and xi + h indicate the positions ofeach couple of points.

There are several different Kriging techniques and all are developed as variations ofsimple Kriging [75]. Among these, we decided to apply Ordinary Kriging and UniversalKriging approaches.

3.1.1. Ordinary Kriging

Ordinary Kriging is one of the most commonly used Kriging techniques. It is definedas the best linear unbiased estimator (BLUE): best because it aims at minimizing thevariance of the errors, linear because its estimates are weighted linear combinations of

Geosciences 2022, 12, 62 7 of 23

the available data, and unbiased since it tries to have the mean residual or error equal tozero [76].

Ordinary Kriging serves to estimate a value at a point of a region for which a variogramis known using data in the neighborhood of the estimation location [77]. In other words,its aim is to predict the value of the random variable z(x0) at an unsampled point x0 of ageographical region [78].

The function z(xi) is composed of a deterministic component µ and a random func-tion e(xi) [79]

z(xi) = µ + ε(xi) (2)

The deterministic component is a constant value for each xi location in each regionof study.

Ordinary Kriging is based on two assumptions:

(1) The global, constant mean µ ∈ R of the random function Z(x) is unknown; and(2) The data derive from an intrinsically stationary random function Z(x) with known

variogram function γ(h).

Thus, the equation that defines the model assumed by Ordinary Kriging is:

z(x0) =n

∑i=1

wiz(xi) (3)

where wi provides the unknown weights corresponding with the influence of the variablez(xi) in the computation of z(x0).

The value of each output pixel is calculated through different steps [80].Firstly, the algorithm determines the input points for each pixel output, calculates the

distances between all of them, and find the semi-variogram value for those distances. Thesemi-variogram values are filed out in the following matrix:

0 γ(h12) γ(h13)γ(h21) 0 γ(h23)γ(h31) γ(h32) 0

γ(h14) · · · γ(h1n)γ(h24) · · · γ(h2n)γ(h34) · · · γ(h3n)

γ(h41) γ(h42) γ(h43)· · · · · · · · ·

γ(hn1) γ(hn2) γ(hn3)

0 · · · γ(h4n)· · · · · · · · ·

γ(hn4) · · · 0

(4)

where

n is the number of the input point;i = 1, 2, . . . . n;j = 1, 2, . . . n;hij(h11, h12, h13, . . . , h21, h22, h23, . . .) is the distance between input point i and inputpoint k; andγ(hij) ((γ(h11), (γ(h12), (γ(h13), . . . , (γ(h21), (γ(h22), (γ(h23), . . .) is the value of thesemi-variogram for the distance between input point i and input point k.

The distances of the output pixel towards all input points are calculated and the semi-variance for these distances extracted from the semi-variogram are filled out in vector D:

γ(hp1)

γ(hp2)

γ(hp3)

γ(hp4)

· · ·γ(hpn)

(5)

where

hpi(hp1, hp2, hp3, . . . , hpn

)is the distance between the output pixel p and input point i and

Geosciences 2022, 12, 62 8 of 23

γ(hpn)((

γ(hp1),(γ(hp2

),(γ(hp3

), . . . ,

(γ(hpn

))is the value of the semi-variogram for

distance hpi between the output pixel p and input point i.

The following matrix equation is solved to calculate the weight factors:

0 γ(h12) γ(h13) γ(h14) · · · γ(h1n) 1γ(h21) 0 γ(h23) γ(h24) · · · γ(h2n) 1γ(h31) γ(h32) 0 γ(h34) · · · γ(h3n) 1γ(h41) γ(h42) γ(h43) 0 · · · γ(h4n) 1· · · · · · · · · · · · · · · · · · 1

γ(hn1) γ(hn2) γ(hn3) γ(hn4) · · · 0 11 1 1 1 1 1 0

·

w1w2w3w4· · ·wnλ

=

γ(hp1)

γ(hp2)

γ(hp3)

γ(hp4)

· · ·γ(hpn)

1

(6)

where wi is a weight factor for input point i andλ is a Lagrange multiplier introduced to minimize possible estimation error.This matrix equation can be written as a set of n + 1 simultaneous equations:

∑i(wi·γ

(hij)) (7)

∑i

wi = 1 (8)

Then, the algorithm calculates the estimated or predicted value Z of the output pixelas the sum of the products of the weight factors and the input point values using theEquation (2).

3.1.2. Universal Kriging

While Ordinary Kriging assumes that the mean µ is constant across the entire region ofstudy, this method assumes that the mean µ(x) has a functional dependence on the spatiallocation and can be approximated by a model with the equation [81]:

µ(x) =k

∑l=1

al fl(x) (9)

where

al is the coefficient to be estimated from the data (generally unknown) and characterizesthe local trend (drift coefficient);fl is the lth basic function (generally polynomials) of spatial coordinates that describes thedrift; andk is the number of functions used in modelling the drift.

For this method, Equation (2) can be rewritten as follow [82]:

z(xi) = µ(xi) + ε(xi) (10)

The local trend or drift can be expressed by a polynomial; in the case of a quadraticexpression, Equation (9) can be rewritten as:

µ(x) = a0 + a1xi + a2yi + a3x2i + a4xiyi + a5y2

i (11)

where

xi , yi are the x, y coordinates of the input points anda0, a1, . . . , a5 are the unknown drift coefficients.

In the case of linear expression, Equation (11) can be rewritten as:

µ(x) = a0 + a1xi + a2yi (12)

Geosciences 2022, 12, 62 9 of 23

As for the Ordinary Kriging and also in this case, the weights must be calculated.Considering 5 input points and a local linear trend, the set of equations in matrix form is:

0 γ(h12) γ(h13) γ(h14) γ(h15) 1 x1 y1γ(h21) 0 γ(h23) γ(h24) γ(h25) 1 x2 y2γ(h31) γ(h32) 0 γ(h34) γ(h35) 1 x3 y3γ(h41) γ(h42) γ(h43) 0 γ(h45) 1 x4 y4γ(h51) γ(h52) γ(h53) γ(h54) 0 1 x5 y5

1 1 1 1 1 0 0 0x1 x2 x3 x4 x5 0 0 0y1 y2 y3 y4 y5 0 0 0

·

w1w2w3w4w5λa1a2

=

γ(hp1)γ(hp2)γ(hp3)γ(hp4)γ(hp5)

1xpyp

(13)

where

n is the number of the input point;i = 1, 2, . . . n;j = 1, 2, . . . n;hij(h11, h12, h13, . . . , h21, h22, h23, . . .) is the distance between input point i and inputpoint k;(

hp1, hp2, hp3, . . . , hpn)

is the distance between the output pixel p and input point i, and isthe value of the semi-variogram for the distance between input point i and input point k;γ(hpn)

((γ(hp1

),(γ(hp2

),(γ(hp3

), . . . ,

(γ(hpn

))is the value of the semi-variogram for

distance hpi between the output pixel p and input point i;xi , yi are the x, y coordinates of the input points;a0, a1, . . . , a5 are the unknown drift coefficients; andxp , yp are the x, y coordinates of the output pixel.

This matrix form must be solved for each output point in the same way as describedfor Ordinary Kriging. The weights are crucial to calculate an estimated or predicted valuein each pixel of the output map using Equation (2).

3.2. Adopted Methodological Approach3.2.1. Model Construction and Global Accuracy Evaluation

For this work, 7 models were generated using Ordinary Kriging and 7 models usingUniversal Kriging, namely one for each selected subset of points. All operations were car-ried out in the GIS environment: Geostatistical Analyst, the extension of ArcGIS that addsadvanced tools for exploratory spatial data analysis and the interpolation process aimedat creating a statistically valid surface [83], allowed us to generate optimal models fromsample data and evaluate predictions for better decision making. Geostatistical Analystpermitted us to apply Ordinary Kriging as well as Universal Kriging; in addition, it allowedus to choose the most performing semi-variogram model that results in both interpolators’stable model. The grid cell was fixed at 1 m taking into account the characteristics of theinitial dataset. However, in addition to the morphology of the seabed, the number ofpoints necessary for the 3D modeling must take into account the resolution of the finalmodel [55]; consequently, if the number of points per km2 is limited, it is necessary to fix avery large cell size. Nevertheless, in our study, the attention was focused on the influenceof the morphology of the seabed on the determination of the number of points needed,therefore the grid cell was set at 1 m in every case and the number of points used formodeling varied from time to time. The choice of the pixel dimension equal to 1 m allowsfor the direct comparison of the constructed models with the starting grid obtained fromthe multibeam survey.

To assess the accuracy of the results, two approaches were carried out:

- Cross validation (CV); and- Direct comparison with the MBD.

Geosciences 2022, 12, 62 10 of 23

In the first case, cross-validation permits the defining of the accuracy level of predictivevalues [84]. In other words, it is a statistical validation technique used to assess how wellan interpolation model performs [85]. Different methods are available in literature for CVapplication [86–90]. Particularly, the leave-one-out (LOO) method is adopted, removingeach point in turn from the dataset and using the other points to estimate a value at thelocation of the removed point [91].

In the second case, the original model obtained with all the points of the initial dataset(MBD) was compared with the seven models obtained both for the Ordinary Kriging andUniversal Kriging depending on the number of points taken into consideration. In otherwords, the difference between the original one and each of the seven obtained models wasperformed using a raster calculator, which is the powerful ArcGIS tool that allows you tocreate and execute an expression of the map algebra [92] that will output a raster.

The performance of both approaches is evaluated using statistical terms such as mean,standard deviation, minimum, maximum, and Root Mean Square Error (RMSE). RMSE isconsidered the most widely used global accuracy measure for evaluating the performanceof DEMs and is calculated in accordance with the formula [55,93,94]:

RMSE =

√∑N

i=1(Zi(x, y)− Z′i(x, y)

)2

N(14)

where

N is the number of the depth points (that, in this case, coincides with the number of thegrid cells);Zi (x,y) is the measured depth at the location i(x, y); andZ′i(x, y) is the interpolated depth at the same location i(x, y).

3.2.2. Morphological Variability and Local Accuracy Evaluation

The impact of the density of the points on the modeling of the seabed differs accordingto the variation of the seabed: the higher the variation, the higher is the number of pointsnecessary for modelling.

The study area chosen for this work presents seabed with variegated slopes, as shownin Figure 4.

The proposed organization of the initial data into 6 subsets, each covering an area(sector) of 200 m × 200 m, is representative of the seafloor configuration since each of themremarks local trends and characterization of the morphology. The identification of thesesectors with the letters A, B, C, D, E, and F is shown in Figure 5.

Each sector has different seafloor patterns; for example, sector B has large variationsin depth while sector D has less abrupt and more linear variations in depth.

Subsequently, the analysis of the residuals between the initial dataset and the individ-ual models was conducted in relation to each of the six sectors into which the area has beendivided so as to highlight the variability of the results also in relation to the specificity ofthe morphology of the seabed.

In literature, different morphometric variables are described for DEM analysis andtheir use is also possible for Digital Depth Model (DDM) analysis. Lozano Parra et al.(2009) [95] considered three parameters to classify landforms: slope gradient (DigitalSlope Model, DSM), curvature (Digital Curvature Model, DCM), and roughness (DigitalRoughness Model, DRM). DSM represents the slope angle and is usually calculated as themaximum rate of change between the altitude of each cell and its neighbors, expressed indegrees; basically, the maximum change in elevation over the distance between the cell andits eight neighbors identifies the steepest downhill descent from the cell [96]. The DCMrepresents the concavity or convexity of a surface and can be calculated using a fourth-degree polynomial function fitted to the surface [97]. The DRM not only expresses the rangeof elevation in the vertical direction (z) but also reflects its variation and irregularity [98]; itcan be defined as the variation rate of the slope in a specific area and no standard method

Geosciences 2022, 12, 62 11 of 23

exists to calculate it [95]. The terrain roughness index (TRI) proposed in [99] calculatesthe quantitative measurement of terrain heterogeneity for every location by summarizingthe change in elevation within the 3 × 3 pixel grid: each pixel contains the difference inelevation from a center cell and the eight cells surrounding it.

Geosciences 2022, 12, x FOR PEER REVIEW 11 of 24

3.2.2. Morphological Variability and Local Accuracy Evaluation The impact of the density of the points on the modeling of the seabed differs accord-

ing to the variation of the seabed: the higher the variation, the higher is the number of points necessary for modelling.

The study area chosen for this work presents seabed with variegated slopes, as shown in Figure 4.

Figure 4. Slopes chart of the initial MBD in UTM/WSG 84 plane coordinates expressed in meters (EPSG: 32632).

The proposed organization of the initial data into 6 subsets, each covering an area (sector) of 200 m × 200 m, is representative of the seafloor configuration since each of them remarks local trends and characterization of the morphology. The identification of these sectors with the letters A, B, C, D, E, and F is shown in Figure 5.

Figure 4. Slopes chart of the initial MBD in UTM/WSG 84 plane coordinates expressed in meters(EPSG: 32632).

Geosciences 2022, 12, x FOR PEER REVIEW 12 of 24

Figure 5. Division of the study area into 6 sectors (A–F), each of 200 m on each side.

Each sector has different seafloor patterns; for example, sector B has large variations in depth while sector D has less abrupt and more linear variations in depth.

Subsequently, the analysis of the residuals between the initial dataset and the indi-vidual models was conducted in relation to each of the six sectors into which the area has been divided so as to highlight the variability of the results also in relation to the specific-ity of the morphology of the seabed.

In literature, different morphometric variables are described for DEM analysis and their use is also possible for Digital Depth Model (DDM) analysis. Lozano Parra et al. (2009) [95] considered three parameters to classify landforms: slope gradient (Digital Slope Model, DSM), curvature (Digital Curvature Model, DCM), and roughness (Digital Roughness Model, DRM). DSM represents the slope angle and is usually calculated as the maximum rate of change between the altitude of each cell and its neighbors, expressed in degrees; basically, the maximum change in elevation over the distance between the cell and its eight neighbors identifies the steepest downhill descent from the cell [96]. The DCM represents the concavity or convexity of a surface and can be calculated using a fourth-degree polynomial function fitted to the surface [97]. The DRM not only expresses the range of elevation in the vertical direction (z) but also reflects its variation and irregu-larity [98]; it can be defined as the variation rate of the slope in a specific area and no standard method exists to calculate it [95]. The terrain roughness index (TRI) proposed in [99] calculates the quantitative measurement of terrain heterogeneity for every location by summarizing the change in elevation within the 3 × 3 pixel grid: each pixel contains the difference in elevation from a center cell and the eight cells surrounding it.

In order to analytically express the heterogeneity of the analyzed surface, a new in-dex is introduced in this work, named the Morphological Variation Index (MVI). This in-dex takes into account two components: the first component (Ixy) is related to the variation of the direction of the slope (ascending or descending) while the second component (Sxy) takes into account the variation of the value of the slope. Both these components are con-sidered in relation to x and y directions.

MVI is given by the following formula: 𝑀𝑉𝐼 = 𝐼 ∙ 𝑆 (15)

Figure 5. Division of the study area into 6 sectors (A–F), each of 200 m on each side.

Geosciences 2022, 12, 62 12 of 23

In order to analytically express the heterogeneity of the analyzed surface, a new indexis introduced in this work, named the Morphological Variation Index (MVI). This indextakes into account two components: the first component (Ixy) is related to the variation ofthe direction of the slope (ascending or descending) while the second component (Sxy) takesinto account the variation of the value of the slope. Both these components are consideredin relation to x and y directions.

MVI is given by the following formula:

MVI = Ixy·Sxy (15)

Regarding the component Ixy, the difference between the consecutive pixel values isconsidered along the x direction (i.e., along each row) and along the y direction (i.e., alongeach column); if this difference always has the same sign, the direction does not change,i.e., the seabed continues to rise or fall. By assigning the value 0 to the conservation of thedirection and 1 to its variation, the sum of all the values divided by the total number of thepixels of the grid gives a percentage value that expresses the variability of the seabed interms of the frequency of the ascend–descent inversion. Since two values are calculated(one for the x direction and the other for the y direction) for each pixel, Ixy is the mean ofthem, referring to all the grid cells.

Regarding the component Sxy, the difference between the consecutive pixel values(depth values) related to the pixel dimension is considered along the x direction as well asthe y direction to calculate slope. Then, the variation of slope is determined and normalizedby dividing each value by the maximum (theoretical) value, which is 180◦. Additionally,in this case, two values were calculated (one for the x direction and the other for they direction) for each pixel, thus Sxy is the mean of them, referring to all the grid cells.

Finally, once the interpolator is fixed (Ordinary Kriging or Universal Kriging), sixvalues of Ixy, six values of Sxy, and, consequently, six values of MVI are calculated, namelyone for each sector.

4. Results and Discussion

The statistic values of all the residuals for each dataset are shown in Table 1 forOrdinary Kriging applications analyzed by cross-validation and in Table 2 for the sameapplications analyzed by direct comparison. Similarly, statistic values of all the residualsfor each dataset are shown in Table 3 for Universal Kriging applications analyzed bycross-validation and in Table 4 for the same applications analyzed by direct comparison.

By examining the RMSE values, the results confirm the efficiency of both consideredKriging methods, remarking the augmentation of the accuracy level related to the incrementof the number of measured points in the same area. Ordinary Kriging applications produceRMSE values that rapidly decrease from the first to the seventh subset (from 4.489 m to0.503 m using cross-validation and from 4.053 m to 0.537 m using direct comparison).Universal Kriging models present similar trends of the residuals related to the measuredpoint density, even if the results are slightly less accurate as testified by the RMSE values(from 5.105 m to 0.628 m using cross-validation and from 4.863 m to 0.647 m using directcomparison).

Table 1. Statistical terms of the residuals supplied by cross-validation for the Ordinary Kriging.

Count Min (m) Max (m) Mean (m) St. Dev. (m) RMSE (m)

24 −13.495 11.618 −0.303 4.479 4.48948 −12.587 8.487 0.139 3.014 3.017

120 −9.895 5.095 −0.134 1.892 1.897240 −9.062 11.159 −0.042 1.331 1.332480 −8.810 8.211 0.021 1.130 1.130

1200 −7.366 5.305 0.002 0.714 0.7142400 −5.945 4.760 −0.002 0.503 0.503

Geosciences 2022, 12, 62 13 of 23

Table 2. Statistical terms of the residuals between each Ordinary Kriging model and initial dataset values.

Count Min (m) Max (m) Mean (m) St. Dev. (m) RMSE (m)

24 −30.799 6.526 −0.213 4.047 4.05348 −17.098 11.578 −0.180 2.052 2.060

120 −12.337 21.097 0.056 2.020 2.020240 −10.694 10.399 −0.110 1.344 1.349480 −11.457 7.816 −0.059 1.087 1.089

1200 −8.067 7.703 0.007 0.663 0.6632400 −14.037 7.720 −0.008 0.537 0.537

Table 3. Statistical terms of the residuals supplied by cross-validation for the Universal Kriging.

Count Min (m) Max (m) Mean (m) St. Dev. (m) RMSE (m)

24 −14.745 12.489 −0.233 5.100 5.10548 −16.002 8.574 −0.297 3.297 3.310

120 −15.212 10.861 −0.017 2.168 2.168240 −11.032 11.916 −0.013 1.581 1.581480 −11.188 7.590 −0.041 1.211 1.212

1200 −6.799 4.830 0.006 0.752 0.7522400 −8.307 5.835 −0.003 0.628 0.628

Table 4. Statistical terms of the residuals between each Universal Kriging model and initial dataset values.

Count Min (m) Max (m) Mean (m) St. Dev. (m) RMSE (m)

24 −32.563 10.076 −0.340 4.851 4.86348 −24.201 21.304 −0.321 3.459 3.474

120 −15.360 31.617 0.098 2.319 2.321240 −12.154 17.509 −0.108 1.566 1.570480 −12.320 17.700 −0.006 1.180 1.180

1200 −15.795 8.532 0.013 0.795 0.7952400 −17.151 8.575 −0.016 0.646 0.647

By analyzing the other statistics of the residuals, it is evident that the direct comparisonproduces very high maximum and minimum values. This effect is due to the particularmorphology of the considered area that is extremely rugged in one of the six sectors (B),with a rapid alternation of ups and downs as well as abrupt changes in slopes.

For this reason, a statistical analysis was carried out for each sector through directcomparison with MBD. In particular, Figures 6 and 7 show the trend of the RMSE as afunction of the number of points obtained in each sector for Ordinary Kriging and UniversalKriging, respectively.

In both cases, the data clearly show that the highest values of RMSE are obtainedin sector B while the lowest in sector D: for this reason, we report in Tables 5 and 6 thestatistical values relating to these two sectors with 2400 points used.

In analyzing the RMSE values as the number of points varies, it is noted that as thepoints increase, the RMSE generally decreases, except for some special cases, due to theparticular conformation of the seabed and by the totally random distribution of the points.Ordinary Kriging provides slightly better results than Universal Kriging in all sectors.Regardless of the adopted method, it is possible to notice how the RMSE values drop below1 m when already using 1200 points, except in sector B. By doubling the number of points,the RMSE values in the second sector still remain above 1 m, even if, in the case of theOrdinary Kriging, the RMSE is only 61 mm above the meter (1.061 m).

Geosciences 2022, 12, 62 14 of 23

Geosciences 2022, 12, x FOR PEER REVIEW 14 of 24

2400 −14.037 7.720 −0.008 0.537 0.537

Table 3. Statistical terms of the residuals supplied by cross-validation for the Universal Kriging.

Count Min (m) Max (m) Mean (m) St. Dev. (m) RMSE (m) 24 −14.745 12.489 −0.233 5.100 5.105 48 −16.002 8.574 −0.297 3.297 3.310 120 −15.212 10.861 −0.017 2.168 2.168 240 −11.032 11.916 −0.013 1.581 1.581 480 −11.188 7.590 −0.041 1.211 1.212

1200 −6.799 4.830 0.006 0.752 0.752 2400 −8.307 5.835 −0.003 0.628 0.628

Table 4. Statistical terms of the residuals between each Universal Kriging model and initial dataset values.

Count Min (m) Max (m) Mean (m) St. Dev. (m) RMSE (m) 24 −32.563 10.076 −0.340 4.851 4.863 48 −24.201 21.304 −0.321 3.459 3.474 120 −15.360 31.617 0.098 2.319 2.321 240 −12.154 17.509 −0.108 1.566 1.570 480 −12.320 17.700 −0.006 1.180 1.180

1200 −15.795 8.532 0.013 0.795 0.795 2400 −17.151 8.575 −0.016 0.646 0.647

By analyzing the other statistics of the residuals, it is evident that the direct comparison produces very high maximum and minimum values. This effect is due to the particular morphology of the considered area that is extremely rugged in one of the six sectors (B), with a rapid alternation of ups and downs as well as abrupt changes in slopes.

For this reason, a statistical analysis was carried out for each sector through direct comparison with MBD. In particular, Figures 6 and 7 show the trend of the RMSE as a function of the number of points obtained in each sector for Ordinary Kriging and Universal Kriging, respectively.

Figure 6. Direct comparison of Ordinary Kriging models with MBD: the variation of RMSE related to the number of points analyzed in each sector. Figure 6. Direct comparison of Ordinary Kriging models with MBD: the variation of RMSE related tothe number of points analyzed in each sector.

Geosciences 2022, 12, x FOR PEER REVIEW 15 of 24

Figure 7. Direct comparison of Universal Kriging models with MBD: the variation of RMSE related to the number of points analyzed in each sector.

In both cases, the data clearly show that the highest values of RMSE are obtained in sector B while the lowest in sector D: for this reason, we report in Tables 5 and 6 the sta-tistical values relating to these two sectors with 2400 points used.

Table 5. Statistical terms of the residuals between the Ordinary Kriging model with 2400 total points and initial dataset values for B and D sectors.

Sector Min (m) Max (m) Mean (m) St. Dev. (m) RMSE (m) B −14.037 7.720 −0.015 1.061 1.061 D −0.405 0.352 −0.003 0.067 0.067

Table 6. Statistical terms of the residuals between the Universal Kriging model with 2400 total points and initial dataset values for B and D sectors.

Sector Min (m) Max (m) Mean (m) St. Dev. (m) RMSE (m) B −14.469 7.499 −0.046 1.214 1.214 D −0.486 0.353 −0.001 0.064 0.064

In analyzing the RMSE values as the number of points varies, it is noted that as the points increase, the RMSE generally decreases, except for some special cases, due to the particular conformation of the seabed and by the totally random distribution of the points. Ordinary Kriging provides slightly better results than Universal Kriging in all sectors. Re-gardless of the adopted method, it is possible to notice how the RMSE values drop below 1 m when already using 1200 points, except in sector B. By doubling the number of points, the RMSE values in the second sector still remain above 1 m, even if, in the case of the Ordinary Kriging, the RMSE is only 61 mm above the meter (1.061 m).

The statistical values obtained for sector B are always the most extreme: it has the highest maximum, standard deviation, and RMSE values, as well as the lowest minimum value. This is due to the particular conformation of this sector, which can be seen in detail in Figure 8. Nevertheless, the higher variability of sector B is remarked also by the MVI; starting from MBD, this index is calculated for all sectors to investigate the level of accu-racy of the Kriging models related to the morphological aspects.

Figure 7. Direct comparison of Universal Kriging models with MBD: the variation of RMSE relatedto the number of points analyzed in each sector.

Table 5. Statistical terms of the residuals between the Ordinary Kriging model with 2400 total pointsand initial dataset values for B and D sectors.

Sector Min (m) Max (m) Mean (m) St. Dev. (m) RMSE (m)

B −14.037 7.720 −0.015 1.061 1.061D −0.405 0.352 −0.003 0.067 0.067

Table 6. Statistical terms of the residuals between the Universal Kriging model with 2400 total pointsand initial dataset values for B and D sectors.

Sector Min (m) Max (m) Mean (m) St. Dev. (m) RMSE (m)

B −14.469 7.499 −0.046 1.214 1.214D −0.486 0.353 −0.001 0.064 0.064

Geosciences 2022, 12, 62 15 of 23

The statistical values obtained for sector B are always the most extreme: it has thehighest maximum, standard deviation, and RMSE values, as well as the lowest minimumvalue. This is due to the particular conformation of this sector, which can be seen in detailin Figure 8. Nevertheless, the higher variability of sector B is remarked also by the MVI;starting from MBD, this index is calculated for all sectors to investigate the level of accuracyof the Kriging models related to the morphological aspects.

Geosciences 2022, 12, x FOR PEER REVIEW 16 of 24

Figure 8. Three-dimensional representation of the seafloor in sector B. The detail permits us to ap-preciate the high variability of the seafloor.

The calculation of the MVI was carried out in successive steps. In particular, the first step consisted of calculating the differences in the depth values of the pixels along x as well as along y, obtaining two images. The image related to the y direction is shown in Figure 9.

Figure 9. Image of the differences of MBD values along y direction.

For the sake of completeness, the statistical values of the depth differences between successive pixels are reported in Table 7 (along x) and Table 8 (along y).

Figure 8. Three-dimensional representation of the seafloor in sector B. The detail permits us toappreciate the high variability of the seafloor.

The calculation of the MVI was carried out in successive steps. In particular, the firststep consisted of calculating the differences in the depth values of the pixels along x as wellas along y, obtaining two images. The image related to the y direction is shown in Figure 9.

Geosciences 2022, 12, x FOR PEER REVIEW 16 of 24

Figure 8. Three-dimensional representation of the seafloor in sector B. The detail permits us to ap-preciate the high variability of the seafloor.

The calculation of the MVI was carried out in successive steps. In particular, the first step consisted of calculating the differences in the depth values of the pixels along x as well as along y, obtaining two images. The image related to the y direction is shown in Figure 9.

Figure 9. Image of the differences of MBD values along y direction.

For the sake of completeness, the statistical values of the depth differences between successive pixels are reported in Table 7 (along x) and Table 8 (along y).

Figure 9. Image of the differences of MBD values along y direction.

Geosciences 2022, 12, 62 16 of 23

For the sake of completeness, the statistical values of the depth differences betweensuccessive pixels are reported in Table 7 (along x) and Table 8 (along y).

Table 7. Statistical terms of the differences of MBD values along x direction.

Sector Min (m) Max (m) Mean (m) St. Dev. (m)

A −5.250 6.540 −0.041 0.260B −8.030 9.050 −0.091 0.503C −5.280 3.890 −0.200 0.242D −0.440 0.270 −0.091 0.040E −2.810 4.240 −0.069 0.090F −2.670 2.310 −0.068 0.063

Table 8. Statistical terms of the differences of MBD values along y direction.

Sector Min (m) Max (m) Mean (m) St. Dev. (m)

A −4.570 7.830 0.258 0.287B −8.690 9.320 0.287 0.552C −3.020 4.280 0.105 0.254D −0.280 0.420 0.033 0.055E −1.830 4.490 0.050 0.138F −1.510 3.330 0.017 0.089

The second step analyzes the variability of the direction of the slope (ascendingor descending) along the x and y directions; in particular the slope inversions betweensuccessive pixels are identified according to the procedure illustrated in Section 3.2.2. Theimage concerning the slope inversions along the y direction is shown in Figure 10.

Geosciences 2022, 12, x FOR PEER REVIEW 17 of 24

Table 7. Statistical terms of the differences of MBD values along x direction.

Sector Min (m) Max (m) Mean (m) St. Dev. (m) A −5.250 6.540 −0.041 0.260 B −8.030 9.050 −0.091 0.503 C −5.280 3.890 −0.200 0.242 D −0.440 0.270 −0.091 0.040 E −2.810 4.240 −0.069 0.090 F −2.670 2.310 −0.068 0.063

Table 8. Statistical terms of the differences of MBD values along y direction.

Sector Min (m) Max (m) Mean (m) St. Dev. (m) A −4.570 7.830 0.258 0.287 B −8.690 9.320 0.287 0.552 C −3.020 4.280 0.105 0.254 D −0.280 0.420 0.033 0.055 E −1.830 4.490 0.050 0.138 F −1.510 3.330 0.017 0.089

The second step analyzes the variability of the direction of the slope (ascending or descending) along the x and y directions; in particular the slope inversions between suc-cessive pixels are identified according to the procedure illustrated in Section 3.2.2. The image concerning the slope inversions along the y direction is shown in Figure 10.

Figure 10. The image of the slope inversions along y direction in the study area (the pixels in which slope inversions occur are colored white).

Those raster files already provide an important indication for themself, namely the number of slope-reversals for each sector. Since the value 1 indicates the slope inversion and 0 the absence of variation, the mean value of all the pixels present in each sector was calculated as a parameter of the variability of the seabed. The two values calculated (one for the x direction and the other for the y direction) were used to achieve Ixy.

However, this is not enough for the purpose of studying the morphology of the sea-bed as an almost flat surface could still have many slope inversions, as in the case of sector D; in fact, in the neighboring pixels, they often have very similar depth values. Conse-quently other steps were necessary for determining the further parameter and Sxy was

Figure 10. The image of the slope inversions along y direction in the study area (the pixels in whichslope inversions occur are colored white).

Those raster files already provide an important indication for themself, namely thenumber of slope-reversals for each sector. Since the value 1 indicates the slope inversionand 0 the absence of variation, the mean value of all the pixels present in each sector was

Geosciences 2022, 12, 62 17 of 23

calculated as a parameter of the variability of the seabed. The two values calculated (onefor the x direction and the other for the y direction) were used to achieve Ixy.

However, this is not enough for the purpose of studying the morphology of the seabedas an almost flat surface could still have many slope inversions, as in the case of sector D;in fact, in the neighboring pixels, they often have very similar depth values. Consequentlyother steps were necessary for determining the further parameter and Sxy was introducedin the MVI to take into account the level of inclination of the seabed and its variation. Thestudy area has both low slope and steep slopes, often alternating with each other even atshort distances, thus high values of abrupt variations were found.

The third step involves calculating the variation of the slope along the x and y di-rections. Initially, using the raster calculator, the values of the depth difference betweenconsecutive pixels were normalized with respect to the grid step size; the arctangentfunction applied to them provides the values of the slope in each pixel. Subsequently, incontinuing to use the raster calculator, the variations of the slope between consecutivepixels were determined. The image concerning the slope variations along the y direction isshown in Figure 11.

Geosciences 2022, 12, x FOR PEER REVIEW 18 of 24

introduced in the MVI to take into account the level of inclination of the seabed and its variation. The study area has both low slope and steep slopes, often alternating with each other even at short distances, thus high values of abrupt variations were found.

The third step involves calculating the variation of the slope along the x and y direc-tions. Initially, using the raster calculator, the values of the depth difference between con-secutive pixels were normalized with respect to the grid step size; the arctangent function applied to them provides the values of the slope in each pixel. Subsequently, in continuing to use the raster calculator, the variations of the slope between consecutive pixels were determined. The image concerning the slope variations along the y direction is shown in Figure 11.

Figure 11. The slope variations (in degree) along y direction.

These values were normalized by dividing by the theoretical maximum limit value of the slope variation (180°). In the end, we derived two images: the first relating to the slope variations along x and the other to the slope variations along y. The average of all the recorded values is Sxy.

From these values, the proposed index MVI was calculated according to Equation (15). The results are shown in Table 9.

Table 9. Morphological variation index (MVI) achieved for each sector.

Sector MVI A 0.0041 B 0.0075 C 0.0026 D 0.0014 E 0.0017 F 0.0011

By comparing those values with the RMSE values, it is evident that, once the number of points to be interpolated is fixed, more accurate results are achieved in the presence of a low level of morphological variation. The Figure 12 highlights this aspect in the case of 2400 points.

Figure 11. The slope variations (in degree) along y direction.

These values were normalized by dividing by the theoretical maximum limit valueof the slope variation (180◦). In the end, we derived two images: the first relating to theslope variations along x and the other to the slope variations along y. The average of all therecorded values is Sxy.

From these values, the proposed index MVI was calculated according to Equation (15).The results are shown in Table 9.

By comparing those values with the RMSE values, it is evident that, once the numberof points to be interpolated is fixed, more accurate results are achieved in the presence of alow level of morphological variation. The Figure 12 highlights this aspect in the case of2400 points.

Geosciences 2022, 12, 62 18 of 23

Table 9. Morphological variation index (MVI) achieved for each sector.

Sector MVI

A 0.0041B 0.0075C 0.0026D 0.0014E 0.0017F 0.0011

Geosciences 2022, 12, x FOR PEER REVIEW 19 of 24

Figure 12. RMSE values in the case of 2400 points using Ordinary Kriging and Universal Kriging for each sector (upper) and MVI values calculated in the six sectors (lower).

At this point of our analysis, we wanted to reconsider the graphs shown in Figures 6 and 7 in relation to the MVI indicator.

In the sectors where the lowest values of the indicator are recorded (sector D, E, and F), the RMSE values of the residuals decrease rapidly as the number of known points in-creases. A density equal to at least 1 point every 1000 m2 is sufficient to make the RMSE values fall below 1 m for both Ordinary Kriging and Universal Kriging. On the contrary, for the sectors where the MVI indicator has the highest values, the RMSE below 1 m is much higher. In sector B, characterized by the highest value of MVI, 1 point per 100 m2 is just enough to obtain an RMSE of approximately 1 m but only in the case of the Ordinary Kriging. In fact, in the case of Universal Kriging, the same concentration determines an RMSE value that remains higher than 1 m (RMSE = 1.214 m).

Finally, we wanted to remark that in the presence of a high MVI, even if RMSE is not higher than 1 m, residuals can reach considerable values in some points, as evidenced by the statistics in sector B, as shown in Table 5 (−14.037 m for Universal Kriging) and Table 6 (−14.469 m for Ordinary Kriging).

5. Conclusions In the case of very varied seabed morphology, it is essential, for different purposes

(e.g., safe navigation or geological studies), to have a detailed and accurate bathymetric model. To achieve these results, particular attention must be given to the interpolation algorithms used to reproduce 3D models, starting from a dataset consisting of a point cloud.

The results of our study validate the efficiency of the Kriging methods, remarking the influence of the dataset location and density on the 3D model accuracy in the case of

Figure 12. RMSE values in the case of 2400 points using Ordinary Kriging and Universal Kriging foreach sector (upper) and MVI values calculated in the six sectors (lower).

At this point of our analysis, we wanted to reconsider the graphs shown in Figures 6 and 7in relation to the MVI indicator.

In the sectors where the lowest values of the indicator are recorded (sector D, E, and F),the RMSE values of the residuals decrease rapidly as the number of known points increases.A density equal to at least 1 point every 1000 m2 is sufficient to make the RMSE values fallbelow 1 m for both Ordinary Kriging and Universal Kriging. On the contrary, for the sectorswhere the MVI indicator has the highest values, the RMSE below 1 m is much higher. Insector B, characterized by the highest value of MVI, 1 point per 100 m2 is just enough toobtain an RMSE of approximately 1 m but only in the case of the Ordinary Kriging. In fact,

Geosciences 2022, 12, 62 19 of 23

in the case of Universal Kriging, the same concentration determines an RMSE value thatremains higher than 1 m (RMSE = 1.214 m).

Finally, we wanted to remark that in the presence of a high MVI, even if RMSE is nothigher than 1 m, residuals can reach considerable values in some points, as evidenced bythe statistics in sector B, as shown in Table 5 (−14.037 m for Universal Kriging) and Table 6(−14.469 m for Ordinary Kriging).

5. Conclusions

In the case of very varied seabed morphology, it is essential, for different purposes(e.g., safe navigation or geological studies), to have a detailed and accurate bathymetricmodel. To achieve these results, particular attention must be given to the interpolationalgorithms used to reproduce 3D models, starting from a dataset consisting of a point cloud.

The results of our study validate the efficiency of the Kriging methods, remarkingthe influence of the dataset location and density on the 3D model accuracy in the case ofthe interpolation process. To investigate this relationship, a new index called MVI wasintroduced as a measurement of the level of variation of seabed morphology; this indexresults from the product of two components: the first component (Ixy) is related to thefrequency of the variation of the slope direction (ascending or descending) and the second(Sxy) is associated to the variation of the value of the slope, both considered for the xdirection as well as y direction.

The experiments demonstrate that by using Universal Kriging as well as OrdinaryKriging, a density equal to at least 1 point every 1000 m2 is sufficient to produce an accuratemodel in areas characterized by a low level of variation of seabed morphology (not onlyRMSE but also minimum and maximum values fall below 1 m). On the contrary, a density10 times greater than that is necessary to produce an accurate model in areas characterizedby a high level of variation of seabed morphology; in this case, even if the RMSE dropsbelow 1 m, there may be strong differences between the predicted and observed depths insome places, thus a further increment of the measured points is required.

MVI is useful to represent the seabed variation as a unique value but to calculateit in an accurate way, the morphology must be already known. In our study, we havedetermined its value because a grid model derived by MB survey is already available.Consequently, we used it to establish the relationship between point density and modelaccuracy; if we want to define a priori the optimal number of depths to measure in anunknown area for a bathymetric survey, we can try to calculate MVI using informationavailable, e.g., previous survey data and/or nautical charts even on a smaller scale. In thiscase, you must know that the calculated MVI is an approximate value and does not providethe exact photograph of reality, thus the advantage of using it may be limited.

Concerning the future developments of this work, further studies will be focused onthe relationship between bathymetric point density and seabed model accuracy. In particu-lar, other datasets representative of different seabed configurations will be considered bothin terms of depth range (in the proposed study, depths not exceeding 108 m were analyzed)and in terms of the extension of the study area. Higher values of point density will beinvestigated to achieve better results also in roughness areas. In addition, further tests willbe carried out to validate the efficiency of the proposed MVI to represent the variation ofseabed morphology.

Author Contributions: C.P. conceived the article and designed the experiments; E.A. and P.P.A. con-ducted the bibliographic research; C.P. organized the data collection and supervised the applications;E.A. and P.P.A. carried out experiments on Kriging applications; C.P. designed the MVI; E.A. andP.P.A. carried out the accuracy tests; and all authors took part in the result analysis and in writing thepaper. All authors have read and agreed to the published version of the manuscript.

Funding: This research study received no external funding.

Institutional Review Board Statement: Not applicable.

Geosciences 2022, 12, 62 20 of 23

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Acknowledgments: We wish to thank the Italian Hydrographic Authority (Istituto Idrografico dellaMarina Militare) for providing the data. In particular, we would like to thank Lieutenant LuigiCarnevale and hydrographic surveyor Nunziante Langellotto for their valuable technical support indata handling.

Conflicts of Interest: The authors declare no conflict of interest.

References1. Monego, M.; Menin, A.; Fabris, M.; Achilli, V. 3D survey of Sarno Baths (Pompeii) by integrated geomatic methodologies. J. Cult.

Herit. 2019, 40, 240–246. [CrossRef]2. Ebolese, D.; Lo Brutto, M.; Dardanelli, G. The integrated 3D survey for underground archaeological environment. Int. Arch.

Photogramm. Remote Sens. Spat. Inf. Sci. 2019, XLII-2/W9, 311–317. [CrossRef]3. Aguilar, F.J.; Fernández, I.; Casanova, J.A.; Ramos, F.J.; Aguilar, M.A.; Blanco, J.L.; Moreno, J.C. 3D coastal monitoring from very

dense UAV-Based photogrammetric point clouds. In Advances on Mechanics, Design Engineering and Manufacturing; Springer:Cham, Switzerland, 2017; pp. 879–887. [CrossRef]

4. Masiero, A.; Tucci, G.; Conti, A.; Fiorini, L.; Vettore, A. Initial evaluation of the potential of smartphone stereo-vision in museumvisits. Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. 2019, XLII-2/W11, 837–842. [CrossRef]

5. Pepe, M.; Costantino, D.; Alfio, V.S.; Zannotti, N. 4D geomatics monitoring of a quarry for the calculation of extracted volumes bytin and grid model: Contribute of UAV photogrammetry. Geogr. Tech. 2020, 16, 1–14. [CrossRef]

6. Baiocchi, V.; Napoleoni, Q.; Tesei, M.; Servodio, G.; Alicandro, M.; Costantino, D. UAV for monitoring the settlement of a landfill.Eur. J. Remote Sens. 2019, 52 (Suppl. 3), 41–52. [CrossRef]

7. Shingare, P.P.; Kale, S.S. Review on digital elevation model. Int. J. Mod. Eng. Res. 2013, 3, 2412–2418.8. Habib, M.; Alzubi, Y.; Malkawi, A.; Awwad, M. Impact of interpolation techniques on the accuracy of large-scale digital elevation

model. Open Geosci. 2020, 12, 190–202. [CrossRef]9. Alcaras, E.; Carnevale, L.; Parente, C. Interpolating single-beam data for sea bottom GIS modelling. Int. J. Emerg. Trends Eng. Res.

2020, 8, 591–597. [CrossRef]10. Arun, P.V. A comparative analysis of different DEM interpolation methods. Egypt. J. Remote Sens. Space Sci. 2013, 16, 133–139.

[CrossRef]11. Tobler, W.R. A computer movie simulating urban growth in the Detroit region. Econ. Geogr. 1970, 46, 234–240. [CrossRef]12. Waters, M.N. Spatial interpolation I, lecture 40. In NCGIA Core Curriculum, Technical Issues in GIS; University of California: Santa

Barbara, CA, USA, 1989.13. Mitas, L.; Mitasova, H. Spatial interpolation. Geogr. Inf. Syst. Princ. Tech. Manag. Appl. 1999, 1, 481–492.14. Chang, T.J.; Wang, W.L. Spatial Analysis Using GIS to Study Performance of Highway Culvers in Ohio. In Pipelines 2008: Pipeline

Asset Management: Maximizing Performance of Our Pipeline Infrastructure; ASCE: Reston, VA, USA, 2008; pp. 1–10. [CrossRef]15. Burrough, P.A.; McDonnell, R.A. Principles of Geographical Information Systems; Oxford University Press: New York, NY, USA,

1998; p. 190.16. Vansarochana, A.; Tripathi, N.K.; Clemente, R. Finding appropriate interpolation techniques for topographic surface generation

for mudslide risk zonation. Geocarto Int. 2009, 24, 313–332. [CrossRef]17. Lam, N.S.N. Spatial interpolation methods: A review. Am. Cartogr. 1983, 10, 129–150. [CrossRef]18. GIS Resources Classification of Interpolation. GIS Resources. Available online: https://gisresources.com/classification-of-

interpolation_2/ (accessed on 27 December 2021).19. ESRI, Deterministic Methods for Spatial Interpolation, ArcGIS Pro 2.9. Available online: https://pro.arcgis.com/en/pro-app/latest/

help/analysis/geostatistical-analyst/deterministic-methods-for-spatial-interpolation.htm (accessed on 27 December 2021).20. Eberly, S.; Swall, J.; Holland, D.; Cox, B.; Baldridge, E. Developing Spatially Interpolated Surfaces and Estimating Uncertainty; United

States Environmental Protection Agency: Washington, DC, USA, 2004; pp. 28–40.21. Bhattacharjee, S.; Mitra, P.; Ghosh, S.K. Spatial interpolation to predict missing attributes in GIS using semantic kriging. IEEE

Trans. Geosci. Remote Sens. 2013, 52, 4771–4780. [CrossRef]22. ESRI. How Inverse Distance Weighted Interpolation Works. In ArcGIS Pro Help; ESRI: Redlands, CA, USA, 2021; Avail-

able online: https://pro.arcgis.com/en/pro-app/latest/help/analysis/geostatistical-analyst/how-inverse-distance-weighted-interpolation-works.htm (accessed on 27 December 2021).

23. Childs, C. Interpolating surfaces in ArcGIS spatial analyst. ArcUser 2004, 3235, 32–35.24. Tabios, G.Q., III; Salas, J.D. A comparative analysis of techniques for spatial interpolation of precipitation 1. JAWRA J. Am. Water

Resour. Assoc. 1985, 21, 365–380. [CrossRef]25. Xie, Y.; Chen, T.B.; Lei, M.; Yang, J.; Guo, Q.J.; Song, B.; Zhou, X.Y. Spatial distribution of soil heavy metal pollution estimated by

different interpolation methods: Accuracy and uncertainty analysis. Chemosphere 2011, 82, 468–476. [CrossRef] [PubMed]

Geosciences 2022, 12, 62 21 of 23

26. Parente, C.; Vallario, A. Interpolation of Single Beam Echo Sounder Data for 3D Bathymetric Model. Int. J. Adv. Comput. Sci. Appl.2019, 10, 6–13. [CrossRef]

27. Erdogan, S. A comparision of interpolation methods for producing digital elevation models at the field scale. Earth Surf. Process.Landf. 2009, 34, 366–376. [CrossRef]

28. Ouma, Y.O.; Owiti, T.; Kipkorir, E.; Kibiiy, J.; Tateishi, R. Multitemporal comparative analysis of TRMM-3B42 satellite-estimatedrainfall with surface gauge data at basin scales: Daily, decadal and monthly evaluations. Int. J. Remote Sens. 2012, 33, 7662–7684.[CrossRef]

29. Chilès, J.P.; Delfiner, P. Geostatistics: Modeling Spatial Uncertainty; John Wiley & Sons. Inc.: Hoboken, NJ, USA, 1999.30. Kristic, M.; Žuškin, S.; Brcic, D.; Valcic, S. Zone of confidence impact on cross track limit determination in ECDIS passage planning.

J. Mar. Sci. Eng. 2020, 8, 566. [CrossRef]31. Latapy, A.; Héquette, A.; Pouvreau, N.; Weber, N.; Robin-Chanteloup, J.B. Mesoscale morphological changes of nearshore sand

banks since the early 19th century, and their influence on coastal dynamics, Northern France. J. Mar. Sci. Eng. 2019, 7, 73.[CrossRef]

32. Kågesten, G.; Fiorentino, D.; Baumgartner, F.; Zillén, L. How Do Continuous High-Resolution Models of Patchy Seabed HabitatsEnhance Classification Schemes? Geosciences 2019, 9, 237. [CrossRef]

33. Iacono, C.L.; Mateo, M.; Gràcia, E.; Guasch, L.; Carbonell, R.; Serrano, L.; Serrano, O.; Dañobeitia, J. Very high-resolutionseismo-acoustic imaging of seagrass meadows (Mediterranean Sea): Implications for carbon sink estimates. Geophys. Res. Lett.2008, 35, 1–5. [CrossRef]

34. El-Hattab, A.I. Single beam bathymetric data modelling techniques for accurate maintenance dredging. Egypt. J. Remote Sens.Space Sci. 2014, 17, 189–195. [CrossRef]

35. Specht, M.; Specht, C.; Mindykowski, J.; Dabrowski, P.; Masnicki, R.; Makar, A. Geospatial Modeling of the Tombolo Phenomenonin Sopot using Integrated Geodetic and Hydrographic Measurement Methods. Remote Sens. 2020, 12, 737. [CrossRef]

36. Sinapi, L.; Lamberti, L.O.; Pizzeghello, N.M.; Ivaldi, R. The Graham Bank: Hydrographic features and safety of navigation. Int.Hydrogr. Rev. 2016, 15, 7–20.

37. Arseni, M.; Voiculescu, M.; Georgescu, L.P.; Iticescu, C.; Rosu, A. Testing different interpolation methods based on single beamechosounder river surveying. Case study: Siret River. ISPRS Int. J. Geo-Inf. 2019, 8, 507. [CrossRef]

38. Wilson, M.F.; O’Connell, B.; Brown, C.; Guinan, J.C.; Grehan, A.J. Multiscale terrain analysis of multibeam bathymetry data forhabitat mapping on the continental slope. Mar. Geod. 2007, 30, 3–35. [CrossRef]

39. Degraer, S.; Moerkerke, G.; Rabaut, M.; Van Hoey, G.; Du Four, I.; Vincx, M.; Henriet, J.-P.; Van Lancker, V. Very-high resolutionside-scan sonar mapping of biogenic reefs of the tube-worm Lanice conchilega. Remote Sens. Environ. 2008, 112, 3323–3328.[CrossRef]

40. Wang, L.; Liu, H.; Su, H.; Wang, J. Bathymetry retrieval from optical images with spatially distributed support vector machines.GIScience Remote Sens. 2019, 56, 323–337. [CrossRef]

41. Liu, Y.; Wu, Z.; Zhao, D.; Zhou, J.; Shang, J.; Wang, M.; Zhu, C.; Luo, X. Construction of high-resolution bathymetric dataset forthe Mariana Trench. IEEE Access 2019, 7, 142441–142450. [CrossRef]

42. Flemming, B.W. Side-scan sonar: A practical guide. Int. Hydrogr. Rev. 1976, 53, 65–92.43. Ceylan, A.; Karabork, H.; Ekozoglu, I. An analysis of bathymetric changes in Altinapa reservoir. Carpathian J. Earth Environ. Sci.

2011, 6, 15–24.44. Stumpf, R.P.; Holderied, K.; Sinclair, M. Determination of water depth with high-resolution satellite imagery over variable bottom

types. Limnol. Oceanogr. 2003, 48, 547–556. [CrossRef]45. Parente, C.; Pepe, M. Bathymetry from worldView-3 satellite data using radiometric band ratio. Acta Polytech. 2018, 58, 109–117.

[CrossRef]46. Carrara, A.; Bitelli, G.; Carla, R. Comparison of techniques for generating digital terrain models from contour lines. Int. J. Geogr.

Inf. Sci. 1997, 11, 451–473. [CrossRef]47. Chung, J.W.; Rogers, J.D. Interpolations of groundwater table elevation in dissected uplands. Groundwater 2012, 50, 598–607.

[CrossRef]48. Sassais, R.; Makar, A. Methods to generate numerical models of terrain for spatial ENC presentation. Annu. Navig. 2011, 18, 69–81.49. Gosciewski, D. The effect of the distribution of measurement points around the node on the accuracy of interpolation of the

digital terrain model. J. Geogr. Syst. 2013, 15, 513–535. [CrossRef]50. Mesa-Mingorance, J.L.; Ariza-López, F.J. Accuracy assessment of digital elevation models (DEMs): A critical review of practices

of the past three decades. Remote Sens. 2020, 12, 2630. [CrossRef]51. Aguilar Torres, F.J.; Aguilar Torres, M.A.; Agüera Vega, F.; Carvajal Ramírez, F.; Sánchez Salmerón, P.L. Efectos de la Morfología

del Terreno, Densidad Muestral y Métodos de Interpolación en la Calidad de los Modelos Digitales de Elevaciones. In Proceedingsof the XIV Congreso Internacional de Ingeniería Gráfica, INGEGRAF, Santander, Spain, 5–7 June 2002.

52. Bakuła, K. Reduction of DTM obtained from LiDAR data for flood modeling. Arch. Fotogram. Kartogr. I Teledetekcji 2011, 22, 51–61.53. Sterenczak, K.; Ciesielski, M.; Balazy, R.; Zawiła-Niedzwiecki, T. Comparison of various algorithms for DTM interpolation from

LIDAR data in dense mountain forests. Eur. J. Remote Sens. 2016, 49, 599–621. [CrossRef]54. Zhang, Z.; Gerke, M.; Vosselman, G.; Yang, M.Y. Filtering photogrammetric point clouds using standard LiDAR filters towards

DTM generation. ISPRS Ann. Photogramm. Remote Sens. Spat. Inf. Sci. 2018, IV-2, 319–326. [CrossRef]

Geosciences 2022, 12, 62 22 of 23

55. Aguilar, F.J.; Agüera, F.; Aguilar, M.A.; Carvajal, F. Effects of terrain morphology, sampling density, and interpolation methods ongrid DEM accuracy. Photogramm. Eng. Remote Sens. 2005, 71, 805–816. [CrossRef]

56. Chaplot, V.; Darboux, F.; Bourennane, H.; Leguédois, S.; Silvera, N.; Phachomphon, K. Accuracy of interpolation techniquesfor the derivation of digital elevation models in relation to landform types and data density. Geomorphology 2006, 77, 126–141.[CrossRef]

57. Dupuy, R.; Makar, A. Analysis of digital sea bottom models generated using ENC data. Annu. Navig. 2011, 18, 27–36.58. Alcaras, E.; Parente, C.; Vallario, A. A Comparison of different interpolation methods for DEM production. Int. J. Adv. Trends

Comput. Sci. Eng. 2019, 6, 1654–1659. [CrossRef]59. Cutroneo, L.; Ferretti, G.; Scafidi, D.; Ardizzone, G.D.; Vagge, G.; Capello, M. Current observations from a looking down vertical

V-ADCP: Interaction with winds and tide? The case of Giglio Island (Tyrrhenian Sea, Italy). Oceanologia 2017, 59, 139–152.[CrossRef]

60. Frezza, V.; Carboni, M.G. Distribution of recent foraminiferal assemblages near the Ombrone River mouth (Northern TyrrhenianSea, Italy). Rev. Micropaléontologie 2009, 52, 43–66. [CrossRef]

61. ESRI (Environmental Systems Research Institute). ArcGIS 10.3; ESRI: Redlands, CA, USA, 2012.62. ESRI (Environmental Systems Research Institute). Geostatistical Analyst, ArcGIS 10.3; ESRI: Redlands, CA, USA, 2012.63. Efron, B. The Jackknife, the Bootstrap and other resampling plans. In CBMS-NSF Regional Conference Series in Applied Mathematics;

Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, USA, 1982.64. Krivoruchko, K. Empirical bayesian Kriging-Implemented in ArcGIS Geostatistical Analyst. ArcUser Fall 2012, 6, 1–5.65. Ceron, W.L.; Andreoli, R.V.; Kayano, M.T.; Canchala, T.; Carvajal-Escobar, Y.; Souza, R.A. Comparison of spatial interpolation

methods for annual and seasonal rainfall in two hotspots of biodiversity in South America. An. Acad. Bras. Ciências 2021, 93,e20190674. [CrossRef] [PubMed]

66. Salekin, S.; Burgess, J.H.; Morgenroth, J.; Mason, E.G.; Meason, D.F. A comparative study of three non-geostatistical methods foroptimising digital elevation model interpolation. ISPRS Int. J. Geo-Inf. 2018, 7, 300. [CrossRef]

67. Gentilucci, M.; Barbieri, M.; Burt, P.; D’Aprile, F. Preliminary data validation and reconstruction of temperature and precipitationin Central Italy. Geosciences 2018, 8, 202. [CrossRef]

68. Bargaoui, Z.K.; Chebbi, A. Comparison of two Kriging interpolation methods applied to spatiotemporal rainfall. J. Hydrol. 2009,365, 56–73. [CrossRef]

69. Wu, T.; Li, Y. Spatial interpolation of temperature in the United States using residual Kriging. Appl. Geogr. 2013, 44, 112–120.[CrossRef]

70. Robinson, T.P.; Metternicht, G. A comparison of inverse distance weighting and Ordinary Kriging for characterising within-paddock spatial variability of soil properties in Western Australia. Cartography 2003, 32, 11–24. [CrossRef]

71. Alcaras, E.; Falchi, U.; Parente, C. Digital Terrain Model Generalization for Multiscale Use. Int. Rev. Civ. Eng. (IRECE) 2020,11, 52–59. [CrossRef]

72. Stein, M.L. Interpolation of Spatial Data: Some Theory for Kriging; Springer Science & Business Media: Berlin/Heidelberg, Germany, 1999.73. Jian, X.; Olea, R.A.; Yu, Y.S. Semivariogram modeling by weighted least squares. Comput. Geosci. 1996, 22, 387–397. [CrossRef]74. Rishikeshan, C.A.; Katiyar, S.K.; Mahesh, V.V. Detailed evaluation of DEM interpolation methods in GIS using DGPS data. In

Proceedings of the 2014 International Conference on Computational Intelligence and Communication Networks, Bhopal, India,14–16 November 2014; IEEE: Hoboken, NJ, USA; pp. 666–671. [CrossRef]

75. Malvic, T.; Ivšinovic, J.; Velic, J.; Rajic, R. Kriging with a small number of data points supported by Jack-Knifing, a case study inthe Sava depression (Northern Croatia). Geosciences 2019, 9, 36. [CrossRef]

76. Kiš, I.M. Comparison of Ordinary and Universal Kriging interpolation techniques on a depth variable (a case of linear spatialtrend), case study of the Šandrovac Field. Rud.-Geološko-Naft. Zb. (Min.-Geol.-Pet. Bull.) 2016, 31, 41–58. [CrossRef]

77. Wackernagel, H. Ordinary Kriging. In Multivariate Geostatistics; Springer: Berlin/Heidelberg, Germany, 2003; pp. 79–88.[CrossRef]

78. Webster, R.; Oliver, M.A. Geostatistics for Environmental Scientists; John Wiley & Sons: Chichester, UK, 2007.79. ESRI. Understanding Ordinary Kriging, ArcGIS 10.3- Help, Redlands, CA, USA. Available online: http://webhelp.esri.com/

arcgisdesktop/9.2/index.cfm?TopicName=Understanding_Ordinary_Kriging (accessed on 26 December 2021).80. ILWIS (The Integrated Land and Water Information System) Team. ILWI Software 3.5—Help; Faculty of Geo-Information Science

and Earth Observation: Enschede, The Netherlands, 2008.81. Kumar, V. Optimal contour mapping of groundwater levels using Universal Kriging—A case study. Hydrol. Sci. J. 2007, 52,

1038–1050. [CrossRef]82. ESRI. Understanding Universal Kriging, ArcGIS 10.3- Help, Redlands, CA, USA. Available online: https://desktop.arcgis.com/

en/arcmap/latest/extensions/geostatistical-analyst/understanding-Universal-Kriging.htm (accessed on 26 December 2021).83. Johnston, K.; Ver Hoef, J.M.; Krivoruchko, K.; Lucas, N. Using ArcGIS Geostatistical Analyst; Esri: Redlands, CA, USA, 2001;

Volume 380.84. Falchi, U.; Parente, C.; Prezioso, G. Global geoid adjustment on local area for GIS applications using GNSS permanent station

coordinates. Geod. Cartogr. 2018, 44, 80–88. [CrossRef]

Geosciences 2022, 12, 62 23 of 23

85. Xiao, Y.; Gu, X.; Yin, S.; Shao, J.; Cui, Y.; Zhang, Q.; Niu, Y. Geostatistical interpolation model selection based on ArcGISand spatio-temporal variability analysis of groundwater level in piedmont plains, Northwest China. SpringerPlus 2016, 5, 425.[CrossRef]

86. Sain, S.R.; Baggerly, K.A.; Scott, D.W. Cross-validation of multivariate densities. J. Am. Stat. Assoc. 1994, 89, 807–817. [CrossRef]87. Browne, M.W. Cross-validation methods. J. Math. Psychol. 2000, 44, 108–132. [CrossRef]88. Refaeilzadeh, P.; Tang, L.; Liu, H. Cross-validation. Encycl. Database Syst. 2009, 5, 532–538. [CrossRef]89. Lam, K.C.; Bryant, R.G.; Wainright, J. Application of spatial interpolation method for estimating the spatial variability of rainfall

in semiarid New Mexico, USA. Mediterr. J. Soc. Sci. 2015, 6, 108. [CrossRef]90. Berrar, D. Cross-Validation. Encycl. Bioinform. Comput. Biol. 2019, 1, 542–545. [CrossRef]91. Fasshauer, G.E.; Zhang, J.G. On choosing “optimal” shape parameters for RBF approximation. Numer. Algorithms 2007, 45,

345–368. [CrossRef]92. Tomlin, C.D. Map algebra: One perspective. Landsc. Urban Plan. 1994, 30, 3–12. [CrossRef]93. Li, Z. On the measure of digital terrain model accuracy. Photogramm. Rec. 1988, 12, 873–877. [CrossRef]94. Yang, X.; Hodler, T. Visual and statistical comparisons of surface modeling techniques for point-based environmental data.

Cartogr. Geogr. Inf. Sci. 2000, 27, 165–175. [CrossRef]95. Lozano Parra, F.J.; Gutiérrez, Á.G.; Fernández, M.P.; Contador, J.L. Classification of morphometric units from digital terrain

models: Applications in land cover classification. Rocz. Geomatyki 2009, 7, 83–89.96. ESRI. How Slope Works, Spatial-Analyst-Toolbox. ArcGIS. Available online: https://desktop.arcgis.com/en/arcmap/10.3/tools/

spatial-analyst-toolbox/how-slope-works.htm (accessed on 26 December 2021).97. Zevenbergen, L.W.; Thorne, C.R. Quantitative analysis of land surfacetopography. Earth Surf. Processes Landf. 1987, 12, 47–56.

[CrossRef]98. Wu, J.; Fang, J.; Tian, J. Terrain Representation and Distinguishing Ability of Roughness Algorithms Based on DEM with Different

Resolutions. ISPRS Int. J. Geo-Inf. 2019, 8, 180. [CrossRef]99. Riley, S.J.; DeGloria, S.D.; Elliot, R. Index that quantifies topographic heterogeneity. Intermt. J. Sci. 1999, 5, 23–27.