Assessing the Accuracy of 3D Modeling of Hydrotechnical Structures Using Bathymetric Drones: A Study of the Karatomara Reservoir

Abstract

In recent years, Kazakhstan has faced the problem of sustainable development in the field of operation of a number of reservoirs: periods of drought lead to a systematic decrease in accumulated fresh water reserves, and the flood of 2024 led to the flooding of a number of settlements. The article raises questions about the real state of the region’s reservoirs (using the example of the Karatomar reservoir), the accuracy of the conducted bathymetric studies, and the correctness of estimating the required step (or distance between the control points being taken) of the tacks (trajectory lines) of the measurement, which was carried out using the Apache 3 bathymetric drone. The study of the patterns of modeling accuracy from the frequency of tacks (trajectory lines) was carried out using kriging methods. Reservoir models were built in QGis and Surfe. When analyzing the coastline, Sentinel-2 space images and Kazvodkhoz (Kazakhstani state enterprise) data were used. The result of the study was an algorithm for determining the step of tacks (trajectory lines) for modern bottom geomorphology. The conducted research has shown that over 78 years of use, the reservoir’s parameters have undergone significant changes. A similar situation of significant deterioration in parameters is characteristic of other hydrotechnical structures in the region.

1. Introduction

Hydrotechnical structures and the reservoirs they create play an important role in the global water cycle, regulating the flow of water from the environment into artificial systems. For example, about 57% of the seasonal variability of the world’s surface fresh water (SFF) is compensated by reservoirs [1].

As a rule, after the commissioning of reservoirs, practically no work is carried out to assess the relief of the bottom of reservoirs, and no work is carried out on a regular bathymetric assessment of the morphology of the bottom of reservoirs. Usually, studies and descriptions of such objects are limited only to assessing the indicators of type, shape, retaining levels, bed size, and design volume of water in reservoirs. At the same time, effective monitoring of the state of reservoirs is necessary to solve a wide range of practical problems, such as modeling and forecasting flood protection measures, planning irrigation and man-made loads, and assessing the long-term consequences of these structures for the global climate [2,3].

Following a series of spring floods in 2024 in Kazakhstan, monitoring the state of water resources and, especially, the actual state of hydrotechnical structures has been included in the list of priority tasks of sustainable development programs at the macroeconomic level. It is carried out at the state level of the Republic of Kazakhstan [4]. This fact served as the main reason for the study of the cascade of reservoirs in the Kostanay region, including the Karatomar reservoir.

Reservoirs, like any hydrotechnical structures, are subject to natural aging processes, leading to changes in the relief and, as a consequence, changes in the parameters of the entire structure.

The main characteristics of the reservoir operation, allowing to predict/model the conditions and plan corrective measures, are derived on the basis of a digital 3D model based on the results of its bathymetry [5].

Modern bathymetry methods offer the use of various computerized methods for collecting an array of primary data, ranging from collecting field test data (manual methods, hydroacoustic methods, and methods based on the use of photometry and LiDAR sensors [5,6]) to satellite altimetry [7,8,9].

However, all methods have their own specific limitations. These limitations are especially significant when surveying the surfaces of underwater relief of reservoirs [10]. The most important of these limitations is the accuracy of measurements and the resulting accuracy of the 3D model.

The work is devoted to three main research questions:

• firstly, what is the real situation with the Karatomar reservoir after 78 years of use (as well as other reservoirs of the cascade);
• secondly, how accurately do the bathymetric studies provide an assessment of the bottom relief of the reservoir after interpolation of bathymetric data;
• thirdly, with what step of bathymetric drone trajectory lines and with what required density of measurements should bathymetric assessment of plain reservoir bowls (cascade of reservoirs on the Tobol River) be carried out to achieve the required accuracy of 3D modeling.

To solve the tasks, the bathymetric assessment survey data of the regulated volume of the Karatomar reservoir over an area of 61 km² was used. The preliminary bathymetry of the reservoir was carried out in July 2024. The calculation of the required research density, detailed bathymetry, and construction of a digital model were completed in August 2024.

Planning of the measurement density (frequency of bathymetric drone tacks or trajectory lines) in the study was carried out based on the data of the reservoir reference polygon by assessing the accuracy of the obtained models (models with different steps of bathymetric drone tacks or trajectory lines). Data interpolation was carried out using three different methods: linear interpolation, planar regression, and simple kriging. Based on the results of assessing the accuracy of the methods, kriging was chosen to build the final 3D model of the reservoir.

The reliability of the results of calculating the dependence of the required measurement density on the parameter of the specified accuracy of modeling was assessed by comparing the accuracy of various models for the test site of the Ayat River (“bay”). The reliability of the research results was checked by comparing the obtained bathymetry/modeling data with the data of control points obtained by manual measurement methods.

The approach proposed is a reliable solution for planning the required bathymetry accuracy by determining the required measurement density. From a practical point of view, the approach allowed establishing more reliable values of the water level-area-volume ratio for the Karatomar Reservoir of the Republic of Kazakhstan. The approach implies the possibility of replicating the research methods for any lowland reservoirs when assessing/forecasting their economic and environmental parameters within the framework of sustainable development policy.

2. Materials and Methods

2.1. Object of Study

The object of the study was the Karatomar reservoir, located in the Kostanay region of the Republic of Kazakhstan. The reservoir was created at the confluence of the Tobol and Ayat rivers. The reservoir is a valley-type hydrotechnical structure. The design parameters of the reservoir in 1961 were: area 88.0–93.7 km², total volume 791 million m³, useful volume—562 million m³, length—72 km, width—4 km, and depth up to 19.8 m.

The reservoir is subject to natural aging processes. At the present time, the issue of updating its real parameters is important.

This algorithm, which is the first in a series of surveys, includes 6 stages (Figure 1).

2.2. Obtaining Initial Data

To understand the processes occurring in the reservoir and to effectively manage it, accurate and up-to-date 3D models constructed based on the research results play a key role. The research methods remain bathymetric studies (obtaining the bottom relief) and studies of the coastline relief [5].

Modern bathymetry methods offer various computerized methods for collecting a large array of primary data, ranging from collecting field test data to satellite altimetry.

Manual methods of depth measurement are currently practically not used due to the low accuracy of the digital models obtained on their basis and the significant time costs of personnel to obtain depth data. Automated bathymetry methods, also based on fieldwork on reservoirs, are built on the use of such tools as single- and multi-beam sonars, depth gauges, and LiDAR sensors [6]. The methods provide high measurement accuracy but also require significant time to obtain the required amount of data during fieldwork. In the pool of satellite remote sensing methods for surveying the relief of a reservoir’s water area, satellite altimetry [1,7], the Shuttle Radar Topography Mission (SRTM) method based on the use of an interferometric radar with a synthetic aperture [8], and the method based on the use of space-based thermal emission and reflection radiometers (ASTER), which relies on pairs of stereoscopic images [9], have recently been singled out. However, these remote methods also have certain limitations. These limitations are especially significant when surveying the surfaces of underwater relief of reservoirs [10].

The fact that it is impossible to study the surfaces of underwater relief makes remote sensing methods of little use for calculating the actual volume of reservoirs. Given the problems voiced, accurate bathymetric data are currently still obtained through more tedious fieldwork on-site using tools such as sonars and LiDAR sensors [10]. Bathymetric measurements at the research site were carried out in July–August 2024. Bathymetric studies were conducted using an Apache 3 autonomous bathymetric drone with a single-beam, single-frequency echo sounder. The technical characteristics of the bathymetric drone and its echo sounder are given in

Table 1. Main characteristics of the Apache 3 autonomous bathymetric drone.

Parameter Name	Value
Echo sounder
Measured depth range, m	from 0.15 to 200
Echo sounder operating frequency, kHz	200
Echo sounder resolution, m	0.01
Echo sounder beam width, °	6.5 ± 1
Root mean square error (RMSE) of depth measurements, m	0.01 + 0.001·H, where H is the measured depth in cm
Location
Number of channels	624
GNSS	GPS NAVSTAR: L1C/A, L1C, L2C, L2P, L5 ΓЛOHACC: L1C/A, L1P, L2C/A, L2P BeiDou: B1, B2, B3 Galileo: E1, E5A, E5B SBAS: WAAS, EGNOS, MSAS, QZSS, GAGAN
RMSE	RTK in plan 8.0 mm + 1.0 mm/km
RMSE	RTK in height 15.0 mm + 1.0 mm/km
RMSE DGPS in plan	0.25 m
RMSE DGPS in altitude	0.5 m
Heading accuracy	0.2° per 1 m baseline
Inertial navigation stability	6° per hour

.

The data from control measurements of the reservoir depths for verification of the results of automated bathymetric measurements/modeling were obtained using a line and a hydrometric rod.

The reliability of the data was analyzed by comparing the measurement results using two methods (manual and using a single-beam echo sounder).

To adjust the accuracy of the bathymetric drone echo sounder, the echo sounder corrections were determined in the water area of the Karatomar Reservoir Bay using the calibration method. The work was carried out using a calibration disk and a hydrometric rod. The accuracy was adjusted by setting the parameters at depths of 1, 3, and 5 m of this reservoir (reservoir bay), plus an additional 10 and 15 m for the reservoir itself.

For additional verification of both the measurement results and 3D modeling, the authors manually collected data on the depths of the reservoir control points using a line and a hydrometric rod. The data from ultrasonic measurements using a bathymetric drone and manual measurement methods coincided with an error of less than 0.1 m. The deviation of the modeling data from manual measurement methods also did not exceed the required 10%. Which confirmed the success of the drone calibration and verified the accuracy and reliability of the measurements/modeling.

The bathymetric drone sonar passed state verification in 2024. The device is certified for use throughout the Republic of Kazakhstan.

To obtain maximum accuracy of primary data and increase the efficiency of using the bathymetric drone, georeferencing was carried out using DGPS. The CNCNAV GPSS station was used. The station received a certificate of conformity from the Republic of Kazakhstan.

The bathymetric drone’s movement routes were built in the AutoPlanner 2.0.11.9341 software product. The WGS84 coordinate system was used. Preliminary processing of the obtained bathymetric data was carried out in the freely distributed GIS package QGis v 3.34.11. In this package, sections were obtained, and a digital model of the reservoir bowl was processed.

The water line was constructed using the superimposed Sentinel-2_L2A_True_color image from 4 July 2024, corresponding to the reservoir bowl being 98% full (according to the Kazvodkhoz Fund). Below is a comparison of the quality of the available space images from 14 June 2024 and 4 July 2024 with a resolution of 30 m/pixel. The quality of the images is determined by the cloudiness on the day of the space survey above the research object. The image from 4 July 2024 has the best quality indicators (Figure 2).

The decrease in the area of the water zone in the Sentinel-2_L2A NDWI (Normalized Difference Water Index) image compared to Google and Sentinel-2_L2A_True_color images is caused by significant overgrowing of the riverbed with reeds within the boundaries of the beginning of the reservoir (despite the maximum rise in the water level in the reservoir over the past 10 years). The length of the water edge line was 93.77 km, which, with a photograph resolution of 30 m/pixel, corresponds to the formed error in delineating the area of the reservoir mirror of no more than 5.63 km².

2.3. Methods of Processing Initial Bathymetric Information

All used digital models of water bodies are based on deterministic and geostatic interpolation methods [11]. Recently, in contrast to the previously dominant methods of mathematical one-dimensional interpolation, multidimensional interpolation methods and interpolation methods based on the use of artificial intelligence apparatus are increasingly used [12].

Firstly, when using interpolation algorithms, a distinction is made between interpolation on a regular [13] and irregular data grid [14,15]. The latter, although more stringent to the interpolation apparatus, are still less demanding on the array of initial data. This allows, in the process of research, the transfer of part of the complexity from fieldwork to office post-processing of data when modeling the object under study, which is even more convenient. Deterministic interpolation methods generate surfaces from discrete measurement values (arrays of data points) based on either the degree of similarity (inverse distance weighted methods) or, for example, the level of smoothing (radial basis functions) [16]. Geostatistical interpolation methods (variations of kriging algorithms, etc.) [17] exploit the statistical properties of the measured points. Geostatistical methods are more flexible with respect to the input data, but within a family of methods such as kriging, different requirements are put forward for the conditions that must be met for the output results to be acceptable. Disjunctive kriging and empirical Bayesian kriging generally give smaller absolute interpolation errors for water bodies than universal kriging and simple kriging [17,18]. However, ref. [18,19] in their studies note that the deterministic radial basis function of interpolation in some cases is comparable to kriging both in accuracy and in productivity.

Interpolation methods based on artificial intelligence apparatus are often reduced to neural network methods [20,21] (multilayer perceptron (MLP) [22,23], radial basis function of neural network (RBFN) [24], back propagation network (FNN) [25], deep neural network (DNN) [26], and recently investigated the applicability of capsule neural networks (CapsNet)) [27].

Interpolation by the initial data and modeling of the surface of the reservoir bowl and the bay are performed in the Surfe software product (version 1.7.2). Data interpolation was carried out using three different methods: linear interpolation, planar regression, and simple kriging. Based on the results of the assessment of the accuracy of the methods, kriging was selected for constructing the final 3D model of the reservoir.

The standard mean square error (MSE) and maximum relative deviation (MRD) are selected as the tools for assessing the quality of interpolation.

MSE allows you to assess the quality of the approximation over the entire interval.

$$MSE={\displaystyle \frac{{\displaystyle \sum _{i=1}^N{(h_{Original}-h_{Forecast})}^2}}{N}}$$

(1)

MRD gives peak deviations and allows you to assess the quality of the approximation at the worst point.

$$MRD=\max \left(\left|{\displaystyle \frac{h_{Original}-h_{Forecast}}{y_{Original}}}\right|\right)$$

(2)

3. Results

3.1. Analysis of Potential Methods for Studying Reservoir Bowls

To understand the capabilities of various research methods within the framework of the task, an analysis of various modern methods of digital modeling of the underwater relief of water bodies was carried out (Table 2).

Manual methods, such as compass survey and visual survey in topographic methods, were not considered due to the fact that at present (like most manual methods) they are gradually falling out of use [28].

Initial attempts to determine the underwater relief of water bodies using remote sensing methods were based on the methods [29,30,31,32]. The issues of modern successful remote sensing methods for assessing the bathymetry of water bodies are systematized in the works [6,8,9,33].

Systematizing the results of the analysis of space remote methods, it can be said that the existing methods for assessing the state of water bodies can be conditionally presented in the form of two large groups [10,34]. The first category combines methods of processing satellite images to track changes in the area of the object’s water surface using radar altimetry. These include methods:

• MODIS [33],
• Landsat [35],
• Sentinel-2 [35].

This group of high-precision methods, however, has limited applicability due to the impossibility of scanning the underwater relief.

The second group of methods is based on an attempt to reconstruct the bathymetry of the reservoir [5,36], which allows us to estimate the full relief of the object. However, this group of methods often relies on overly simplified assumptions for reconstructing the bathymetry, which leads to a decrease in accuracy in conditions of complex reservoir relief [10].

Table 2. Comparative review of methods of bathymetric and topographic research of reservoirs based on literary sources.

Link	Name/Brief Description of the Method	Required Equipment	Advantages	Disadvantages
Manual methods of depth measurement and topographic survey of the relief of the coastal zone of the reservoir
-	The method is based on measuring the length of the released lead line from the side of the vessel.	Hand lot, vessel	Affordability of single measurements	Low speed and accuracy of water body depth measurements
[28]	Theodolite survey	Theodolite, steel measuring tape (or optical rangefinder)	Affordability of single measurements	Low measurement speed, Limited sizes of theodolite traverses (polygons), Difficulty of carrying out work on hard-to-reach riverbed slopes
	Tacheometric survey	Tacheometer
	Tablet survey	Measuring plate		Difficulty of carrying out work on riverbed slopes
	Surface leveling (vertical or altitude survey)	Level	High precision	Need for additional planning and reference work of reference points
	Phototheodolite survey of coastline relief (terrestrial)	Phototheodolite
Methods of automated bathymetry and topographic survey of the relief of the coastal zone of the reservoir
[37]	Hydroacoustic sounding of the underwater part of the reservoir	Echo sounder, underwater measurement vessel/drone, attitude sensors	Possibility of capturing large volumes of data with high accuracy	Impossibility of taking measurements on overgrown areas of the reservoir
[38,39]	Phototheodolite survey of the coastline relief (aerial photography)	Aircraft, aerial camera	Possibility of obtaining plans of large areas.	Need for additional planning and reference work of reference points Sensitivity of measurement accuracy to the density of vegetation cover.
[40]	Laser scanning of the coastline relief	LiDAR, optional aerial survey aircraft + attitude sensors	Obtaining topographic plans of complex profiles. High accuracy. Measurement speed.	Sensitivity of measurement quality when working with reflective surfaces
Methods of remote satellite sensing and digital modeling of water bodies
[33,35]	Group of space radar altimetry methods	Requires satellite LiDAR data and field survey data to determine depth	Obtaining topographic plans of complex profiles. High accuracy. Measurement speed	Dependent on cloudiness and weather conditions. The complexity of constructing a model of the underwater part of a water body with high accuracy
[5,36]	A group of methods for reconstructing and forecasting bathymetry using space data and global models.	Space photography data is required	Possibility of constructing the full relief of a reservoir bowl	Often does not take into account the complex bathymetry of a water body

Table 2. Comparative review of methods of bathymetric and topographic research of reservoirs based on literary sources.

Link	Name/Brief Description of the Method	Required Equipment	Advantages	Disadvantages
Manual methods of depth measurement and topographic survey of the relief of the coastal zone of the reservoir
-	The method is based on measuring the length of the released lead line from the side of the vessel.	Hand lot, vessel	Affordability of single measurements	Low speed and accuracy of water body depth measurements
[28]	Theodolite survey	Theodolite, steel measuring tape (or optical rangefinder)	Affordability of single measurements	Low measurement speed, Limited sizes of theodolite traverses (polygons), Difficulty of carrying out work on hard-to-reach riverbed slopes
	Tacheometric survey	Tacheometer
	Tablet survey	Measuring plate		Difficulty of carrying out work on riverbed slopes
	Surface leveling (vertical or altitude survey)	Level	High precision	Need for additional planning and reference work of reference points
	Phototheodolite survey of coastline relief (terrestrial)	Phototheodolite
Methods of automated bathymetry and topographic survey of the relief of the coastal zone of the reservoir
[37]	Hydroacoustic sounding of the underwater part of the reservoir	Echo sounder, underwater measurement vessel/drone, attitude sensors	Possibility of capturing large volumes of data with high accuracy	Impossibility of taking measurements on overgrown areas of the reservoir
[38,39]	Phototheodolite survey of the coastline relief (aerial photography)	Aircraft, aerial camera	Possibility of obtaining plans of large areas.	Need for additional planning and reference work of reference points Sensitivity of measurement accuracy to the density of vegetation cover.
[40]	Laser scanning of the coastline relief	LiDAR, optional aerial survey aircraft + attitude sensors	Obtaining topographic plans of complex profiles. High accuracy. Measurement speed.	Sensitivity of measurement quality when working with reflective surfaces
Methods of remote satellite sensing and digital modeling of water bodies
[33,35]	Group of space radar altimetry methods	Requires satellite LiDAR data and field survey data to determine depth	Obtaining topographic plans of complex profiles. High accuracy. Measurement speed	Dependent on cloudiness and weather conditions. The complexity of constructing a model of the underwater part of a water body with high accuracy
[5,36]	A group of methods for reconstructing and forecasting bathymetry using space data and global models.	Space photography data is required	Possibility of constructing the full relief of a reservoir bowl	Often does not take into account the complex bathymetry of a water body

The issues of methodology of “gluing” interpolation data of underwater and coastal relief in the construction of digital maps of water bodies are considered [41,42].

The key point in the construction of 3D models adequate to the tasks remains the selection of interpolation methods optimal for a given water body and dataset and the determination of the minimum sufficient number of points of these initial data. The use of overly simplified approaches for interpolation can nullify the research results due to a significant loss of accuracy.

One of the most important problems of geodetic research in the study of a water body using bathymetry data [15,28] is the definition of the purpose of developing an interpolation model.

The tools of deterministic interpolation and geostatistical analysis offer many different interpolation methods, each of which has its own unique features and often gives significantly different results.

Firstly, all methods used in geodata interpolation can be divided into exact (ORB, RBF) and inexact—most often stochastic (LPI, KSB, Kriging). In exact rigid methods, at each output position the interpolated surface will have exactly the same value as the value of the input data, while some are non-rigid, with the possibility of deviation of the interpolated value from the value determined at a given point [43]. It should also be taken into account that for some decisions it is important to consider not only the interpolated values but also the uncertainty (variability) associated with this interpolation [44]. Interpolation methods also differ in complexity, which can be measured by the number of assumptions that must be made to check the model, and in computational complexity, which is characterized by the volume of mathematical transformations and elementary machine operations. In connection with the conducted analysis for interpolation of bathymetric data of the reservoir, in our opinion, the method of linear interpolation and kriging (more precisely, the method of simple kriging) is of interest.

The method of linear interpolation is the simplest and most frequently used. The method does not depend on the number of nodes (when adding nodes, there is no need to recalculate all the values). Of the obvious advantages, it is necessary to note fairly accurate results in cases where the change in the values of the interpolated depths is relatively uniform and predictable.

However, the method of linear interpolation has a significant drawback—the graph of the function is not smooth and often significantly does not coincide with the true values of the depths.

In comparison with the studied methods, kriging gives the best linear unbiased prediction of intermediate values with the correct chosen a priori assumptions. Also, kriging minimizes the variance of the measurement error.

Thus, firstly, kriging provides the required smoothness of the interpolated relief, typical for slowly flowing flat rivers; secondly, the ability to take into account potential noise and missing data; and, thirdly, it has acceptable indicators of computational complexity (the maximum time of bathymetric calculations on a computer based on an Intel i7-13700 K processor with 8 GB of RAM did not exceed 30 min).

3.2. Analysis of Hydrogeology of the Tobol and Ayat Rivers in the Karatomar Reservoir Zone

The Karatomar reservoir is located at 52°53′40″ N 63°01′45″ E in Kazakhstan, in the Beimbet Mailin district of the Kostanay region (Figure 3). The need for a large reservoir for the sustainable development of this territory was due to the development of the Sokolov, Sarbay, and Kachar iron ore deposits and the need to provide water to the cities of Rudny and Kachar under construction [45].

The reservoir is located in the valley between the Trans-Ural and Turgay plateaus. It was formed by a dam built in 1966 and filled with water in 1967–1969.

The reservoir was built on the Tobol River, and the Ayat River flows into the reservoir from the west. A valley-type reservoir with the following planned parameters: area 88.0–93.7 km², total volume 791 million m³, useful volume—562 million m³, length—72 km, width—4 km, depth up to 19.8 m, water discharge for river flooding 1.3 m³/s. In 2024, when the reservoir was 99% full during the spring flood due to melting snow, the emergency discharge of the reservoir exceeded 3250 m³/s. The catchment area of the reservoir is 28.5 thousand km², and the length of the coastline is 102 km. A large tributary is the Ayat River. The reservoir was built using the long-term flow regulation type. Intra-annual fluctuations in water level reach 11 m. Water mineralization is 200–400 mg/L; the waters belong to the sodium hydrocarbonate type. The operating mode of the Karatomar reservoir is intermediate-flow, as it is controlled by the volume of filling and discharge from the Verkhnetobol reservoir.

The reservoir is an additional flow regulator and a reserve reservoir for replenishing the downstream flow reservoirs, as well as a source of drinking water supply for a number of settlements connected to the Kostanay water pipeline. In addition to water supply for settlements, water is used for irrigation of agricultural lands and fisheries.

Water exchange in the Karatomar reservoir is more active than in the Verkhnetobol reservoir due to the constant inflow of relatively fresh water from the Ayat River and constant discharge for flooding the Tobol River and replenishment of the downstream Sergeev and Amangeldy reservoirs. Filling of the reservoirs is carried out mainly in the spring months: April, May, due to the runoff of the spring flood.

The accumulative type of the reservoir relief was represented by gently sloping and horizontally stepped surfaces of terraced plains, the morphological expression of which is very weak. By genesis, the accumulative plain is alluvial and is found along rivers.

The Tobol River flows through the territory of Kazakhstan and Russia and is the left and most abundant tributary of the Irtysh River. The length of the river is 1591 km, and the area of the drainage basin is 426,000 km². The difference in altitude between the source and the mouth is 237 m. The location of the Karatomar Reservoir under study is 1257 km from the Tobol River (the confluence with the Ayat River). The Tobol is formed at the confluence of the Bozbie River with the Kokpektysay River on the border of the eastern spurs of the Southern Urals and the Turgai tableland of the country (51°28′00″ N 61°00′31″ E). The middle and lower reaches of the river are within the West Siberian Plain in a wide valley with a winding channel. The right bank rises above the left, as the Tobol flows over a deep fault in the earth’s crust and separates the Kurgan synclinorium and the Tobol-Ubagan uplift.

The relative height of the floodplain terraces above the water level increases downstream from 12 to 15 m in the area of the Karatomar reservoir to 35 m north of the city of Kostanay.

The maximum width of the floodplain terrace in the area of the reservoir reaches several kilometers. This terrace is separated from the floodplain surface by a sharply expressed ledge.

The riverbed in the area of the studied reservoir is located in a wide floodplain composed of modern sandy deposits. The width of the channel is 10 to 50–100 m, and the depth is 4–8 m. The current in low water is 0.1 m/s, in flood up to 3 m/s, and the slope is 0.0003. The left bank of the river is often steep. The riverbed is winding with meanders and oxbow lakes that are flooded only during floods, almost everywhere surrounded by willow and reed thickets. The river flow is regulated by reservoirs. The river is fed mainly by snow, while downstream the share of rain increases. The flood occurs from the first half of April to mid-June in the upper reaches and until the beginning of August in the lower reaches. The average annual water flow is 26.2 m³/s in the upper reaches (898 km from the mouth) and 805 m³/s at the mouth (maximum 348 m³/s and 6350 m³/s, respectively). Average turbidity is 260 g/m³, and annual sediment runoff is 1600 thousand tons. It freezes in the lower reaches in late October–November and in the upper reaches in November and opens up in the second half of April–the first half of May.

As a result of the above, it can be said that the hydrogeology of the Tobol River includes the following aspects:

• the hydromorphological situation is represented by alternating shallow riffles with shallow and medium-depth pools. The depth of the pools can reach up to 5 m, and in some cases up to 10 m or more;
• the river bottom is sandy-silty, rocky in places;
• the width of the channel varies from 50 to 400 m;
• the banks are mainly loamy, overgrown with small bushes, and slightly intersected by dry stream beds. The banks are steep, in places precipitous, 5–6 m high, and at the confluence with the slopes of the valley, they reach up to 30 m;
• snow waters are predominant in the river’s nutrition (70–90%). In winter, rivers are fed by underground waters, in summer—by underground waters, less often by rain;
• the water regime is characterized by a pronounced spring flood (up to 85–96% of the annual flow) and a long low water period;
• the mineralization of water during the spring flood period is 100–200 mg/L, and the hardness is 0.5–1.25 millimoles/L. In summer, the mineralization of water increases, and the water becomes sulfate or weakly hydrocarbonate.

The Ayat is a river in Russia and Kazakhstan, flowing through the Varna district of the Chelyabinsk region of Russia (22 km from the source) and the Taranovsky, Fyodorovsky Kostanay districts of the Kostanay region of Kazakhstan (95 km). The Ayat is formed by the confluence of the Karataly-Ayat and Archagli-Ayat rivers. The river flows into the Tobol River in the area of the Karatomar reservoir. The length of the river is 117 km, and the area of the drainage basin is 13,300 km².

Geomorphological features of the Ayat River:

• the soils of the basin are mainly sandy and loamy, sometimes salty. Alluvial channels are located in a well-defined river valley;
• the channel is gently winding, stretching, and located in a well-defined river valley;
• the bottom of the river is sandy-silty, and rocky in places;
• the width of the riverbed varies from 5 to 20 m;
• the maximum water depth is 2 m;
• the water regime is unstable, almost the entire annual flow occurs during the spring flood;
• due to large fluctuations in the water level, there are spits, islands-middens, and shoals on the river.

A significant part of the Ayat River basin is formed by 383 endorheic lakes (total area 208 km²). During the observation period, the river froze to the bottom 5 times in winter.

3.3. Evaluation of the Boat’s Tack Pitch to Ensure the Required Measurement Accuracy

When conducting a study of a reservoir using the instrumental bathymetry method, the issue of choosing the measurement step (in the case of hydroacoustic measurements—the tack step of the vessel for taking points) is important for achieving the required accuracy. In terms of the measurement step on the course, such a problem does not arise, since the vessel’s equipment made it possible to obtain points with an excess density—more than 10 points per meter of the distance traveled. Due to the rather difficult hydrometeorological conditions at the time the research began (July 2024 was characterized by average wind speeds from 13 to 25 m/s, or from 6 to 8 points on the Beaufort scale [46]), it was decided to conduct an initial assessment of the required tack step of the bathymetric flights of the bathymetric drone to ensure the accuracy of measurements in the bay of the reservoir (more precisely, at the mouth of the channel of a seasonally drying-up stream flowing into the reservoir and called the “bay” by the local population). The “Bay” (Figure 4) was chosen as a reference site for bathymetric studies.

The “Bay”, like the reservoir, is also a flowing one, but the flow rate and (most importantly) the wave height are significantly less than in the main basin of the reservoir, which made it possible to reduce risks and facilitate bathymetric surveys. To assess the applicability of the proposed method for calculating the optimal pitch of boat tacks or trajectory lines, an assessment was made of the correlation of the bottom profile of the bay with the bottom profile of the Tobol and Ayat reservoir channels [47].

Initially, data on the bay were taken from the entire area of the reservoir with a tack step of 50 m across the current. Only the shallow upper part (about 10% of the area), abundantly overgrown with reeds, remained unexplored.

The studied object (bay) is characterized by a length of about 900 m, a width of 100–140 m, and a depth interval of up to 6.4 m. The slope of the bay is about 7.11‰.

Data interpolation for building a digital model and constructing a bathymetric map was performed using 3 different mathematical methods: linear interpolation (mean segment method), linear regression Z = AX + BY + C, and simple kriging.

The isobath map of the bay is shown in Figure 5.

The results of the comparison of methods are presented in

Table 3. Estimation of bay depths by different interpolation methods.

Parameter	Depth (Minimum)				Depth (Maximum)
Run step	10	25	50	100	10	25	50	100
Interpolation method
Linear	0.800	0.400	0.130	1.540	5.540	5.250	5.490	5.360
Planar Regression: Z = AX + BY + C	0.1353	0.009	0.016	0.001	3.054	1.073	1.096	0.707
Kriging	1.573	1.141	1.007	2.727	5.550	5.419	5.919	5.348

. The results of the comparison of methods based on the processing of primary flood data with the summation of data from all runs allow us to state that the regression method for this dataset gives the worst result. Kriging with an increase in the step reduces the error value, which characterizes it as more preferable.

To determine the optimal tack pitch of the boat for bathymetric studies, a rectangular polygon was selected in the center of the bay near the Parallel tourist base (Figure 6).

The size of the additional research polygon in the bay is 80 × 130 m, where 130 m is the side across the bay bed. The minimum measured depth according to the bot is 3.55 m, and the maximum is 4.98 m.

Based on the results of measurements in the polygon of the section locations, the bay bottom slope in the direction of the current is 2.6–5.8‰, across the bay channel—6.08‰ towards the left bank.

Based on the relief shape of the sections, it is clear that with an increase in the track pitch of the bathymetric boat to 50 m, local extrema of the bay bottom (an unidentified submerged object) begin to be lost. Although with a fairly flat bottom and the absence of underwater objects, the Pearson correlation coefficient is about 0.9.

The constructed depth profiles for sections with different steps are shown in Figure 7, Figure 8, Figure 9 and Figure 10.

The accuracy of the interpolation methods was assessed based on identifying the method with the lowest MSE. The analysis data are summarized in

Table 4. Estimation of average depth and error of interpolation methods.

Parameter	Depth (Mean)				MSE (Relative Mean Diff)
Run step	10	25	50	100	10	25	50	100
Interpolation method
Linear	4.225	4.216	4.274	4.301	0.112	0.106	0.105	0.161
Planar Regression: Z = AX + BY + C	2.245	0.313	0.320	0.288	0.164	0.359	0.301	0.342
Kriging	4.298	4.261	4.343	4.367	0.119	0.102	0.107	0.105

.

The obtained data array was evaluated within the confidence interval. To improve the quality of the initial data, preliminary processing was carried out, and pulsation data that did not satisfy the 3-sigma rule were removed from the input array (the probability of erroneous removal of a given pulsation does not exceed 0.28%).

The depth values for the investigated areas with different steps are presented in

Table 5. Values of interpolated depths for sections of the bay polygon.

	Section P1				Section P2				Section P3
Run step	10	25	50	100	10	25	50	100	10	25	50	100
Point	Depth
0	3.82	3.79	3.80	3.81	4.18	4.18	4.18	4.18	4.51	4.49	4.52	4.52
10	3.79	3.78	3.80	3.81	4.22	4.22	4.22	4.19	4.52	4.52	4.52	4.51
20	3.55	3.68	3.79	3.8	4.27	4.23	4.23	4.19	3.33	4.52	4.58	4.60
30	3.76	3.69	3.74	3.83	4.27	4.24	4.25	4.24	4.61	4.55	4.65	4.71
40	3.87	3.85	3.72	3.86	4.27	4.26	4.26	4.28	4.57	4.63	4.69	4.82
50	3.90	3.89	3.84	3.89	4.27	4.30	4.28	4.29	4.64	4.71	4.71	4.92
60	3.94	3.94	3.87	3.93	4.29	4.32	4.31	4.31	4.73	4.74	4.80	4.99
70	4.00	4.01	4.00	4.00	4.32	4.32	4.33	4.33	4.83	4.81	4.74	5.03
80	4.03	4.04	4.02	4.04	4.33	4.36	4.34	4.36	4.98	4.99	4.67	5.03

.

The data for cross-section P4 were obtained in a similar manner.

In order to prove that the curves representing the bottom relief geometry for different runs are similar, in addition to the correlation coefficient, it was decided to use the curve similarity coefficient. For this purpose, the probability that one relief cross-section can be transformed into another by scaling along the depth axis was checked, i.e., to check that the sections are geometrically similar.

For this purpose, a modified formula was used to prove the similarity of curves:

$$y_1=k\cdot y_2$$

(3)

where y₁ and y₂—are the depths of the reference and compared sections;

k—is the curve similarity coefficient.

All linear dimensions of one curve should be proportional to similar dimensions of another curve with the same similarity coefficient. Then, mathematically, this can be expressed through the correspondence of the coordinate points of the two curves:

$$k=\min (y_1/y_2)$$

(4)

For a set of cross-section points of two reliefs, the authors selected the worst similarity coefficient, which characterizes the worst similarity of the possible options in the comparison area. And the run with a step of 10 m is again used as a reference. The results are shown in

Table 6. Pearson correlation coefficient and similarity coefficient of relief curves, MSE and MRD measurements from the step of runs.

Parameter	P1			P2			P3			P4
Run step	25	50	100	25	50	100	25	50	100	25	50	100
MSE	0.00	0.01	0.01	0.00	0.00	0.00	0.16	0.19	0.21	0.00	0.00	0.01
MRD	0.02	0.04	0.00	0.01	0.01	0.02	0.01	0.06	0.00	0.00	0.01	0.04
Pearson correlation coefficient	0.93	0.69	0.85	0.90	0.94	0.87	0.60	0.44	0.57	1.00	0.99	0.96
Similarity coefficient	0.99	0.97	0.96	0.99	0.98	0.99	0.99	0.94	0.92	0.98	0.97	0.95

.

Thus, based on the calculations performed (Figure 11), it can be stated that in order to ensure the declared accuracy of at least 10%, it is necessary to conduct a field survey of the reservoir water area with a tack step of no more than 200 m. To verify the results, the authors carried out bathymetry of individual polygons of the Ayat Riverbed in the area of the village of Naberezhnoye with a tack density of 50, 100, 200, 300, 400, and 500 m (the data for which were not included in the final 3D model of the reservoir). Analysis of the accuracy of the data obtained confirmed the previously obtained conclusions about the minimum required density of measurements with a step of no more than 200 m.

3.4. Evaluation of the Compliance of the Tobol River Bed with the Conditions Taken as the Standard “Bay”

To check the correspondence of the reservoir to the bay, a comparative analysis of the obtained average sections of the bay with the obtained average sections of the reservoir was carried out (

Table 7. Interpolation data and Pearson correlation coefficients for sections P1 (Ayat), P2 (Tobol), and three sections (P1b Z2b Z3) of the reservoir “bay”.

		Depth, m									Pirson Ayat	Pirson Tobol
Distance, m		0	10	20	30	40	50	60	70	80
r. Ayat	P1	3.55	−3.55	3.56	3.56	3.56	3.57	3.57	3.57	3.58
r. Tobol	P2	5.44	5.45	5.45	5.46	5.46	5.47	5.47	5.48	5.48
“Bay”	P1	3.82	3.79	3.55	3.76	3.87	3.90	3.94	4.03	4.00	0.74	0.74
	P2	4.18	4.22	4.27	4.27	4.27	4.27	4.29	4.32	4.33	0.93	0.93
	P3	4.51	4.52	3.33	4.61	4.57	4.64	4.73	4.83	4.98	0.54	0.54

). The correlation coefficient and the curve similarity index were used for the assessment.

The layout of profiles P1-P4 on the Karatomar reservoir scheme and the geometry of the cross-section of the Tobol and Ayat river beds at their confluence (profiles P3 and P4) are shown in Figure 12.

The constructed profile P1 along the Ayat riverbed allows us to speak of a bottom slope of 0.5‰ (Figure 13). The bottom slope of the bay exceeds the slope of the Ayat River within the reservoir by 5 times. The relief is rich in whirlpools. The characteristic repeating peaks of depth minima are most likely caused not by shallows but by an error in the interpolation method due to insufficient initial data (gaps between polygons, which explains their periodicity, coinciding with the zones of absence of drone tacks or trajectory lines) and require additional field bathymetric surveys (planned for the warm period of summer 2025).

According to the table data, we see a high correlation between the values of the bottom relief (depths) in the middle of the Ayat River (within the boundaries of the reservoir) and the depths in the middle (flowing part of the bay), which allows us to talk about the applicability of the proposed method for assessing the accuracy of the tack pitch for the Ayat River.

The constructed profile along the Tobol River bed allows us to state that the current bottom slope is 0.8‰ (Figure 14). The bottom slope of the bay exceeds the slope of the Tobol River within the reservoir by 3 times. The relief is quite smooth. The existing peak may also correspond to a break in the drone’s measurement runs (additional field bathymetric surveys are required, planned for the warm period of 2025).

The correlation of the Tobol River data with the depths of the bay is identical to the Ayat data.

Based on the results of the studies, it can be stated that for riverbeds with smoothed bottom reliefs corresponding to the flow of the Ayat and Tobol Rivers within the boundaries of the Karatomar Reservoir, the dependence of the measurement error on the tack pitch is of the following nature:

$$\delta \approx 0.0092\cdot L+0.0455$$

(5)

where L is the tack pitch across the river flow.

3.5. Modeling the State of the Kartomar Reservoir

As noted above, the zero level of the reservoir was constructed using the superimposed Sentinel-2_L2A_True_color image from 4 July 2024 (corresponds to 98% filling of the reservoir volume). The processing of photographs from the Sentinel satellite only, carried out by the authors, made it possible to achieve the required accuracy. If necessary, the shoreline can be refined using the natural discontinuities method (Jenks method) [48].

The constructed map of the Karatomar reservoir in QGis based on bathymetric measurements performed with a step of 200 m is shown in Figure 15.

Based on the constructed 3D map, reservoir depth polygons were modeled with a depth step of 1 m (if necessary, it is possible to construct a map with any depth step) (

Table 8. Results of interpolation of depth polygons of current values of the Karatomar reservoir.

Depth, m	Mirror Area by Depth		Bowl Volume by Depth
	km2	% of Maximum	km3	% of Maximum
0	61.466	100.00	367.049	100.00
1	55.254	89.89	305.582	83.25
2	50.716	82.51	250.329	68.20
3	45.473	73.98	199.613	54.38
4	38.947	63.36	154.140	41.99
5	31.860	51.83	115.192	31.38
6	24.899	40.51	83.332	22.70
7	20.794	33.83	58.433	15.92
8	16.750	27.25	37.638	10.25
9	11.618	18.90	20.889	5.69
10	5.739	9.34	9.271	2.53
11	2.137	3.48	3.531	0.96
12	0.878	1.43	1.394	0.38
13	0.309	0.50	0.516	0.14
14	0.129	0.21	0.208	0.06
15	0.061	0.10	0.079	0.02
16	0.018	0.03	0.018	0.00

). The reservoir mirror area and current (corresponding to a filling of 98% as of 14 July 2024) reservoir volumes within the boundaries selected from the space image were also calculated.

A comparative analysis of the planned indicators of the reservoir (for the period 1966) and those obtained as a result of the bathymetric study is given in

Table 9. Results of comparison of the main planned and current values of the Karatomar reservoir.

Parameters	Data 1966	Data 2024	Deviation
Reservoir volume,	791 mil. m3 (when the bowl is 100% full	367 mil. m3 (at 98% bowl filling	−53.60%
Useful reservoir volume,	562 mil. m3	Not rated	-
Water surface area,	93.7 km2	61.47 km2 (based on digitized polygon) ± 5.63 km2	−34.44 ± 6.01%
Maximum depth, m	19.8	16.0	−18.99%

.

In our opinion, significant changes in the reservoir parameters are related to silting of the reservoir bowl (over the past 25 years, there has been a significant decrease in the discharge of floodwaters, which did not contribute to the natural washing of the Tobol and Ayat riverbeds in the reservoir bowl from alluvial rocks, and dredgers did not carry out work to clean and deepen the bottom) and overgrowing of the banks with vegetation (the reservoir surface area in the upper part of the Tobol River has been significantly reduced) [49,50]. A study of the bottom sediments of the Karatomar Reservoir is planned for the period spring-summer 2025.

The capacity of the environment to compensate for the impact of the Karatomar reservoir dam on the hydrology of the Ayat and Tobol river basins has been exhausted. For further sustainable development of the region, both in economic terms (preservation of the most important source of fresh water) and in environmental terms (preservation of the region’s biodiversity and the only major waterway), timely continuous monitoring of the bathymetry of hydrotechnical structures, identification of the causes of the reduction in reservoir volumes, and development of compensatory measures are necessary.

4. Discussion

As a result of the research, we received answers to the questions posed.

First, a comparative analysis of interpolation methods on the data of the “bay” of the Karatomar reservoir allows us to state the preference for using the simple kriging method over the linear interpolation and regression methods with an increase in the frequency of the bathymetric drone tack step.

The correlation of the geomorphology parameters of the “bay” with the geomorphology of the Tobol and Ayat rivers in the reservoir zone allows us to state the possibility of transferring the obtained dependencies to the water areas of these rivers.

Achieving the required accuracy of bathymetric studies, with an error of less than 10%, with the current geomorphology of the bottom of the Karatomar reservoir (as a plain-type reservoir with a bottom slope of about 0.5–7.11‰) is possible if the drone tack step does not exceed 200 m. However, even if this condition was met, during the study of the water area, areas with significant deviations in relief were obtained, the clarification of which requires additional field studies.

Secondly, over 78 years of operation, the actual parameters of the Karatomar reservoir have undergone significant changes (the total volume of the reservoir has decreased by more than 50%, the area of the water surface has decreased by more than 30%, and the maximum depth has decreased by more than 18%). A similar picture of a significant deterioration in the characteristics of a hydrotechnical structure is characteristic not only of the Karatomar reservoir but also of the Verkhnetobol reservoir, studied by the authors of the article (it can be assumed that other hydrotechnical structures in the region are in a similar situation).

These changes are presumably caused by significant overgrowing of shallow sections of the Tobol River with reeds in the upper zone of the reservoir and silting of the reservoir bowl.

Such unaccounted changes in the condition of hydrotechnical structures may lead to additional difficulties and risks in managing the 2024 flood, and a significant reduction in water reserves may additionally affect the prospects for sustainable development of the region.

Inaccurate data on the assessment of both freshwater reserves during the dry period and an erroneous idea of the available free volumes of existing reservoirs can lead to significant environmental and socio-economic consequences. Only in the context of the consequences of the flood in the spring of 2024, according to public sources:

• 3.3 thousand affected families received a one-time social payment for a total of 1.1 billion tenge;
• 1149 houses were surveyed and assessed. More than 700 of them were recognized as unsuitable for habitation;
• more than 270 families received compensation for the repair and restoration of their houses for a total of 821.1 million tenge.

The environmental consequences have not yet been fully determined, since the flood zone (in addition to crop fields) also included objects posing significant threats.

In connection with the significant influence of the Karatomar reservoir as one of the main reservoirs of the Kostanay region on the economy of Kazakhstan, we believe that it is necessary to conduct additional research to study the nature of siltation and develop measures to restore the parameters of the hydrotechnical structure under study.

In connection with the significant influence of the Karatomar reservoir as one of the main reservoirs of the Kostanay region on the economy of Kazakhstan, we believe that for the sustainable development of the region, it is necessary to conduct additional research to study the nature of siltation and develop measures to restore the parameters of the hydrotechnical structure under study.