Method for stereophotography of bottom topography of water body and apparatus for realising said method

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.

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The invention relates to the field of hydrography and can be used for stereo imaging of bottom waters hydroacoustic means (GUS), and search for underwater objects, located on the bottom surface of the waters.

There is a method of stereo imaging of bottom waters (US patent No. 3781775, CL 340/3R [1]), which involves moving two carrier gas, given the tacks, which antennas are spaced apart in a horizontal plane by a specified amount, the determining component of the basis of stereoizer, when this antenna GUS emit illuminating pulses, which, as the distribution sequentially irradiate the bottom surface waters receive reflected from the bottom surface signals, measuring the time from the moment of radiation of each of the probe pulse until the reception of each reflected from the surface of the bottom pulse with a continuous recording their intensity, determination of the parallactic displacement between the corresponding entries images of the topography of the area or wrecks on the echograms two recorders GUS arising from the separation of their antennas, and their geodetic coordinates and compilation of data obtained stereocard of bottom waters.

There is a method of stereo imaging surface of the bottom waters ([1]) is implemented through a device containing two GUS, to the which include functionally United two acoustic antenna, two transceiver, two Electromechanical Registrar, photogrammetric instrument, space navigation receiver system, the computing unit parallactic displacement between records images of the terrain on the echograms two recorders GUS and their geodetic coordinates and display stereocity bottom relief of the surface of the waters.

The calculation parallactic displacement is calculated as the difference of parallaxes (Δ) two points, spaced vertically in accordance with addictions

$$\Delta P=\{[g+(l-d)]-[g+d]\}-\{[gH+(l-d)]-[gH-d]\},(1)$$

where$$g=\frac{Z}{B}$$- coefficient of landforms;

Z - depth of the lowest point;

In the basis of stereoizer;

$$d=\frac{d\text{'}}{B}$$ the normalized position of the lower point;

H=i-h - conditional excess;

$$h=\frac{h\text{'}}{z}$$the normalized excess of the upper point above the bottom,

where h' is the excess of the lower point.

The disadvantages of this method of stereo imaging of bottom waters and device for its implementation, based on their essence, are:

- the complexity of computing Δ by the formula (1);

- the presence of errors in the determination Δ from the angles of roll, pitch and yaw of the carrier gas;

the dependence of stereoizer on the size of the basis, which limits the depth of the stop-motion on the water at a constant basis.

In addition, when solving applied problems, for example, associated with the construction of underwater pipelines at great depths, it is very important that all forms of relief or artificial underwater objects were identified during the bathymetric instrumental shooting on the measured depths of the sound signals generated, in particular, high frequency by multi-beam echo sounders to get a detailed picture of the bottom relief.

When shooting topography by multi-beam echo sounders depth in the horizontal plane are measured (formed) with a certain resolution, which is associated with the angle direction is s beam, a method of beam forming, frequency multibeam sonar resolution beam forming. Moreover, this resolution is in General a function of depth L=f(H).

For example, for high-frequency multibeam echo sounder type EAT 100 used to carry out bathymetric instrumental shooting for project works for laying underwater pipelines for transportation of hydrocarbons, the horizontal resolution of the distribution of depths at equal distance in the shallow mode is equal to L=6.3% Of N, where N is the depth, the distribution of depths at equal angles is the distance on the lateral rays is increased in comparison with the Central rays. This leads to the fact that with increasing depth of capture possible threat to pipeline landforms. From the point of view of the design parameters of the pipeline, the gas pipeline crossing this dangerous form leads to an increase of the free span of the pipe and increase the load at the point of contact of the pipe with relief dangerous forms. When designing a pipeline based on the bathymetric profile, and the lack of fixation on the profile of a dangerous depth in real conditions can result in exceeding the allowable loads on the pipe and, consequently, to damage, so the problem of determining the probability of making the ka dangerous forms of relief when conducting bathymetry is very important.

Similar problem for GUS, made in the form of hydrographic echograph side-view type GABA - 100, designed for the production of hydrographic work in order to identify the nature of the bottom topography or locating underwater objects representing a navigational danger (instructions for using hydrographic echograph GABA - 100. The Ministry of defense of the USSR, head Department of navigation and Oceanography. Admiralty No. 9125, 1985, pp.5-45 [2]).

The recovery of the shape of the bottom topography on discrete measurements performed by integral transformations based on combinatorial analysis method geospatial field spot measurements. In this case, one of the important preparatory task is the task of building a digital elevation model (DEM). This task is carried out either on maps or on the original measurements of the depths. In the latter case, to obtain a DEM apply computational procedures that are built into software packages and geographic information systems (GIS). However, GIS technology does not allow to obtain a consistent DEM. This is due to two main factors.

The first circumstance is associated with distortion of sense when using mathematical terms in Geoinformatics. This logical error is the sophistic techniques, based on a substitution of concepts. For example, in Geoinformatics used mathematical methods interpolation and approximation, as the prediction function is unknown, the unknown and the class of approximated functions. Whereas in mathematics, these terms apply to either the specified function or to a given class of functions. In other words, in terms of Geoinformatics application of these terms, as well as the corresponding mathematical methods, logically incorrect.

The second circumstance is due to the fact that algorithms can be used, whose properties are not adequately correspond to the properties of the terrain. For example, a heuristic method for interpolation kriging based uses artificial assumption of relief as a stationary random function, which contradicts one of the principles of measurement theory, which asserts that the measured value may not be random (Lechneve CENTURIES, the so-CALLED Orphaned., Doubet LI fundamentals of theory of measurement of physical quantities. - SPb.: Publishing house Etu "LETI", 2004. - 310 C.). In contrast, another method - triangulation - does not take into account the basic property of landscape variability. Triangulation takes into account only property relative position coordinate points and the values of the heights (depths) are not used. The relief of the DEM describes a selected area of a surface in the fixed protrans venom scale. When this remain unknown and algorithms to obtain a DEM, and agreement algorithms DEM with other scales of this site or bordering on it. In this approach, no idea about the terrain as a single integral structure that includes all the scales geomorfologica forms on the entire surface of the Earth. In these conditions, to construct a logically correct technology for the automated processing of hydrographic measurements to build the bottom topography is almost impossible.

The resolution of this problem situation requires the development of descriptions of the terrain as a single formal object with an explicit list of its properties. Only the availability of this information will allow the development of constructive automated technology of building of bottom topography on hydrographic measurements.

The deficiencies inherent and other analogues (patent RU №2292062 C2, 20.01.2007 [3], patent RU No. 2272303 C1, 20.03.2006 [4], patent RU No. 2340916 C1, 10.12.2008 [5], patent RU No. 2326408, 10.06.2008 [6], patent RU No. 1829019 A1, 27.07.1993 [7], JP patent No. 10325871 A, 08.12.1998 [8], JP patent No. 94372890 A, 25.12.1992 [9]).

The objective of the proposed technical solution is to increase the reliability of the recovery of bottom relief when performing stereo imaging surface of bottom waters through GUS.

The problem is solved due to the fact that the way of the stop-motion Rel the FA of the bottom waters including moving hydroacoustic equipment, given the shallower waters through a survey vessel equipped with a measuring speed and direction, depth gauge, plementation satellite navigation system and/or plementation radionavigation system connected with the ship's computer, using antennas hydroacoustic equipment, made in the form of hydrographic echograph side view, emit the excitation pulses, as they spread sequentially irradiate the bottom surface waters receive reflected from the bottom surface signals, measuring the time from the moment of radiation of each of the probe pulse and continuous recording of their intensity, determination of the parallactic displacement between the respective records images of bottom waters on the echograms two recorders and their geodetic coordinates and the data obtained drafting stereocard of bottom waters, the pre-form digital map of bottom waters on archival records emitting dangerous forms of relief, hydroacoustic antenna means is placed in a vertical plane on one Board the survey vessel, the radiation of the probe pulses is performed synchronously for each antenna at different frequencies, is reflected from the bottom surface signals take directions forming on a vertical plane, perpendicular to the diametrical plane of the hydrographic vessel, two symmetric specified angle, the received discrete measurements build a digital map of bottom topography, perform a topological analysis of the terrain, thus form a graph Kronrod-RIBA and complexes, Morse-Smale for each piecewise-linear surface, perform an assessment of the fractal parameters of the relief, in the device for implementing the method of the stereo imaging of the topography of the area, made in the form of hydroacoustic equipment containing functionally United two piemeslotie antenna, two Electromechanical Registrar, photogrammetric unit, the unit determine the parallactic displacement between the respective records images of the terrain on the recorders Electromechanical registrars, stereocarto elevation of the bottom of the water area of the shooting and information connected with the ship's computer, inputs of the function block, its inputs connected to the outputs of the two Electromechanical registrars and their input-output connected to the input-output shipboard computer, inertial measurement module, coupled to plementation satellite navigation system, electronic map navigation system is connected to an input-output input-output is m ship's computer, two piemeslotie antenna mounted on the chassis, equipped with inertial measurement module connected to plementation satellite navigation system.

Novel features of the proposed solution lies in the fact that the pre-form digital map of bottom waters on archival records emitting dangerous forms of relief, hydroacoustic antenna means is placed in a vertical plane on one Board the survey vessel, the radiation of the probe pulses is performed synchronously for each antenna at different frequencies reflected from the bottom surface signals take directions, forming with the vertical plane perpendicular to the diametrical plane of the hydrographic vessel, two symmetric specified angle, the received discrete measurements build a digital map of bottom topography, perform a topological analysis of the terrain, thus form a graph Kronrod-RIBA and complexes, Morse-Smale for each piecewise-linear surface, perform an assessment of the fractal parameters of the relief, in the device for implementing the method of the stereo imaging of the topography of the area, made in the form of hydroacoustic equipment containing functionally United two piemeslotie antenna, two Electromechanical Registrar pictures the metric unit, block determine the parallactic displacement between the respective records images of the terrain on the recorders Electromechanical registrars, stereocarto elevation of the bottom of the water area of the shooting and information connected with the ship's computer, inputs of the function block, its inputs connected to the outputs of the two Electromechanical registrars and their input-output connected to the input-output shipboard computer, inertial measurement module, coupled to plementation satellite navigation system, electronic map navigation system is connected to an input-output with the input-output shipboard computer, two piemeslotie antenna installed on indirect stabilized platform, equipped with inertial measurement module connected to plementation satellite navigation system.

When moving hydrographic vessel equipped with a gas with the installation of two transmitting antennas in a vertical plane on one Board a survey vessel with radiation probe pulses synchronously for each antenna at different frequencies, the reception of the reflected from the bottom surface of the signals in the directions, forming with the vertical plane perpendicular to the diametrical plane of the hydrographic vessel, two symmetric set the GLA allow to sequentially illuminate each element of the relief of the bottom of the first one, and then another acoustic beam, which causes the occurrence of longitudinal parallax in the direction of movement of the survey vessel. The role of linear basis plays a corner solution directivity antennas (acoustic rays), which is the angular basis stereobar.

Pre-shaping digital terrain maps a given water area according to archival data with the release of hazardous forms of relief can be prevented from crossing dangerous forms of relief when conducting bathymetry.

The application of the methods of description of relief with the help of Morse functions, graphs Kronrod-RIBA and complexes, Morse-Smale provides the ability to:

topological encoding of landforms;

cartographic generalization;

- recognition geomorphological objects;

formal classification of geomorphological objects;

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

- hierarchically to evaluate the similarity of two maps in relief, representing one region, in one or multiple scales;

- assess the validity of selected landforms taking into account the error and the power of the source information;

- assessing the adequacy of a set of point measurements for vos is Stanovlenie relief with the specified detail.

To implement these features in a device for implementing the method introduced functional block that implements the following list of basic algorithms:

- relief reconstruction from discrete measurements using topological;

- formation graph Kronrod-RIBA for piecewise-linear surface;

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

simplify piecewise linear surface using received for her structures graph Kronrod-RIBA and complexes, Morse-Smale;

- estimation of fractal parameters of the relief on the basis of the given structures graph Kronrod-RIBA and complexes, Morse-Smale, allowing the restoration of the complex structures of the relief, such as caves, tunnels, etc. on the basis of theory of homology.

The essence of the proposed technical solution is illustrated by the drawings (figures 1, 2, 3). Figure 1. The block diagram of the device for implementing the method of the stop-motion relief. The block diagram contains hydrographic vessel 1 equipped with speed measuring 2, course 3, the depth measuring device 4, the receiver-indicators 5 and 6 satellite and radionavigation systems, respectively, sonar tool, made in the form of hydrographic echograph 7 side view, the ship computer 8, a functional block 9, power is ing map navigation and information system (ECDIS) 10, inertial measuring unit 11.

Figure 2. Block diagram hydrographic echograph 6 side view. The block diagram includes functionally United two piemeslotie antenna 12, two transceiver 13, the two Electromechanical Registrar 14, photogrammetric unit 15, block 16 determine the parallactic displacement between the respective records images of the bottom topography on the recorders Electromechanical Registrar 14.

Figure 3 (a, b, C, d). The schematic side view of the bottom. Positions marked: hydrographic vessel 1, the direction of the beams 17, 18 formed respectively priekaistaudami antennas 12, the bottom 19 of the waters of the shooting, the location of the shooting of 20 gals, stereocarto 21, V is the speed of the survey vessel 1, x, y, z - axes, a and C is an arbitrary point on the bottom, R - parallax arbitrary points a and C on the echograms Electromechanical registrars 14, X_AX_Cand$$X_A^{\text{'}}$$,$${\text{X}}_{\text{C}}^{\text{'}}$$the abscissa of the points a and C, taken from echograms Electromechanical registrars 14, respectively, by means of photogrammetric unit 7, α is the angle between the vertical and the direction of the point of reflection, the h - relation is sustained fashion the excess points a and C.

Sonar tool to capture topography made in the form of hydrographic echograph 6 side view and includes functionally United two piemeslotie antenna 12, two Electromechanical Registrar 14, photogrammetric unit 15, block 16 determine the parallactic displacement between the respective records images of the terrain on the recorders Electromechanical registrars 14 with stereocarto elevation of the bottom of the water area of the shooting and information connected with the ship's computer 8, a functional block 9, which has its inputs connected to the outputs of the two Electromechanical registrars 14, and its input-output connected to the input-output shipboard computer 8, two piemeslotie antenna 12 mounted on the chassis, equipped with a measuring inertial module 11 connected to plementation 5 satellite navigation system.

Similar hydrographic echograph 6 side view is the hydrographic echography, GABA-100 [2].

The velocity meter 2, course 3, the depth measuring device 4, the receiver-indicators 5 and 6 satellite and radionavigation systems are regular means hydrographic vessel 1 to provide solutions to navigation problems, and bathymetry.

Inertial measuring unit 11 is a strapdown inertial MEAs the measuring module, based on fiber-optic gyroscopes and miniature accelerometers and micromechanical installed in a single enclosure and connected to plementation 5 satellite navigation systems GPS/GLONASS, which analogue is a miniature integrated inertial/satellite navigation system and the orientation of the Mini-navigation-1" (JSC "Concern "CSRI "Elektropribor") and is intended to provide angles of pitch and yaw, heave and components of angular velocity as a hydrographic vessel 1, and the setup priekaistaudami antennas 12.

E-map navigation and information system (ECDIS) 10 is an ECDIS type "soenke 4000-19" (Shipbuilding, No. 4, 2010, p.54) and is intended for control of movement of the hydrographic vessel 1 in the production of bathymetry and displaying information on the video plotter, analog is a panoramic sonar-video - plotter type sew-K (Vasakronan, Spearazon, Whitebalance. Hydroacoustic parametric system. Rostov-on-don. "Rostedt". 2004, s-307).

Functional block 9 is a hardware block and consists of a CPU, graphics accelerators, object graphical engine type OGRE, software modules like PhysX, Hydrax, Skyx and ANSYS AQWA. As a graphic engine who is you can also use commercial engines type CRY ENGINE, VALVE or similar.

The proposed method for stereo imaging of bottom waters is as follows.

When moving survey vessel 1 equipped with appropriate measuring equipment (1, 2) to ensure the tasks of navigation and bathymetry, scheduled shooting gals 20, under control of electrical signals produced in the Electromechanical registrars 14, transceivers 13 and priekaistaudami antennas 12 are incident on different frequency ultrasonic probing signals to the bottom surface 19 of the waters of the shooting and receiving reflected from the data pulses according to the directions, forming with the vertical plane perpendicular to the diametrical plane of the hydrographic vessel 1, two symmetrical specified angle α and the Desk reflected from the bottom surface of the signals on the mechanical recorders recorders 14.

Longitudinal parallax P of arbitrary points A and C in the echograms Electromechanical registrars 14 is determined in accordance with the relationship:

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

where X_AX_Cand$${\text{X}}_{\text{A}}^{\text{'}},$$$${\text{X}}_{\text{C}}^{\text{'}}$$the abscissa of the points a and C, taken from the echogram Electromechanical registrars 14, respectively photogrammetric device 15. The difference of parallaxes can be calculated by the formula

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

or by the formula$$\Delta P=2Mtg\alpha h,\left(4\right)$$

where M is the scale of the Desk;

h is the relative excess points a and C,

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

By the formulas(3), (4), (5) in block 16 determine the parallactic displacement between the respective records images of the bottom topography on the recorders Electromechanical registrars 14 is constructed stereocarto of bottom waters.

Upon detection of the dangerous forms of relief perform additional tacks.

Next, on the obtained discrete measurements build a digital map of bottom topography, perform a topological analysis of the terrain, thus form a graph Kronrod-RIBA and complexes, Morse-Smale for each piecewise-linear surface, perform an assessment of the fractal parameters of the relief by means of a functional block 9, which implements matematicheski the nd apparatus topology, in part, elements of theory of differential and algebraic (combinatorial) topology. The use of this apparatus allows uniformly mathematically to describe all forms of relief regardless of the size and specific forms to display on a map or display.

Algebraic topology provides the link between geometry and algebra, between continuous and discrete description of the topography, structural description topography: points, minima, maxima, lines network talugu and watersheds. Differential topology provides the basis for the identification and coordination of the global properties of the surface topography with a set of its local structural features. All this allows implementation of constructive algorithms for the computer.

In addition, the apparatus of differential and algebraic topology allows one to approximately restore the surface by a set of point data, to compare the degree of closeness of two views of the surface relief of a fixed region for one scale and for different scales. To obtain a logically reasoned way to simplify relief for purposes of generalization, and to remove noise measurements.

Algebraic topology provides a set of tools that allow you to capture and describe the shape of the surface is large, to determine what surface are the same or different. In addition, in algebraic topology there are classic tools, such as Morse theory, homotopy and homology, which are suitable to address a number of issues related to the shape of the surface (J. Milnor. Morse Theory. - M.: Publishing house of LCG, 2011. - 184 S.).

Morse theory provides the basis for describing the set of critical points of a smooth function defined on a manifold. Using Morse theory, you can define a way to describe the shape of the surface based on the evolution of the surface contour levels, reflecting the function. This method, based on the comparison of levels of critical points on the surface, generally regarded as one of the easiest ways of describing the surface geometry. Each function is associated to some one-dimensional continuum, its one-dimensional tree. The study of some properties of the function is reduced to the study of the properties of the corresponding functions on the one-dimensional tree. The separation properties of functions of two variables on a "one-dimensional" and "two-dimensional" is the fundamental fact. From this point of view, the introduction of a one-dimensional tree just significantly: with it especially clearly distinguished one-dimensional properties of two-dimensional functions.

Denoting by f^-1(r) the full type of the scalar values of r is the function f defined on the surface S ²(f:S²→R)as a regular function values, i.e. the values in the prototype which there are no critical points, then f^-1(a) is always smooth subvariety of S²due to well-known theorems on implicit functions. We denote the critical values of the function, i.e. the values in the type which has at least one critical point. Let f be a Morse function on a compact smooth manifold S². Consider an arbitrary surface level f^-1(a) and its connected components, which we call layers. As a result, the diversity is broken in the merge layers, the resultant lamination with features. We emphasize that each layer is connected by definition. Declaring each layer one point and introducing a natural factor-topology in the space of layers G, we obtain a factor-space. It can be considered as the base of this lamination. For a Morse function space G is a graph. A graph G is called a graph of Kronrod-RIBA.

One-dimensional properties of functions of two variables are described using graph Kronrod-RIBA, and the properties of the higher dimension with the complexes, Morse-Smale (Sharko V.V. About Kronrod-Reeb Graph of a Function on a Manifold // Methods of Functional Analysis and Topology, Vol.12 (2006), no. 4, pp.389-396. Pontryagin PS foundations of combinatorial topology. - M.; Science, 1976. - 136 C.) for a Morse function f on m is ooopsie S ². The top graph Kronrod-RIBA call point, to meet the specific layer of the function f, i.e. the connected component of the level that contains the critical point of the function. The top graph Kronrod-RIBA call limit, if it is the end of exactly one edge in the graph. All other vertex is internal. End vertices of the graph Kronrod-RIBA one-to-one correspond to local minima and maxima of the function. Internal vertex of Kronrod-RIBA one-to-one meet the special layer functions containing a saddle critical point.

If you know in advance that the target surface is oriented or non-orientable, then the graph Kronrod-RIBA arbitrary simple functions it enables you to recover the topology of the surface. Graphs Kronrod-RIBA considered up to isomorphism of oriented graphs. Two Morse functions on the oriented surface layers are equivalent if and only if their graphs Kronrod-RIBA isomorphic.

Generalized mechanism graph construction Kronrod-RIBA is the following. Let the diversity function set Morse. Two critical points are connected by an edge in the graph Kronrod-RIBA if and only if there exists a monotone smooth path on the manifold that connects these points and does not intersect the critical layers (in addition to their konzo is). Under monotonic path refers to the path along which the function is increasing. Two saddles are connected by two edges in the graph Kronrod-RIBA then and only then, when there are two monotone smooth path on the manifold that connect these saddles and do not cross critical layers (except the ends), and these paths cannot be connected to the permanent way on the diversity (i.e. the inner points of the paths are in different layers).

Thus, a one-dimensional tree Morse function consists of a set of end points plus not more than a countable number of simple arcs intersecting in no more than one point, which is, moreover, a bifurcation point. It should be noted that for a degenerate Morse functions (presence of a degenerate critical point) arbitrarily small perturbations of the functions of juice you can make at every critical level (i.e. the set of points p for which f(p)=C) lay exactly one critical point. In other words, the critical point, got on one level, you can make on similar levels. A Morse function with exactly one critical point on each critical level, are called simple.

For a nondegenerate Morse function graph Kronrod-RIBA is a tree with T triple branching points, K=T+2 end points and P=2T+1 edges connecting K+T=2T+2 vertices of the graph.

the La functions Morse, the topology of numerous levels associated with critical points and a field gradient of the function. This relationship provides the opportunity for a formal description of the surface in algebraic form. Unlike other methods, topology, based on, for example, trees Kronrod-RIBA, the use of a complex of a Morse-Smale provides a description of two-dimensional and multidimensional properties of the surface that allows you to get an idea of the local topology of smooth functions and segmentation of the surface into regions with homogeneous" field gradient. In other words, the geometry of the smooth surface is displayed in a simple geometric images (simplicial complexes), the analysis of which allows to describe the structural features of the original surface, different dimensions: zero-, one - and two-dimensional. These structural features easily interpreted in geomorphological terms, for example, nomernym objects correspond to peaks (peaks) and depressions (pits), one-dimensional line network Talbakov, watersheds, two-dimensional - monotonic slopes. This ensures that the link geometric properties through structural features with geomorphological semantics.

Recently, Morse theory was 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 Dotor of Philosophy. The University of California. 2008.). This extension to piecewise linear functions and the discrete grid allows you to use Morse theory to solve practical problems with real data sets.

The critical point of a Morse function are those points on the two-dimensional surface, where the function is stationary. To fully describe a Morse function, we must allocate its structural features. You need to determine the vector field, called the gradient.

The gradient of a Morse function is a vector field on S². Will proindeksiruem this vector field in order to carry out the decomposition of S²in regions with homogeneous flows. Curve l(t) is called the line integral of 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 the 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 zero, the integral of the line passing through this point can be found by tracing forward and backward along the gradient vector field. The following ODA the division determines upper and lower limits of the integral points of the line.

Limit$$\underset{s\to -\infty }{\lim }l\left(t\right)$$called the source line integral l(t) and is denoted by org(t).

Limit$$\underset{s\to +\infty }{\lim }l\left(t\right)$$is called the receiver line integral 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 line cover all S²,

3) the sources and sinks of integral lines are the critical points of f.

Integral line monotonous and therefore org(l)≠dest(l). These properties ensure that each point of S²has exactly one integral line through it. All points on S²can be classified as sources or sinks.

These properties follow from the standard differential calculus.

Integral lines that connect the maximum and saddle, or the minimum and the saddle, called lines of separatrices. In geomorphology line of separatrices that connect the minima and saddle, usually called ravines, or what enemy valleys, and those that connect the seat and the highs are called lines of the ridges.

Stable/unstable manifolds. Let p be a critical point of the function f:S²→R. the Unstable manifold of a point p is the set of points belonging to a line integral, for which the source is p,

U(p)={p}∪){x∈S²|x∈im(l), org(l)=p}. Stable manifold of a point p is the set of points belonging to a line integral, for which the receiver is p,

S(p)={p}∪{x∈S²|x∈im(l), dest(l)=p}. Here im(l) is the mapping curve l∈S².

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

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

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

Connected components of U(p)∩s(q) for all critical points p, q∈S²are called cells of the Morse-Smale. We are talking about the cells of dimension 0, 1 and 2, the vertices, arcs and regions, respectively.

The set of cells of the Morse-Smale forms a complex Morse-Smale. One-dimensional skeleton of complex Morse-Smale consists of critical points and lines of separatrices. This frame is called the critical chain.

In this case U(p)∩S(p)={p}, and if p≠q, then U(p)∩s(q) is the set of regular points r∈S²that lie on the integral lines l org(l)=p and dest(l)=q. It is possible that the intersection of the stable and unstable manifolds consists of more than one component.

Each vertex in the set of Morse-Smale 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-diversity high, and unstable 2-manifolds minimum. Note that each cell of the set of Morse-Smale has a simple geometry - almost monotone function is well approximated with a polynomial equation.

Cell complex Morse-Smale triangularity to go to simplicial complexes combinatorial algebraic topology. If given a linearly independent set of k+1 points {e₀, ..., e_k}The R ^kthen the convex hull, built on these points is called a k-simplex. The point of the set of points called vertices.

In the proposed method, the construction of digital maps of topography based on logical relationships between:

- fractal form of the actual topography of the Land and mathematical objects and methods of fractal functions, which, in turn, are represented by systems iterisosik functions and wavelets;

cartographic representation of the elevation of the Land and the methods of algebraic topology: function Morse, Earl of Kronrod-RIBA and complexes, Morse-Smale.

The use of the proposed method of stereo imaging of bottom waters and device for its implementation provides:

- simplifies calculating the difference of parallaxes;

- elimination of error due to the influence of roll, pitch and yaw hydrographic vessel during bathymetric surveys;

- eliminates the dependence of stereoizer on the amount of basis that does not limit the depth of the stop-motion on the water at a constant basis.

In addition, methods of description of relief with the help of Morse functions, graphs Kronrod-RIBA and complexes, Morse-Smale provide the ability to:

topological encoding of landforms;

cartographic generalization;

- recognition geomorphological volume of the LLC;

formal classification of geomorphological objects;

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

- hierarchically to evaluate the similarity of two maps in relief, representing one region, in one or multiple scales;

- evaluation of the validity of selected landforms taking into account the error and the power of the source information;

- the assessment of the degree of adequacy of a set of point measurements for the recovery of relief with the specified detail.

Industrial implementation of the proposed method of stereo imaging of bottom waters and device for realization of technical difficulties does not represent the composition of the necessary equipment can be formed from standard equipment survey vessel and equipment available on the market.

The sources of information.

1. The US patent No. 3781775, CL 340/3R.

2. Instructions for using hydrographic echograph GABA - 100. The Ministry of defense of the USSR, head Department of navigation and Oceanography. Admiralty No. 9125,1985, pp.5-45.

3. Patent RU No. 2292062 C2, 20.01.2007.

4. Patent RU No. 2272303 C1, 20.03.2006.

5. Patent RU No. 2340916 C1, 10.12.2008.

6. Patent RU No. 2326408, 10.06.2008.

7. Patent RU No. 1829019 A1, 27.07.1993.

8. The JP patent No. 10325871 A, 08.12.1998.

9. The JP patent No. 4372890 A, 25.12.1992.

1. The way the stop-motion of the relief of the and bottom waters including moving hydroacoustic equipment, given the shallower waters through a survey vessel equipped with a measuring speed and direction, depth gauge, plementation satellite navigation system and/or plementation radionavigation system connected with the ship's computer, using antennas hydroacoustic equipment, made in the form of hydrographic echograph side view radiate probe pulses, as they spread sequentially irradiate the bottom surface waters receive reflected from the bottom surface signals, measuring the time from the moment of radiation of each of the probe pulse and continuous recording of their intensity, determination of the parallactic displacement between the respective records images of the topography of the area to the echograms two recorders and their geodetic coordinates and the data obtained drafting stereocity bottom relief area, wherein the pre-form digital map of bottom waters on archival records emitting dangerous forms of relief, hydroacoustic antenna means is placed in a vertical plane on one Board the survey vessel, the radiation of the probe pulses is performed synchronously for each of the antennas is at different frequencies, reflected from the bottom surface signals take directions, forming with the vertical plane perpendicular to the diametrical plane of the hydrographic vessel, two symmetric specified angle, the received discrete measurements build a digital map of bottom topography, perform a topological analysis of the terrain, thus form a graph Kronrod-RIBA and complexes, Morse-Smale for each piecewise-linear surface, perform an assessment of the fractal parameters of relief.

2. The device for implementing the method of the stereo imaging of the topography of the area, made in the form of hydroacoustic equipment containing functionally United two piemeslotie antenna, two Electromechanical Registrar, photogrammetric unit, the unit determine the parallactic displacement between the respective records images of the terrain on the recorders Electromechanical registrars, stereocarto elevation of the bottom of the water area of the shooting and information connected with the ship's computer, wherein the added function block, its inputs connected to the outputs of the two Electromechanical registrars and their input-output connected to the input-output shipboard computer, inertial measurement module, coupled to plementation satellite navigation system, electronic whom I map navigation system, connected to an input-output input-output shipboard computer, two piemeslotie antenna mounted on the chassis, equipped with inertial measurement module connected to plementation satellite navigation system.