A comparison of interpolation methods on the basis of data

9 The bathymetric survey of Lake Vrana included a wide range of activities that were 10 performed in several different stages, in accordance with the standards set by the International 11 Hydrographic Organization. The survey was conducted using an integrated measuring system 12 which consisted of three main parts: a single-beam sonar Hydrostar 4300, GPS devices 13 Ashtech Promark 500 – base, and a Thales Z-Max – rover. A total of 12 851 points were 14 gathered. 15 In order to find continuous surfaces necessary for analysing the morphology of the bed of 16 Lake Vrana, it was necessary to approximate values in certain areas that were not directly 17 measured, by using an appropriate interpolation method. The main aims of this research were 18 as follows: a) to compare the efficiency of 14 different interpolation methods and discover the 19 most appropriate interpolators for the development of a raster model; b) to calculate the 20 surface area and volume of Lake Vrana, and c) to compare the differences in calculations 21 between separate raster models. The best deterministic method of interpolation was RBF 22 multiquadratic, and the best geostatistical ordinary cokriging. The mean quadratic error in 23 both methods measured less than 0.3 metres. 24 The quality of the interpolation methods was analysed in 2 phases. The first phase used only 25 points gathered by bathymetric measurement, while the second phase also included points 26 gathered by photogrammetric restitution. 27 The first bathymetric map of Lake Vrana in Croatia was produced, as well as scenarios of 28 minimum and maximum water levels. The calculation also included the percentage of flooded 29 areas and cadastre plots in the case of a 2-metre increase in the water level. The research 30

The distance between the base and referential devices had to be determined in advance, in 24 order to achieve an adequate degree of precision. This was named the base line and its 25 maximum value was 50 kilometres. The distance between the base GPS and the UHF 1 transmitter had to be a minimum of 10 m. Since the UHF signal was rather weak throughout the lake, three base points were determined 6 using the Ashtech Promark 500 and CROPOS system: 1) coordinates λ=5 541 365.709, φ=4 connected by a benchmark and measuring gauge at the Prosika location. A base GPS device 12 was set at those points, depending on the phase of the survey, and connected to a UHF 13 transmitter (with all components) in order to achieve a connection (signal) with the mobile 14 GPS installed on the inflatable boat.

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A dual-frequency probe was fixed to this support with a rover GPS device submerged 20 cm 16 below the water level (Fig. 2c). This arrangement was necessary due to the shallow water of 17 the lake and low water level at the northwest end. Since the Hydrostar 4300 sonar supports 18 depth recording simultaneously at two frequencies, the survey was conducted at two  The measurement process was conducted in two phases (Fig. 5): 1) from 10-12 May 2012, 20 and 2) from 7-9 June 2012.

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The first phase took two days, and included a survey of 14.351 km 2 of the northern part of 22 Lake Vrana. The total length of the measured profiles was 71.3 km, and the total amount of 23 points gathered was 5643. In the first phase of investigation the water level measured at the 24 Prosika station was 0.42 m. The limiting factors for the survey in this part of the lake were the 25 dense grassy vegetation on the bottom, the shallow water and the lush surface-level vegetation 26 which hindered navigation. Measurement was cancelled in these parts, based on previously 27 established profiles, while the shallow water was measured using a plumb-line. As a result, 28 this survey cannot be classified as systematic. It is nevertheless very important in relation to 29 the part of the lake that was measured, since the terrain there is flat or minimally inclined. An 30 acceptable level of interpolation is possible in areas featuring an irregular layout of profiles. The second phase featured negligible limiting factors, so the survey was conducted according 1 to plan. The water level at the Prosika station was 0.37 m. A total area of 15.514 km 2 was 2 surveyed in the southern part of the lake. The total length of the measured profiles was 82.5 3 km, and the total amount of points gathered was 7208. The data obtained from measurement was transferred to a PC via the Juniper System-Allegro 10 controller and the Fast Survey programme package for further processing and interpolation.

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During measurement, the controller creates a separate file with information regarding the 12 point coordinates, time obtained, and depth recorded. Data processing included filtering out 13 noise, calibrating the checked depths to a common referential level, and interpolation. The  Since parts of Lake Vrana are quite difficult to survey, measurements taken by ultrasound 18 showed some background noise. In simple terms, the ultrasound beam bounces off the first 19 obstacle it encounters, so the echo sounder calculates the distance to that obstacle and 20 represents it as a depth measurement. However, such obstacles are not always at the bottom of 21 the lake, and indeed, random noise may be generated by floating matter, plankton, fish, or

Interpolation Methods 16
The spatial interpolation methods have been applied to many disciplines where the most  The most appropriate methods have been chosen, based on seven statistical parameters: 30 minimum value, maximum value, range, sum value, mean value, variance and standard 31 deviation. Of these, standard deviation, or mean quadratic error, is especially worth 32 mentioning, since it is the most used method world-wide for determining the precision of    Table 3 shows that the output results regarding the standard deviation do not reveal significant 28 differences. For example, the difference between automatic and manually found standard Points gathered by the bathymetric survey did not include the entire surface of the lake, since 28 the echo sounder could not gather data in areas above -0.5 m. Since that resulted in a lack of 29 data at the edges of the lake, the modeling toolset poorly extrapolated the surfaces (Fig. 6).

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Visually compared, the methods generally show the greatest differences in the smoothness of 31 isobaths, which is logical since the differences between the chosen parameters are essentially 32 negligible. A more detailed analysis indicates the results of certain methods (appearance of 1 continuous surfaces at micro levels).

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In order to develop a digital model of the lake that would enable various simulations, such as 3 changes in the water level, it is necessary to consider the data that refers to the surrounding 4 terrain (height data, gathered by aero-photogrammetry). The combination of precisely 5 obtained data on heights and depths enables the interpolation for the areas that were not 6 directly included in the survey. The output results turned out well, since the lake features 7 mostly low, flattened shores.

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Due to curious output results in the first phase, the comparison of methods of interpolation 9 was repeated for 30 233 points within the Lake Vrana Nature Park (Table 4). Of those points,   The differences between the four best methods of interpolation are visible in the two-3 dimensional ( Fig. 10a-b-c-d) and three-dimensional graphic representations. Figures 8a and   4 8b show the more vertically dissected part of the lake, with an AB profile, and a length of 5 1500 m, which was used as a further testing sample for the four best interpolation methods.

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The profile line was drawn so as to cover 6 bathymetrically measured points.  The volume of a lake can be efficiently calculated by a regular grid obtained by using a 1 certain interpolation method. The calculation process was relatively simple, since the number 2 of pixels was known (18 714), as well as the surface (40 m x 40 m = 1600 m 2 ) and the height 3 (z) within the coordinate system. A pixel in this case represents a three-dimensional object 4 (cube or a quadratic prism) based on which the volume can be calculated.

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In order to compare it with other algorithms, the volume was calculated for the regular spatial  (Table 5). The output results of volume calculation depend primarily on the spatial resolution; 10 the lower the resolution, the more precise the calculation, because the leaps in values between 11 pixels become less.

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In order to calculate the volume, three more complex Newton-Cotes formulae were used: 1)   Table 5 shows calculated values for the volume derived from Newton-Cotes formulae, 1 applied to five different methods of interpolation. Since every method displays a certain level 2 of error in the approximation of the volume, arithmetical means for the three methods were 3 also calculated.

4
The border of the lake for all the models was an isobath at 0.4 metres, obtained by 5 interpolating bathymetrically measured depth data and terrain elevation data obtained by aero-6 photogrammetry. The isobath was converted into a polygon, which was used to determine a  (Table 5).  The surface area of Lake Vrana, in relation to its water level (which annually oscillates by 16 1.93 m) varies by almost 4 km 2 (Table 6). It can be obtained by manual vectorisation based on 17 a geo-referential digital orthophoto (29.412 km2). The process is relatively simple, and the 18 contour of the lake is represented by the border between the water and land, defined by 19 subjective visual approximation. However, 4.6% of the lake's surface area is covered in dense 20 vegetation (Phragmitetalia), which makes determining the surface area a more complex task.  Table 5 30 Deleted: also shows that the  The water level map at 2 m was overlaid with the map of habitats for Lake Vrana Nature Park