RU2487368C1 - Method for stereophotography of bottom topography of water body and apparatus for realising said method - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: method for stereophotography of the bottom topography of a water body involves moving sonar equipment by a hydrographic ship which is fitted with devices for measuring speed and heading, a depth metre, a receiver-indicator of a satellite navigation system and/or a receiver-indicator of a radio navigation system connected to the ship computer. The sonar equipment is in form of a hydrographic side-scanning echograph which radiates probing pulses and receives signals reflected from the bottom surface, whose intensity is continuously recorded, parallactic shift between corresponding records of images of the bottom topography of the water body on echograms of two loggers and their geodetic coordinates are determined and stereo maps of the bottom topography of the water body are constructed based on the obtained data. A digital map of the bottom relief of the water body is first formed based on archival data. Antennae of the sonar equipment are placed in the vertical plane, each on board of the hydrographic ship. The obtained discrete measurements are used to construct a digital map of the bottom relief; Topographic analysis of the topography is carried out to plot a Kronrod-Rib graph and Morse-Smale complexes for each piecewise linear surface and fractal parametres of the topography are estimated. The apparatus has two receive-transmit antennae, two electromechanical recorders, a plotting device, a unit for determining parallactic shift between corresponding records of images of the topography on loggers of the electromechanical recorders, a stereo map of the bottom topography of the water body and data-connected to the ship computer; the apparatus further includes a functional unit, an inertial measurement module connected to the receiver-indicator of the satellite navigation system and an electronic cartographic navigation system.

EFFECT: high accuracy of reconstructing bottom topography during stereophotography of a microtopography using sonar equipment.

2 cl, 6 dwg

Description

The invention relates to the field of hydrography and can be used for stereo surveying the topography of the bottom of the water area with a hydroacoustic means (GAS), as well as searching for underwater objects located on the bottom surface of the water area.

A known method of stereo surveying the topography of the bottom of the water area (US patent No. 3781775, class 340 / 3R [1]), which includes the carrier moving two HACs along predetermined tacks, the antennas of which are spaced horizontally by a predetermined amount, determining the basis component of the stereo survey, while the antenna GAS emit sounding pulses, which sequentially irradiate the bottom surface of the water area as they propagate, receive signals reflected from the bottom surface, measuring the time from the moment of emission of each probe pulse to the moment of reception of each pulse reflected from the bottom surface with continuous recording of their intensity, determining the parallactic displacement between the corresponding images of the bottom topography of the water area or sunken objects on the echograms of two GAS recorders, arising from the spacing of their antennas, and their geodetic coordinates and compiling a stereo relief map from the received data bottom of the water area.

The known method of stereo shooting of the microrelief of the bottom of the water area ([1]) is implemented by means of a device containing two HAS, which include functionally connected two acoustic antennas, two transceivers, two electromechanical registrars, a photogrammetric device, a receiver for a space navigation system, a unit for calculating the parallactic displacement between records of relief images on echograms of two GAS recorders and their geodetic coordinates and display of a stereo map of the relief of the bottom of the surface of the water area.

In this case, the calculation of the parallactic displacement is calculated as the difference of parallaxes (ΔР) of two points spaced vertically in accordance with the dependence

$$\Delta P=\{[g+(l-d)]-[g+d]\}-\{[gH+(l-d)]-[gH-d]\},(\mathrm{one})$$

- коэффициент формы рельефа;Where $$g=\frac{Z}{B}$$

- relief shape factor;

Z is the depth of the lower point;

B is the basis for stereo viewing;

- нормированная позиция нижней точки; $$d=\frac{d\text{'}\text{'}}{B}$$

- normalized position of the lower point;

H = i-h - conditional excess;

- нормированное превышение верхней точки над нижней, $$h=\frac{h\text{'}\text{'}}{z}$$

- normalized excess of the upper point over the lower,

where h 'is the excess of the lower point.

The disadvantages of the known method of stereo surveying the topography of the bottom of the water area and a device for its implementation, based on their nature, are:

- the complexity of calculating ΔP by the formula (1);

- the presence of an error in determining ΔР from the angles of heel, trim and yaw of the HAS carrier;

- the dependence of the stereo survey on the value of the basis, which limits the depth of stereo shooting in the water with a constant basis.

In addition, when solving applied problems, for example, related to the construction of underwater pipelines at great depths, it is very important that all landforms or artificial underwater objects are identified during bathymetric instrumental surveys by measured depths with sound signals generated, in particular, by high-frequency ones multi-beam echo sounders to obtain a detailed picture of the bottom topography.

When surveying the bottom topography with multi-beam echo sounders, depths in the horizontal plane are measured (formed) with a certain discreteness, which is associated with the angle of the beam direction, the method of beam formation, the frequency of the multi-beam echo sounder, the resolution of the beam formation. Moreover, this discreteness is generally a function of the depth L = f (H).

For example, for a high-frequency multipath echo sounder of the EM 100 type, used during bathymetric instrumental surveys during design works when laying underwater trunk pipelines for hydrocarbon transportation, the horizontal discreteness of the depth distribution at an equal distance in shallow mode is L = 6.3% N, where N is depth, when the depths are distributed at equal angles, this distance on the lateral rays increases compared to the central rays. This leads to the fact that with an increase in the depth of shooting, it is possible to skip the relief form, which is dangerous for the pipeline. From the point of view of designing the parameters of the pipeline, the intersection of such a dangerous form by the pipeline leads to an increase in the free span of the pipe and an increase in the load at the point of contact of the pipe with the relief of the dangerous shape. When designing the parameters of the pipeline, the basis is the bathymetric profile, and the absence of fixation on the profile of dangerous depth in real conditions can lead to exceeding the permissible loads on the pipe and, accordingly, to its damage, therefore, the task of determining the probability of missing a dangerous shape of the relief during bathymetric surveying is very important .

A similar problem is also characteristic of the hydro-logging system, made in the form of a hydrographic echograph of a side view, type GEBO-100, designed to perform hydrographic work in order to identify the nature of the bottom topography or to search for underwater objects that pose a navigational hazard (Instructions for using the hydrographic sonograph GEBO-100. Ministry of Defense USSR, General Directorate of Navigation and Oceanography, Admiralteysky No. 9125, 1985, p.5-45 [2]).

In this case, the restoration of the shape of the bottom topography from discrete measurements is performed by integral transformations based on the combinatorial method of analyzing geospatial fields from point measurements. In this case, one of the important preparatory tasks is the task of constructing a digital elevation model (DEM). This task is carried out either according to cartographic materials, or according to the initial data of depth measurements. In the latter case, to obtain DEM, computational procedures are used that are built into the software packages of geographic information systems (GIS). However, GIS technologies do not allow obtaining logically sound DTMs. This is due to two main circumstances.

The first circumstance is associated with distortions of meaning when using mathematical terms in geoinformatics. This logical error refers to sophisticated techniques based on the substitution of concepts. For example, in geoinformatics, mathematical methods of interpolation and approximation are used, while the interpolated function is unknown, the class of approximated functions is also unknown. Whereas in mathematics these terms are applicable either to a given function or to a given class of functions. In other words, in terms of geoinformatics, the application of these terms, as well as the corresponding mathematical methods, is logically incorrect.

The second circumstance is connected with the fact that algorithms are used whose properties do not adequately correspond to the properties of the relief. For example, the heuristic method of interpolation - cricketing - basically uses an artificial assumption of relief as a stationary random function, which contradicts one of the tenets of measurement theory, which argues that the measured value cannot be random (Lyachnev V.V., Syraya T.N. , Dovbeta LI Fundamentals of the theory of measurements of physical quantities. - SPb .: Publishing house of SPbGETU "LETI", 2004. - 310 p.). On the contrary, another method - triangulation - does not take into account the main property of the relief - variability. Triangulation takes into account only the property of the relative position of the coordinates of the points, and does not use the values of heights (depths) in them. The relief in the form of a DEM is described by a certain area of the surface at a fixed spatial scale. At the same time, the algorithms for obtaining DEMs and the algorithms for matching DEMs with other scales of the territory or bordering it remain unknown. With this approach, there is completely no understanding of the relief as a single integral structure, including all scales of geomorphological forms on the entire surface of the Earth. Under these conditions, it is practically impossible to build a logically correct technology for the automated processing of hydrographic measurements to build the bottom topography.

The resolution of this problem situation requires the development of a description of the relief as a single formalized object with an explicit listing of its properties. Only the availability of such information will allow the development of a constructive automated technology for constructing the bottom topography by hydrographic measurements.

The identified shortcomings are inherent in other analogues as well (patent RU No. 2292062 C2, 01.20.2007 [3], patent RU No. 2272303 C1, 03.20.2006 [4], patent RU No. 2340916 C1, 12/10/2008 [5], patent RU No. 23232408 , 10.06.2008 [6], patent RU No. 1829019 A1, 07/27/1993 [7], JP patent No. 10325871 A, 12/08/1998 [8], JP patent No. 94372890 A, 12/25/1992 [9]).

The objective of the proposed technical solution is to increase the reliability of the restoration of the bottom topography when performing stereo imaging of the microrelief of the bottom of the water area by means of a GAS.

The problem is solved due to the fact that in the method of stereo surveying the topography of the bottom of the water area, which includes moving the hydroacoustic means along predetermined tacks in the water area using a hydrographic vessel equipped with speed and heading meters, depth gauge, receiver-indicator of the satellite navigation system and / or receiver-indicator of the radio navigation system, connected with a ship computer, with the help of antennas of hydroacoustic means made in the form of a hydrographic echograph of a lateral survey they emit sounding pulses, which, as they propagate, sequentially irradiate the surface of the bottom of the water area, receive signals reflected from the bottom surface of the bottom, measure the time from the moment of emission of each probe pulse and continuously record their intensity, determine the parallactic displacement between the corresponding records of the bottom topography images on the echograms two recorders and their geodetic coordinates and according to the data compiled a stereo map of the relief of the bottom of the water area, previously a digital relief map of the bottom of the water area is formed according to archive data with the identification of dangerous landforms, the hydroacoustic means are placed in the vertical plane on one side of the hydrographic vessel, the radiation of the probe pulses is performed synchronously for each antenna at different frequencies, the signals reflected from the bottom surface are received in the directions forming with a vertical plane perpendicular to the diametrical plane of the hydrographic vessel, two symmetrical predetermined angles, obtained from discrete and a digital map of the bottom topography is built up for measurements, topological analysis of the topography is performed, while the Kronrod-Rieb graph and Morse-Smale complexes are formed for each piecewise-linear surface, the fractal parameters of the topography are evaluated in a device for implementing the method of stereo surveying of the topography of the bottom of the water, made in the form hydroacoustic means containing functionally connected two transceiver antennas, two electromechanical recorders, photogrammetric device, parallactic cm detection unit the communication between the corresponding records of the relief images on the recorders of the electromechanical recorders, the stereo map of the relief of the bottom of the shooting area and the information computer connected to the ship computer, a function block is additionally introduced, connected to the outputs of two electromechanical recorders by its inputs, and connected to the input-output of the ship computer by its input / output, inertial measuring module connected to the receiver-indicator of the satellite navigation system, electronic cartographic navigation system, connected by its input-output to the input-output of a ship's computer, two receiving-emitting antennas are mounted on a chassis equipped with a measuring inertial module connected to the receiver-indicator of the satellite navigation system.

New distinctive features of the proposed technical solution are that they preliminarily form a digital map of the relief of the bottom of the water area according to archive data with the identification of dangerous landforms, the antennas of the hydroacoustic means are placed in the vertical plane on one side of the hydrographic vessel, the radiation of the probe pulses is performed synchronously for each antenna on different the frequencies reflected from the bottom surface, the signals are received in directions forming with a vertical plane perpendicular to the diameter by the horizontal plane of the hydrographic vessel, two symmetric given angles, from the obtained discrete measurements, construct a digital map of the bottom topography, perform topological analysis of the topography, and form the Kronrod-Rieb graph and Morse-Smale complexes for each piecewise linear surface, evaluate the fractal parameters of the topography, into a device for implementing the method of stereo surveying the topography of the bottom of the water area, made in the form of hydroacoustic means containing functionally connected two receiving-emitting antennas, two an electromechanical recorder, a photogrammetric device, a unit for determining the parallactic displacement between the corresponding relief image records on the electromechanical recorders, a stereo topographic map of the bottom of the shooting area and information connected to the ship's computer, a functional unit is additionally introduced, connected to the outputs of two electromechanical registrars with its inputs, and its input, output connected to the input-output of the ship's computer, an inertial measuring module connected to satellite navigation system indicator, an electronic chart-based navigation system connected by its input-output to the input-output of a ship's computer, two receiving-emitting antennas are mounted on an indirect stabilized platform equipped with an inertial measuring module connected to the receiving-indicator of the satellite navigation system.

When moving a hydrographic vessel equipped with a GAS with the installation of two transceiver antennas in the vertical plane on one side of the hydrographic vessel, with the emission of probe pulses synchronously for each antenna at different frequencies, the reception of signals reflected from the bottom surface in directions forming with a vertical plane perpendicular to the diametric the plane of the hydrographic vessel, two symmetrical predetermined angles make it possible to sequentially illuminate each element of the bottom topography first with one, and then another acoustic beam, which causes the occurrence of longitudinal parallax in the direction of movement of the hydrographic vessel. The role of the linear basis is played by the angular solution of the directivity characteristics of the antennas (acoustic rays), which is the angular basis of the stereo survey.

Preliminary formation of a digital elevation map of a given water area from archive data with the identification of dangerous landforms eliminates the possibility of missing a dangerous landform during bathymetric surveys.

Application of relief description methods using Morse functions, Kronrod-Reeb graphs, and Morse-Smale complexes provides the possibility of:

- topological coding of relief forms;

- cartographic generalization;

- recognition of geomorphological objects;

- formal classification of geomorphological objects;

- tiling the surface of the relief with a family of parametrized (polynomial) functions defined on Morse-Smale cells;

- hierarchically assess the degree of similarity of two terrain maps representing the same area, at the same or different scales;

- assessment of the reliability of the selected relief forms, taking into account the error and power of the source information;

- assessing the degree of sufficiency of a set of point measurements to restore the relief with a given detail.

To implement the above features, a functional block has been introduced into the device for implementing the method, which implements the following list of basic algorithms:

- relief reconstruction by discrete measurements using topological;

- the formation of the Kronrod-Reeb graph for a piecewise linear surface;

- the formation of Morse-Smale complexes for a piecewise linear surface;

simplification of a piecewise linear surface using the structures of the Kronrod-Reeb graph and Morse-Smale complexes obtained for it;

- assessment of fractal relief parameters based on the given structures of the Kronrod-Rieb graph and Morse-Smale complexes, which allows the restoration of complex relief structures, such as “caves”, “tunnels”, etc. based on the position of the theory of homology.

The essence of the proposed technical solution is illustrated by drawings (figure 1, 2, 3). Figure 1. The block diagram of the device for implementing the method of stereo relief. The block diagram contains a hydrographic vessel 1 equipped with a speed meter 2, a direction indicator 3, a depth gauge 4, receiver-indicators 5 and 6 of satellite and radio navigation systems, respectively, a hydroacoustic device made in the form of a hydrographic sonograph 7 of a side view, a ship computer 8, a functional unit 9, electronic cartographic navigation information system (ECDIS) 10, inertial measuring module 11.

Figure 2. The block diagram of the hydrographic ultrasound system 6 side view. The block diagram includes functionally connected two transceiver antennas 12, two transceivers 13, two electromechanical registrars 14, a photogrammetric device 15, a block 16 for determining the parallactic displacement between the corresponding records of bottom relief images on the recorders of electromechanical registrars 14.

, $${\text{X}}_{\text{C}}^{\text{'}}$$

- абсциссы точек А и С, снятые с эхограмм электромеханических регистраторов 14, соответственно посредством фотограмметрического прибора 7, α - угол между вертикалью и направлением на точку отражения, h - относительное превышение точек А и С.Figure 3 (a, b, c, d). Scheme of side view of the bottom. The positions indicate: hydrographic vessel 1, the directions of the rays 17, 18 formed respectively by the receiving-emitting antennas 12, the bottom 19 of the shooting area, the location of the survey tack 20, the stereo map 21, V - the speed of the hydrographic vessel 1, x, y, z - coordinate axes, A and C - arbitrary points at the bottom of the water area, P - parallax of arbitrary points A and C on the echograms of electromechanical recorders 14, X _A , X _C and $$X_A^{\text{'}\text{'}}$$

, $${\text{X}}_{\text{C}}^{\text{''}}$$

- abscissas of points A and C taken from echograms of electromechanical recorders 14, respectively, by means of a photogrammetric device 7, α is the angle between the vertical and the direction to the reflection point, h is the relative excess of points A and C.

The hydroacoustic means for capturing the bottom topography is made in the form of a hydrographic echograph 6 of the side view and includes functionally connected two transceiving antennas 12, two electromechanical registrars 14, a photogrammetric device 15, a parallactic displacement determination unit 16 between the corresponding records of the relief images on the recorders of the electromechanical registrars 14 with a stereo relief map the bottom of the shooting area and information connected to the ship's computer 8, a functional unit 9, which with its inputs connected to the outputs of two electromechanical recorders 14, and its input-output connected to the input-output of the ship's computer 8, two transceiver antennas 12 are mounted on a chassis equipped with a measuring inertial module 11 connected to the receiver 5 of the satellite navigation system.

An analogue of the hydrographic echograph 6 of the side view is the hydrographic echograph GEBO-100 [2].

Speed meter 2, heading indicator 3, depth gauge 4, receiver-indicators 5 and 6 of satellite and radio navigation systems are standard means of hydrographic vessel 1 for solving navigation problems and bathymetric surveys.

The inertial measuring module 11 is a strapdown inertial measuring module built on fiber optic gyroscopes and miniature accelerometers and a microcomputer installed in a single housing and connected to the receiver-indicator 5 of the GPS / GLONASS satellite navigation system, the analogue of which is a miniature integrated inertial navigation / satellite / satellite system orientation type "Mini-Navigation-1" (JSC "Concern" Central Research Institute "Elektropribor") and is designed to develop pitching and trot angles kaniya, vertical rolling and angular velocity components of both the hydrographic vessel 1 and at the installation site of the receiving-emitting antennas 12.

The Electronic Chart Navigation Information System (ECDIS) 10 is an ECDIS of the "SOENKI 4000-19" type (Shipbuilding, No. 4, 2010, p. 54) and is designed to control the movement of hydrographic vessel 1 during bathymetric surveying and displaying information on a video plotter , analogue is a panoramic echo sounder-video - plotter type PEV-K (V.A. Voronin, S.P. Tarasov, V.I. Timoshenko. Hydroacoustic parametric systems. Rostov-on-Don. "Rostizdat". 2004, p. 302-307).

Function block 9 is a hardware-software block and consists of a processor, graphics accelerators, an object-graphics engine such as OGRE, software modules such as PhysX, Hydrax, Skyx and ANSYS AQWA. As a graphic engine, it is also possible to use commercial engines such as CRY ENGINE, VALVE or similar.

The proposed method of stereo surveying the relief of the bottom of the water area is as follows.

When moving a hydrographic vessel 1, equipped with appropriate measuring equipment (Fig. 1, 2) to provide navigation and bathymetric surveying tasks, the planned electrical survey signals 20 generated by electromechanical recorders 14, transceivers 13 and transceiving antennas 12 emit radiation at different frequencies of ultrasonic sounding signals to the bottom surface 19 of the shooting area and receiving data pulses reflected from it in directions forming a vertical plane perpendicular to the diametral plane of the hydrographic vessel 1, two symmetric predetermined angles α and registration of signals reflected from the bottom surface on the recorders of mechanical recorders 14.

The longitudinal parallax P of arbitrary points A and C on the echograms of the electromechanical recorders 14 is determined in accordance with the dependence:

$$\begin{array}{l}{P}_A=X_A-X_A^{\text{'}\text{'}}\\ P_C=X_C-X_C^{\text{'}\text{'}}\end{array}\},\left(2\right)$$

$${\text{X}}_{\text{C}}^{\text{'}}$$

- абсциссы точек А и С, снятые с эхограммы электромеханических регистраторов 14 соответственно фотограмметрическим прибором 15. Разность параллаксов может быть вычислена по формулеwhere X _A , X _C , and $${\text{X}}_{\text{A}}^{\text{''}},$$

$${\text{X}}_{\text{C}}^{\text{''}}$$

- abscissas of points A and C, taken from the echogram of the electromechanical registrars 14, respectively, by a photogrammetric device 15. The difference in parallax can be calculated by the formula

$$\Delta P=\left(X_A-X_A^{\text{'}\text{'}}\right)-\left(X_C-X_{{}_C}^{\text{'}\text{'}}\right),\left(3\right)$$

or according to the formula $$\Delta P=2Mtg\alpha h,\left(\mathrm{four}\right)$$

where M is the registration scale;

h is the relative excess of points A and C,

$$h=\frac{\left(X_A-X_A^{\text{'}\text{'}}\right)-\left(X_C-X_{{}_C}^{\text{'}\text{'}}\right)}{2Mtg\alpha }.\left(5\right)$$

According to formulas (3), (4), (5), in the block 16 for determining the parallactic displacement between the corresponding records of the bottom relief images on the recorders of the electromechanical recorders 14, a stereo bottom topography map is constructed.

If dangerous landforms are detected, additional tacks are performed.

Then, based on the obtained discrete measurements, a digital map of the bottom topography is constructed, topological analysis of the topography is performed, and the Kronrod-Rieb graph and Morse-Smale complexes are formed for each piecewise-linear surface, the fractal parameters of the topography are evaluated using a functional block 9 in which the mathematical apparatus is implemented - topology, in terms of elements of the theory of differential and algebraic (combinatorial) topology. The use of this apparatus allows one to mathematically describe all relief forms irrespective of the scale and specific forms for displaying on a map or display.

Algebraic topology provides a connection between geometry and algebra, between a continuous and discrete description of the relief, a description of the structural features of the relief: minima, maxima, network lines of talwegs and watersheds. Differential topology provides the basis for identifying and reconciling the global properties of the relief surface with a set of its local structural features. All this allows the implementation of constructive algorithms for computers.

In addition, the apparatus of differential and algebraic topology makes it possible to approximately reconstruct the surface from a set of point data, to compare the degree of proximity of two representations of the relief surface of a fixed region for both one scale and for different scales. Get a logically reasoned way to simplify the terrain for both generalization and removal of measurement noise.

Algebraic topology provides a set of tools that allow you to capture and describe the shape of the surface, to determine in which surfaces coincide or differ. In addition, in algebraic topology, there are classical tools, such as Morse theory, homotopy and homology, which are suitable for solving a number of problems related to the surface shape (Milnor J. Morse Theory. - M.: Publishing House of LCI, 2011. - 184 p. )

Morse theory establishes the basis for describing the set of critical points of a smooth function given on a manifold. Using Morse's theory, we can determine a method for describing the shape of a surface based on the evolution of the surface of level isolines that displays a function. This method, which consists in comparing the levels of critical points on the surface, is usually considered as one of the simplest ways to describe the geometry of the surface. Each one-dimensional continuum, its one-dimensional tree, is associated with each function. The study of a number of properties of the function itself is reduced to the study of the properties of the corresponding function on a one-dimensional tree. The separation of the properties of the function of two variables into “one-dimensional” and “two-dimensional” seems to be a fundamental fact. From this point of view, the introduction of a one-dimensional tree is just essential: with its help, one-dimensional properties of a two-dimensional function are particularly clearly distinguished.

Denoting by f ^-1 (r) the full inverse image of the value r of the scalar function f given on the surface S ² (f: S ² → R), and by a the regular values of the function, i.e. values in the inverse image of which there is not a single critical point, then in this case f ^-1 (a) is always a smooth submanifold in S ² due to the well-known implicit function theorem. Let c denote the critical values of the function, i.e. such values in the prototype of which there is at least one critical point. Let f be a Morse function on a compact smooth manifold S ² . Consider an arbitrary level surface f ^-1 (a) and its connected components, which we call layers. As a result, the variety is divided into a union of layers, a foliation with features is obtained. We emphasize that each layer is connected by definition. Declaring each layer as one point and introducing the natural factor topology in the space of layers Γ, we obtain a certain quotient space. It can be considered as the base of this foliation. For the Morse function, the space Γ is a graph. Count G is called the Count of Kronrod-Reeb.

One-dimensional properties of the function of two variables are described using the Kronrod-Reeb graph, and higher-dimensional properties using Morse-Smale complexes (Sharko VV About Kronrod-Reeb Graph of a Function on a Manifold // Methods of Functional Analysis and Topology, Vol.12 (2006), no. 4, pp. 389-396. LS Pontryagin, Fundamentals of combinatorial topology. - M .; Nauka, 1976. - 136 pp.) For the Morse function f on the manifold S ² . The vertex of the Kronrod-Reeb graph is the point corresponding to the special layer of the function f, i.e. a connected component of the level containing the critical point of the function. A vertex of the Kronrod-Reeb graph is called terminal if it is the end of exactly one edge of the graph. All other vertices are called internal. The end vertices of the Kronrod-Reeb graph correspond uniquely to the local minima and maxima of the function. The inner vertices of the Kronrod-Rieb graph correspond one-to-one to special layers of the function containing saddle critical points.

If it is known in advance that the surface under study is orientable or non-orientable, then the Kronrod-Rieb graph of an arbitrary simple function on it allows one to restore the surface topology. Kronrod-Reeb graphs are considered up to isomorphism of oriented graphs. Two Morse functions on an orientable surface are layerwise equivalent if and only if their Kronrod-Reeb graphs are isomorphic.

The generalized mechanism for constructing the Kronrod-Reeb graph is as follows. Let a Morse function be given on a manifold. Two critical points are connected by an edge on the Kronrod-Reeb graph if and only if there exists a monotone smooth path on the manifold that connects these points and does not intersect the critical layers (except for its ends). A monotonous path is a path along which a given function increases. Two saddles are connected by two edges on the Kronrod-Reeb graph if and only if there are two monotone smooth paths on the manifold that connect these saddles and do not intersect critical layers (except for their ends), and these paths cannot be connected in a constant way on the manifold (then there are internal waypoints that belong to different layers).

Thus, the one-dimensional tree of the Morse function consists of a set of end points, plus no more than a countable number of simple arcs that pairwise intersect at no more than one point, which is also a branch point. It should be noted that for a degenerate Morse function (the presence of a degenerate critical point), an arbitrarily small stirring of the Morse function can be achieved so that at each critical level c (i.e., on the set of points p for which f (p) = c) lies exactly one critical point. In other words, critical points that fall on one level can be divided into close levels. Morse functions having exactly one critical point at each critical level are called simple.

For a non-degenerate Morse function, the Kronrod-Riba graph is a tree with T triple branch points, K = T + 2 end points, and P = 2T + 1 edge connecting K + T = 2T + 2 vertices of the graph.

For a Morse function, the topology of many levels is associated with critical points and the gradient field of the function. This relationship provides an opportunity for a formal description of the surface in algebraic form. Unlike other topology methods based, for example, on Kronrod-Reeb trees, the use of the Morse-Smale complex provides a description of two-dimensional and multidimensional surface properties, which allows one to obtain a representation of the local topology of smooth functions and segment the surface into regions with a "uniform" gradient field . In other words, the geometry of a smooth surface is mapped into simple geometric images (simplicial complexes), the analysis of which allows us to describe the structural features of the original surface, of different dimensions: zero, one, and two-dimensional. These structural features are easily interpreted in geomorphological terms, for example, peaks (peaks) and troughs (pits) correspond to zero-dimensional objects, one-dimensional lines of a network of thalwegs, watersheds, two-dimensional - monotonous slopes. This ensures the connection of geometric properties through structural features with geomorphological semantics.

More recently, Morse theory has been extended to piecewise linear functions (Gyulassy A.G. Combinatorial Construction of Morse-Smale Complexes. Dissertation Submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy. University of California. 2008.). Such an extension to piecewise linear functions and discrete grids allows the use of Morse theory to solve practical problems with real data sets.

The critical points of the Morse function are those points on a two-dimensional surface where the function is stationary. In order to fully describe the Morse function, we must highlight its structural features. To do this, you need to define a vector field called a gradient.

для всех t∈R. Другими словами интегральная линия есть путь, для которого касательный вектор параллелен градиенту в каждой точке пути.The gradient of the Morse function is a vector field on S ² . We integrate this vector field in order to decompose S ² into regions with homogeneous flows. A curve l (t) is called an integral line f if $$\frac{\partial }{\partial s}l\left(t\right)=df\left(l\left(t\right)\right)$$

for all t∈R. In other words, the integral line is a path for which the tangent vector is parallel to the gradient at each point of the path.

Integral lines represent the flow along the gradient between critical points. At any point where the gradient is not equal to zero, the integral line passing through this point can be found by tracing forward and backward along the gradient vector field. The following definition defines the upper and lower limits of the points of the integral line.

называется источником интегральной линии l(t) и обозначается org(t).Limit $$\underset{s\to -\infty }{\lim }l\left(t\right)$$

is called the source of the integral line l (t) and is denoted by org (t).

называется приемником интегральной линии l(t) и обозначается dest(l).Limit $$\underset{s\to +\infty }{\lim }l\left(t\right)$$

is called the receiver of the integral line l (t) and is denoted by dest (l).

Integral lines on smooth functions have the following properties:

1) two integral lines either intersect or coincide;

2) integral lines cover all S ² ,

3) sources and receivers of integrated lines are critical points f.

Integral lines are monotonic and, therefore, org (l) ≠ dest (l). These properties provide the condition that each point S ² has exactly one integral line passing through it. All points on S ² can be classified as sources or receivers.

The listed properties follow from the standard differential calculus.

The integral lines that connect the maximum and the saddle, or the minimum and the saddle, are called separatrix lines. In geomorphology, the lines of separatrices that connect the minima and saddles are usually called gullies, or the lines of the valleys, and those that connect the saddles and maxima are called the lines of the ridges.

Stable / unstable manifolds. Let p be some critical point of the function f: S ² → R. An unstable manifold for a point p is the set of points that belong to the integral line for which p is the source,

U (p) = {p} ∪) {x∈S ² | x∈im (l), org (l) = p}. A stable manifold for a point p is the set of points that belong to the integral line for which p is the receiver,

S (p) = {p} ∪ {x∈S ² | x∈im (l), dest (l) = p}. Here im (l) is the map of the curve l∈S ² .

The stable manifold S (p) of the critical point p with index i = i (p) is an open cell of dimension dim (S (p)) = i.

Note that unstable manifolds of a function f are stable manifolds of a function - f, since d (-f) = - df. Thus, two types of manifolds have the same structural properties. Therefore, unstable manifolds of the function f are also open cells, but with the dimension dim (U (p)) = 2-i, where i is the index of the critical point.

The Morse function is called the Morse-Smale function if stable and unstable manifolds intersect only transversally. In two dimensions, this means that stable and unstable 1-manifolds intersect at angles close to straight lines. Their intersection point is necessarily a saddle; their intersection at a regular point would be contrary to property 1) for integral lines.

The connected components U (p) ∩s (q) for all critical points p, q∈S ² are called Morse-Smale cells. We are talking about cells measuring 0, 1 and 2, as vertices, arcs and regions, respectively.

A set of Morse-Smale cells forms a Morse-Smale complex. The one-dimensional skeleton of the Morse – Smale complex consists of critical points and separatrix lines. This skeleton is called a critical network.

Moreover, U (p) ∩S (p) = {p}, and if p ≠ q, then U (p) ∩s (q) is the set of regular points r∈S ² that lie on the integral lines l with org ( l) = p and dest (l) = q. It is possible that the intersection of stable and unstable manifolds consists of more than one component.

Each vertex of the Morse-Smale complex is a critical point, each arc is half of the stable or unstable 1-manifold of the saddle, and each region is one of the components of the intersection of the stable 2-manifold of the maximum, and the unstable 2-manifold of the minimum. We note that each cell of the Morse – Smale complex has a simple geometry — an almost monotonic function that can be well approximated by a polynomial equation.

The cells of the Morse-Smale complex are triangulated to go to simplicial complexes - the combinatorial basis of algebraic topology. If a linearly independent collection of k + 1st points {e ₀ , ..., e _k } ∈R ^{k is given} , then the convex hull constructed on these points is called the kth simplex. Points from a set of points are called vertices.

In the proposed method, the construction of a digital map of the bottom topography is based on the logical relationships between:

- the fractal form of the Earth’s real relief and mathematical objects and methods of fractal functions, which, in turn, are represented by systems of iterative functions and wavelets;

- cartographic representation of the Earth's relief and methods of algebraic topology: Morse function, Kronrod-Reeb graph and Morse-Smale complexes.

Using the proposed method for stereo surveying the relief of the bottom of the water area and a device for its implementation provides:

- simplifies the calculation of the difference in parallax;

- the exception of the error due to the influence of the heel, trim and yaw of the hydrographic vessel during the bathymetric survey;

- the dependence of the stereo survey on the value of the basis is excluded, which does not limit the depth of stereo shooting in the water with a constant basis.

In addition, relief description methods using Morse functions, Kronrod-Reeb graphs, and Morse-Smale complexes provide the ability to:

- topological coding of relief forms;

- cartographic generalization;

- recognition of geomorphological objects;

- formal classification of geomorphological objects;

- tiling the surface of the relief with a family of parametrized (polynomial) functions defined on Morse-Smale cells;

- hierarchically assess the degree of similarity of two terrain maps representing the same area, at the same or different scales;

- assessment of the reliability of the selected relief forms, taking into account the error and power of the source information;

- assessment of the degree of sufficiency of a set of point measurements to restore the relief with a given detail.

The industrial implementation of the proposed method for stereo surveying the topography of the bottom of the water area and the device for its implementation is not of technical complexity, the composition of the necessary equipment can be formed from the standard equipment of the hydrographic vessel and the equipment on the market.

Information sources.

1. US patent No. 3781775, cl. 340 / 3R.

2. Instructions for the use of the hydrographic sonograph GEBO - 100. The Ministry of Defense of the USSR, the Main Directorate of Navigation and Oceanography. Admiralty No. 9125.1985, pp. 5-45.

3. Patent RU No. 2292062 C2, 01.20.2007.

4. Patent RU No. 2272303 C1, 03.20.2006.

5. Patent RU No. 2340916 C1, 12/10/2008.

6. Patent RU No. 23232408, 06/10/2008.

7. Patent RU No. 1829019 A1, 07.27.1993.

8. JP Patent No. 10325871 A, 12/08/1998.

9. JP patent No. 4372890 A, 12.25.1992.

Claims (2)

1. The method of stereo surveying the topography of the bottom of the water area, including moving the hydroacoustic means over predetermined tacks in the water area using a hydrographic vessel equipped with speed and heading meters, depth gauge, receiver-indicator of the satellite navigation system and / or receiver-indicator of the radio navigation system connected to the ship's computer, while using the antennas of the hydroacoustic means made in the form of a hydrographic echograph of lateral viewing emit sounding pulses that As they propagate, they sequentially irradiate the bottom surface of the water area, receive signals reflected from the bottom surface, measure the time from the moment of emission of each probe pulse and continuously record their intensity, determine the parallactic displacement between the corresponding records of the bottom topography images on the echograms of two recorders and their geodetic coordinates and the obtained data compilation of a stereo map of the relief of the bottom of the water area, characterized in that they pre-form a digital map the bottom surface of the water area according to archive data with the identification of dangerous landforms, the hydroacoustic antenna is placed in the vertical plane on one side of the hydrographic vessel, the radiation of the probe pulses is performed synchronously for each antenna at different frequencies, the signals reflected from the bottom surface are received in the directions forming with the vertical plane perpendicular to the diametrical plane of the hydrographic vessel, two symmetric predetermined angles are constructed digitally from the obtained discrete measurements relief map bottom relief operate topological analysis, wherein the graph is formed Kronrod Riba and Morse-Smale complexes for each piecewise linear surface relief operate estimate fractal parameters. 2. A device for implementing the method of stereo surveying the relief of the bottom of the water area, made in the form of hydroacoustic means containing functionally coupled two receiving-emitting antennas, two electromechanical registrars, a photogrammetric device, a unit for determining the parallactic displacement between the corresponding records of the relief images on the recorders of electromechanical recorders, a stereo map of the relief of the bottom of the shooting area and information connected to the ship's computer, characterized in that it is additionally introduced a functional unit connected to the outputs of two electromechanical recorders with its inputs and connected to the input / output of a ship's computer with its input / output, an inertial measuring module connected to the receiver-indicator of the satellite navigation system, an electronic map navigation system connected to its input-output with input-output ship computer, two receiving-radiating antennas are mounted on the chassis, equipped with a measuring inertial module connected to the receiver-receiver of satellite navigation hydrochloric system.

Images