Forming a Method for Determining the Coordinates of Sound Anomalies Based on Data From a Computerized Microphone System

Special features in the process of determining the coordinates of sound anomalies according to the sound series were considered. Sound anomalies have been shown to be a source of information about events, phenomena that are already occurring, or are their harbingers. It has been stated that audio interception systems complement thermal imagers and provide savings in financial and human resources when used in combination with the benefits of unmanned aerial vehicles. The methods facilitating solution of the problem of surveillance and prediction by finding coordinates of sound anomalies were presented. Indirect methods for solving problems of searching for sound anomaly coordinates with three microphones in accordance with a linear scheme of approximants and linear and quadratic approximation were proposed. Solutions were reduced to analytical complete expressions that make it possible to calculate coordinates according to the input conditions for three or four microphones. Also, a problem of finding coordinates of a sound anomaly for three and four microphones was set and solved by direct methods. Solutions were presented as expressions that make it possible to calculate coordinates of sound anomalies. Numerical experiments were performed during which coordinates of sound anomalies, the absolute error of their determination at each iteration, and total time spent for calculation were found. Systems with coordinates of microphones and sound sources almost identical or coinciding have been shown to have the greatest error. Under these conditions, for direct methods, values of the equation coefficients decreased to almost zero or turned to zero and the difference of values of the sought coordinates between iterations increased sharply which slowed down the process of solution convergence. It was shown that the application of approximate methods to the search for coordinates by solving the minimization problems with the involvement of the recurrent approximation method makes it possible to construct simple algorithms. Their implementation to solve the problems of numerical experiments has given quick and practically accurate coordinate values. It was found, that the application of algorithms to the methods of logical analysis and rules of logical inference reduces the number of iterations and the total calculation time.


Introduction
Advantages of integrated video and audio surveillance and the new technical capabilities of sound anomaly recording open up new prospects for widespread monitoring including emergency automated management systems (AMS) [1].
Sound anomalies found in surveying from drones are a source of additional meaningful information about events or phenomena [2] that are occurring or are precursors of subsequent events [3]. In the city, steppe, mountainous or forested areas, sound anomalies are sometimes the only source of information [4]. In this regard, recording, storage, and analysis of audio information [5] are of interest for a systemic collection, processing by various institutions, including government institutions and dual-purpose services [6]. Lack of financial and human resources is one of the reasons explaining the growth of scientific and applied interest in designing computerized systems for the collection and processing of audio information [7,8]. However, further design and development of applied systems for recording, recognizing, documenting and mapping sound anomalies and constructing information flow AMS [9,10] should be preceded by a process of theoretical analysis and a search for new theoretical bases and patterns of calibration and training [11,12]. The methods of collecting, rapid processing, and recognition of potentially dangerous events or preceding events evidenced by sound anomaly have become a topical subject of prospective scientific studies [13,14].
However, innovative development of water economy, problems connected with gamekeeper control, unlicensed lumbering and hunting, road accidents, fires, disturbance of silence, and efficiency of corresponding services depend directly on accuracy and speed of position determination [15]. The solution of the latter and construction of models capable of providing valid and accurate information on sound anomaly coordinates for AMS operation has become a subject of recent studies [16]. In this regard, the development of an effective method of coordinate determination which will extend the functionality of such services and improve the efficiency of their work is an urgent task. Undoubtedly, the latter can be achieved through the technical improvement of tools including the improvement of methods, models, and algorithms of their work [15][16][17].
At present, the construction of models and development of methods for determining coordinates required for the sound anomaly recording systems of the cannon sound reconnaissance requiring immediate development and improvement in view of the real actions and threats from the aggressor is the problem of particular urgency [18].

Literature review and problem statement
Recently, data about new possibilities of technical means of recording acoustic waves propagating in the air are increasingly reported in the literature [1]. Sensitivity and resolution of such systems suggest that terrestrial audio interception systems, such as Trembita M, will replace or informationally supplement thermal imagers and night vision devices [1]. Also, according to [5], innovative approaches to building systems of switching sensors for sensory processing which have previously been tested in adaptive robotic means of flexibly restructuring enterprises [7] are further developed today. The proposed approach based on programmable logic controllers (PLC) [19] is well adaptable for the case when multiple sensors of the same type are switched to the processing module [20]. However, the advantage of the new proposed systems [19] implies the possibility of further expansion in accordance with requirements to the quality of sensors and realization of tasks in VHDL language which is implemented in various PLC procedures. Thus, the technical preconditions permit further innovative improvement of both fixed and mobile systems operating on the basis of a mobile device in cooperation with unmanned aerial vehicles (UAV) [15]. As shown in [21], an increase in the number of sound receivers through the use of UAV swarms can expand the technology capabilities. At the same time, it was also proved that such a solution further increases the number of problems to be solved [21]. The method presented in [22] makes it possible to choose UAV operation zones for effective solution of surveillance and forecasting problems. However, the influence of the features of the present-day formation and terrain topology complicates their solution [22]. Application of the multiagent event isolation method [23] based on multicharacter sets and use of machine learning algorithms, obviously, can be further implemented after additional improvements to solve problems of surveillance and forecast of sound anomalies [24]. Accuracy of recording the sound anomaly, separating it from superimposed signals, and recording the start and end time of a single event remains the task of improving means, algorithms and sensors of UAV [15,20].
The idea of propagation of a spherical acoustic wave regardless of the sound origin, be it sound from a sniper position or explosion of a cannon projectile from a closed position was used in [25]. There are studies in which bursts of self-propelled self-homing missiles are considered as sources of spherical acoustic waves propagating at a constant speed [26] in conditions of imposing sounds of other events. The latter is interference in describing processes and construction of models and algorithms of determining coordinates of the source of sound anomaly [27]. The task of determining coordinates in the context of excessive information shows that, regardless of the binaural effect, the problem of functional minimization is solved by direct or numerical methods. However, the use of the fastest descent method provided convergence regardless of the choice of initial conditions but the Newton method did not do this. Such conclusions can be explained by the presence of irrationality and not simple roots in which denominator of the recurrent formula tends to zero and the initial functional oscillates. As shown in [15], the necessity of recording the sound anomaly in such a model of the phenomenon requires an increase in the number and location of several spatially separated microphones [28]. By its essence, this technical solution makes the functional uniquely positive for all microphones. However, the location of microphones on quadrocopters generates an error in coordinates because of the error in the microphone coordinates. Influence on the error of measuring the target angle was studied in [28] but the algorithm given there did not present a method of correcting the methodical error that arises. The methodological error determination algorithm proposed in [29] makes it possible to estimate it under the conditions of a simplified model and accurate solutions available. Thus, for the tasks in which initial approximation is selected from the conditions of the expected events that requires for its use equipment that uses a binaural effect, it will also require refinement of both the model with a functional and a solution method. Search for and implementation of design solutions that realize the binaural effect [29] requires prediction of the event direction which cannot be predicted for civilian tasks. Under these conditions, the problem of determining the coor dinate of sound anomaly is usually reduced to a problem of either minimizing the functional [30] or solving the system of nonlinear algebraic equations [31].
Theoretical generalizations of the studies devoted to the use of direct and indirect methods of solving the problem of determining coordinates of existing models are presented in [32]. Analysis of the studies [30][31][32] shows that the operation of square-rooting both for indirect gradient methods [30] and the methods of quadratic programming with constraints [31] causes ambiguity. A comparison of the results for direct and indirect methods presented in the applied mathematics study [32] just confirmed the assumption of non-singularity. The latter is a major cause of oscillations between solution iterations and, as a consequence, an increase in time, and discrepancies for some conditions. Thus, the presence of a square root as a source of non-singularity is a major not solved problem and therefore the first reason for searching for and studying other forms of representation of initial mathematical models.
Further development of systems for management of emergencies, traffic on highways, etc., requires joint application of geoinformation systems and mapping tools which was substantiated and presented in [16] as a field of the current study. The contradiction between mapping systems that work with a continuous scale and the numerical methods of finding coordinates [30][31][32] using finite increments is about averaging of the first and second derivatives. The continuity requirement creates the need for interpolation and as a consequence generates an error.
Thus, the requirement of continuity of functions and their derivatives and discreteness inherent in numerical methods contradict. The latter is the second major reason for finding effective applied mathematical methods for determining coordinates of sound anomalies. However, the practical implementation of deterministic models for solving the mapping problem is also a problem of completeness and solution of the quadratic equation system. As substantiated and demonstrated in [33], its simplification implies a recurrent approximation of operators which will be effective for continuous vector functions or algebraic nonlinear items of differential models [34]. Search for the ways that can provide fast calculations or represent values of physical quantities by means of analytic membership functions is demonstrated in [35]. The advantages of adding capabilities of the apparatus of mathematical analysis demonstrated for management problems are convincing in its reality [36] in the presence of analytical solutions. In addition, its application requires additional numerical and physical experiments [37] and a special assessment of the possibilities of providing the specified accuracy and magnitude of errors. Another way to simplify a mathematical problem by applying indirect methods consists in reducing it to a recurrent sequence that allows rapid calculations [38]. However, for its practical application, this methodology requires a concerted action: the reformation of the model, proposal of new analytical methods and algorithms, and study of their convergence in the course of numerical experiments which is an unrelated task as well.

The aim and objectives of the study
The study objective is to develop a method that makes it possible to determine coordinates of a sound anomaly with a given accuracy.
To achieve this objective, the following tasks were set: -to construct a mathematical model that relates coordinates of sound anomaly according to the data of sound series to such parameters as the speed of sound propagation in air, the distance between microphones and their number and location; -to construct an approximate analytic solution of the problem of finding coordinates of a sound anomaly for three microphones using a scheme of linear approach and a linear and quadratic approximation; -to substantiate the algorithms of search by methods of indirect and direct solution of the problem of calculating the sound anomaly coordinate for the model form with a selection of three and four microphones; -to conduct a numerical experiment to determine coordinates of a sound anomaly with a specified accuracy.

1. Description of the computerized microphone system and a model of determining coordinates of sound anomalies based on sound series
Let us consider a system consisting of three or four microphones and controllers. The system is capable of receiving and recording in time audio signals previously converted into electrical signals. Suppose that the system is provided with channels for receiving and transmitting data from a controller of each microphone to the processor. The latter generates frame data for a protocol storage, prior to forming a series from all microphones. According to a command formed according to the algorithm, it inquires, processes and saves the processed data in the time of recording the sound anomaly by individual microphones. Denote the source of sound anomaly and the sound receiver by letters c and r, respectively, in the subindex. Also, denote the speed of the sound wave propagation via v n . Under these conditions, denote the point and coordinates of the signal receiver М j (x rj , y rj , z rj ) in the Cartesian coordinate system. Suppose that coordinates of the microphone on the j-th vehicle, that is, UAV or at a stationary surveillance station are defined at this point. Denote by t сi and t rj , respectively, the time of occurrence and recording of the sound anomaly arising at the point C i with coordinates of the location (x ci , y ci , z ci ) and recorded by the microphone at point М j (x rj , y rj , z rj ) in the same time recording system. Suppose that the sound wave is spherical and its front is recorded as an instantaneous impulse. Under these assumptions and notations, reduce the original mathematical model to a system of equations which essentially are conditions of equality of distance between the source of sound anomaly and the microphone which are determined by their coordinates and the path passed by the spherical wave to that microphone. For example, write for the case of use of three microphones:  (1) Such a system contains three unknown coordinates and a fourth unknown, that is, the time of sound anomaly appearance. In essence, it can be considered as a problem with a parameter or requiring the search for an additional fourth equation.

2. Approximate analytical solution of the problem of finding coordinates of the sound anomaly for three microphones
The formed system (1) is nonlinear, search for its solution is not obvious and unique. Let us explore possible ways of the search for its solution and analyze and compare them. To construct the approximate method, form a functional The items under the sum sign in the functional expression are always positive because they are squares of real numbers, that is, the values of coordinates. In addition, the error in determining coordinates is a bounded quantity determined by the problem content. Denote it by Δl [ ] . Thus, formulate the original problem (1) as a problem with a minimization parameter with constraints: ( ) Record the required condition of minimum for all sought coordinates of a sound anomaly: after direct differentiation, it can be presented as follows: The formulated problem can be solved (including the method of recurrent approximation) in two ways.

2. 1. The solution of the problem of finding the sound anomaly coordinates by a linear scheme of approximants and a linear approximation
Expand the system (4) and reduce the problem (3) to a linear system of algebraic equations using the method of recurrent approximation [33,35] and limiting ourself by a linear scheme of approximants and a linear approximation: Introduce notation of respective coefficients for increments of the sought unknowns and derive their expressions by direct differentiation:   3 Electronic copy available at: https://ssrn.com/abstract=3708254 and also introduce the increment for the n+1-st and n-th approximants: Δx Thus, system (4) will be transformed to an approximant system, which will be a compact representation (5) and rapidly convergent at conditions of boundedness of the second derivatives or coefficients: It is easy to simplify it by the method of exclusion if Δх сі is first excluded: Next, exclude a a a a a a z ci Δ .
The solution of the latter makes it possible to consistently calculate increments which can be generalized for the n-th approximation:  a a a a  a a a a a a a a a 31 Thus, the solution of the system will be reversed for the following approximation: Further, for the next coordinate, the following approximation is similarly given as: a a a a .
The third coordinate, also in the following approximation, is calculated by coordinates of the n-th approximation: The obtained increments make it possible to check the fulfillment of the inequality constraints in (6) and calculate the relative errors: If the modulus of relative error is greater than or equal to the specified error for all three coordinates, that is (|ε x |³[ε] and |ε y |³[ε] and |ε z |³[ε]), it is necessary to move on to the next calculation of coordinates of the sound anomaly point for the next approximation.

2. 2. The solution of the problem of finding coordinates of the sound anomaly by the linear scheme of approximants and a quadratic approximation
Let us expand system (4) and reduce the problem (3) to a nonlinear system of algebraic equations using the method of recurrent approximation [30,32] and restricting to a linear scheme of approximants and a quadratic approximation: Implementation of such a scheme assumes presence at each iteration of the values of approximation of increments of sought values which are zeroed at the first iteration. Expressions of the third derivatives of the selected functional necessary for the implementation of such a solution are as follows: Next, calculate other derivatives    but its coefficients will vary from approximation to approximation. It should be noted that subindices n and n+1 indicate the approximation number which means that in all values of the functional and its partial derivatives, coordinates of the sound anomaly and their increments are determined namely for these approximations. Thus, the approximate solution of the problem of finding coordinates of the sound anomaly for three microphones according to linear scheme approximants and linear or quadratic approximation can be combined into a single algorithm.

3. Direct solution of the problem of finding the sound anomaly coordinates
To provide a direct solution, consider the original system (1). Simplify the solution of the original nonlinear system (1) by reducing it to a linear one. Based on the properties of difference between squares of two numbers and possible combinations of two numbers of three, show the property of representing the difference of squares as a linear form with respect to each of the three numbers. To do this, subtract the equation written for a similar condition of the other microphone from the equation-condition of the signal reaching each microphone. When realizing this, write a new system of three equations: It is easy to make sure that now it is linear with respect to the desired coordinates x ci , y ci , z ci of the i-th sound anomaly and t ci is a parameter. Call it a combination-difference form.

3. 1. Algorithm for direct solving a problem of three microphones
The problem is reduced to considering the wave propagation from a sound anomaly with unknown coordinates to the receivers with known coordinates and known time of its registration. It is also stated by the original conditions that the computer system has three microphones. According to these conditions, problem (1) will be represented by a system of three equations after algebraic transformations of the combination-difference form: Thus, the direct calculation has given coordinates of sound anomaly according to the data for three microphones. It also should be noted that at such a system setting, its independence from the time of the sound anomaly is observed and accuracy will be determined by the relative positioning of microphones and the distance between them. A conclusion can be drawn from the solution that distance between microphones determines the solution error and that microphones themselves should not be in the same plane and at the same height. It is impossible to ensure the fulfillment of these conditions for three microphones in all cases at an arbitrary location of the sound anomaly. In the cases where the wave propagation time to two microphones is practically the same or the distance between the microphones is practically the same, the error in determining the coordinates is greatest. To reduce it, it is necessary to increase the number of microphones and distance between them and change their position in space.

3. 2. An algorithm of direct solution of the problem for four microphones
The initial system is formed on the basis of equality of distance between the source of sound anomaly and the path passed by a spherical wave for four microphones: To simplify the solution of the original nonlinear system (19), it is necessary to convert it to a linear combination-difference form. To do this, subtract the equation with a similar condition from the equation-condition of the signal reaching each of the microphones. When implementing the above, write: Enter notations: a a a b a b   a a  a a a b  Then the system will be simplified to a system of two equations with two unknowns: = ; a a 32 33 = ; a a 13 12 = .
Thus, parameter D 1 turns to zero for individual values of coordinates of microphone locations which in turn blocks the operation of the algorithm unless it is corrected by the logical transition to another equation of system (21).

Conducting the numerical experiment
The data on conditions for conducting the numerical experiment characterizing coordinates of the four microphones are given in Table 1. Coordinates for the numerical experiment were selected based on recommendations given in [15]. Table 2 presents additional parameters for conducting the experiment: the time of recording an abnormal event by microphones. Besides, coordinates of the event and its absolute error E, iteration number n, and total time τ of the calculation obtained during the experiment are presented. Table 1 Coordinates of microphones of the computerized sound anomaly recording system The experiment was conducted on a computerized system capable of recording audio anomalies with four microphones. This variant of the system from many numerical experiments was chosen based on the need to simultaneously determine three coordinates and time of the event. The simulation results are presented in Table 2. Column 1 shows the experiment number. Columns 2, 3, 4, 5 show the time of recording the audio event. Coordinates of the calculated sound anomaly are presented in columns 6, 7, 8, respectively. In the course of the numerical experiment, calculations were performed until the absolute error of 0.01 m was reached which is reflected in Table 2.
The calculations were continued to provide an absolute error of 0.0001 m. These two values of n and N (total number of iterations) are given in the ninth column by first and second numbers, respectively, separated by a slash. The total calculation time τ is the time for which the error reaches a value less than the specified value of 0.0001 m. This parameter is shown in Table 2 after the second slash in the ninth column.
The highest amount of spent time 98, 91, 158, 105 ms was in experiments 11, 16, 17, 20, respectively. In order to reduce cell size in Table 2 when error values were less than 0.01 m, the value was written as 0. It should be noted that even the largest of these values is sufficiently small (0.158 s) and capable of satisfying the requirements for operation of AMS when solving the list of problems [1,9,10,15,16,[24][25][26][27][28][29] based on the idea of determining coordinates of sound anomaly sources. Thus, summarizing the results of numerical simulation, it should be added that the greater the distance between microphones, the smaller the number of iterations in the process of finding the final result for the chosen calculation accuracy. The placement of microphones along the perimeter of the scanned area (a square) is optimal.
When the microphone placement is compact, the values of their coordinates of the same name must not be the same. In addition, the elevation of placement is always greater than the vertical coordinate of the sound anomaly. Compliance with these requirements reduces both the number of iterations and the total time spent in calculating the sound anomaly coordinates.

Discussion of results obtained in studying the applicability of indirect and direct methods for determining the sound anomaly coordinates
The obtained results show that the constructed model of recording the sound anomaly of a single point of the spherical wave's front makes it possible to make analytical solutions both by direct and indirect methods. Solutions of the problem of finding coordinates of sound anomaly according to the linear approximant scheme and by linear and quadratic approximations that were constructed and represented by expressions (11)- (13) and (14), (15) are analytic recurrent expressions. Their analyticity is a major advantage over the results of the studies determining coordinates of the sound anomaly by numerical methods [9,10,15,16,[27][28][29]. Thanks to such property as analyticity, it is relatively simple to build an algorithm of calculating coordinates of a sound anomaly. The difference between the current results and the results obtained in [27][28][29] can be explained by the proposed elimination of irrationality from the model composition in the course of its formation, departure from the methods of [30,31] and application of the recurrent approximation method (RAM) [33][34][35] for the solution. The data of numerical experiments given in Tables 1 and 2 also demonstrate stable algorithm operation and good enough convergence. The number of iterations (10)(11)(12)(13)(14)(15)(16)(17), accuracy selected and adjusted and total calculation time (maximum 0.158 s) privilege the suggestions obtained.
The advantage of these results over those obtained by direct and indirect numerical methods [30][31][32] based on the Newton-Kantorovich method includes the advantages of the RAM itself. This method eliminates the main problem of optimization when gradient drops when approaching the optimum point leading to a step-like growth in the algorithm of root search. The results of numerical experiments show that the greatest error is made by systems in which coordinates of microphones and sound sources are almost identical or coincide. As expected from the analysis of expressions (11)- (13) and (14), (15) and is obvious from the analyticity of solutions can be explained by the following causes. First, under these conditions, coefficients of the equations reduce their values to almost zero or turn to zero. Secondly, the main determinants of intermediate systems tend to small quantities, and the difference between values of the sought coordinates increases sharply from iteration to iteration since it is inversely proportional to the result tending to zero because of the operator action which slows down convergence of solutions.
The proposed transformation of a system of nonlinear equations to a linear combination-difference form (16) has made it possible to obtain a simple algorithm of finding coordinates. However, since it has been reduced to the solution of a system of algebraic equations of the first order, it has limitations on the principal determinant of the system. The latter leads to a limitation on the design parameters of the microphone system, that is, the distance and location which must be taken into account when implementing it.
As evidenced by the results of the numerical experiment (Table 2), the way out of such a position implies the spatial separation of microphones and an increase in their number combined with selective data sampling to calculate the sound anomaly coordinates. The idea of redundancy [27] and selection can also produce new positive results when combined with the solutions obtained. Therefore, the proposed solution to the problem with four microphones is one of the advantages of using the proposed solutions. The conclusion drawn from an analysis of the results obtained in the numerical experiment, regardless of the methods used, is as follows. In contrast to the proposals of studies [7,14], in order to reduce the error, it is necessary to provide spatial separation of microphones between themselves and relative to the plane of two of them. This conclusion is important for practitioners and the implementation of systems because it reduces the number of microphones and improves the accuracy of the coordinate determination.
However, accuracy is not regulated for direct methods. Its value is determined by the accuracy of calculating the coefficients and the items that do not contain unknowns. In the cases of the orientation of the sound source to the microphones when distances become equal, the right parts approach zero. In this case, because of the error of recording the time of the leading wave edge, the solutions lose the property of unambiguity and contain an error of uncertain value which is unadjustable.
Applying the approximate method to coordinate search by solving the minimization problem with the involvement of RAM makes it possible to build simple algorithms of analytical solutions. Their implementation to solve the problems of numerical experiments quickly gives practically accurate values of coordinates being the feature and advantage of using the proposed method of solution. Thus, the constructed model does not contain the flaws produced by the irrationality of the equation systems and the limitations of Newton's method regarding the absence of not simple roots and the functional oscillations.
During the numerical experiments with the proposed algorithms based on the problem solutions by direct methods, it was found that the system coefficients turn to zero at a certain coincidence of parameters. Applying the methods of logical analysis and rules of inference to the construction of algorithms makes it possible to avoid division by infinitesimal values or zero. This approach reduces the number of iterations which in turn reduces the overall search time. Simultaneous application of the offered algorithms of the search for the sound anomaly coordinates and mapping tools obviously forms the basis for the construction of automated emergency monitoring and control subsystems [1,15,16].
This study drawback is its focusing on a simplified model for which the fact of sound anomaly is considered as an instant impulse without interferences and the tasks are related to constructing the model and the aspects of constructing solutions and algorithms of determining the anomaly coordinates. However, real sound series contain an aggregate of resulting sound images along with interferences. The choice of types of sounds of the amplitude-frequency description of anomaly sounds and characteristic points on them enables the application of the obtained models and methods of determining coordinates for the surveillance systems. Due to this, the main drawback, as well as the task of further improvement of the system, is the necessity of developing a method of severance of the anomaly image from the aggregate spectrum and recording the moment of time. The solution to this problem forms the direction of improvement and implementation of AMS for recording and mapping of events of general purpose.

Conclusions
1. The idea of propagation of spherical acoustic waves from point sources the front of which is recorded as instantaneous impulses by spatially separated microphones makes it possible to construct a mathematical model that does not contain irrationality. The relation between coordinates of the sound anomaly and the speed of sound propagation in air, the distance between microphones, their number and location are represented by recurrent systems that adjust the set accuracy.
2. The solution of the problem of finding coordinates of the sound anomaly by indirect methods for three microphones obtained by a linear scheme of approximants and a linear and quadratic approximation was reduced to an analytic recurrent form. Their implementation provides simple algorithms for calculating coordinates of a sound anomaly that are rapidly convergent with a specified or adjustable accuracy and the number of iterations.
3. Conversion of the system of nonlinear equations to the proposed linear combination-difference form makes it possible to obtain a simple algorithm of searching by methods of direct solution of the problem of calculating coordinates of a sound anomaly in a single iteration for three and four microphones.
4. The numerical experiment with determining coordinates of a sound anomaly with a specified accuracy has demonstrated the workability of the algorithms built for this model by indirect and direct methods. The ability of the algorithms to controlled adjustment of accuracy reduces the number of iterations which is especially important for the cases of worse convergence. The maximum total coordinate calculation time was 0.158 s which satisfies the AMS requirements to speed capability. The algorithms are simple, fast converging, with adjustable accuracy and a short calculation time to complete convergence.