A New Algorithm for Reconstructing Two-Dimensional Temperature Distribution by Ultrasonic Thermometry

Temperature, especially temperature distribution, is one of the most fundamental and vital parameters for theoretical study and control of various industrial applications. In this paper, ultrasonic thermometry to reconstruct temperature distribution is investigated, referring to the dependence of ultrasound velocity on temperature. In practical applications of this ultrasonic technique, reconstruction algorithm based on least square method is commonly used. However, it has a limitation that the amount of divided blocks of measure area cannot exceed the amount of effective travel paths, which eventually leads to its inability to offer sufficient temperature information. Tomake up for this defect, an improved reconstruction algorithm based on least squaremethod andmultiquadric interpolation is presented. And then, its reconstruction performance is validated via numerical studies using four temperature distribution models with different complexity and is compared with that of algorithm based on least square method. Comparison and analysis indicate that the algorithm presented in this paper has more excellent reconstruction performance, as the reconstructed temperature distributions will not lose information near the edge of area while with small errors, and its mean reconstruction time is short enough that can meet the real-time demand.


Introduction
Temperature, especially temperature distribution is one of the most fundamental and vital parameters for theoretical study and control of various industrial applications, particularly of which related to burning and heating.Typical examples can be seen from applications of industrial furnaces.Temperature information is essential for the development of combustion theory and optimization of combustion systems, since every part of a combustion event, such as ignition, burnout, and evolution of emissions, is related to temperature [1,2].Information about temperature distribution is helpful for detecting and correcting hot spots which may cause safety accidents like explosions and spontaneous ignitions; meanwhile, it also contributes to designing combustion systems of low   , as presence of hot spots always increases   generation [3,4].Furthermore, temperature distribution always immediately influences the combustion efficiency of pulverized coal, the structure and state of reaction, and the safety of operation [4][5][6].Apart from these applications of industrial furnaces, temperature distribution and its reconstruction during combustion are of great interest in automotive and aerospace industries as well, because engine properties need to be monitored via studying flame dynamics [7,8].In particular, as flames in high-temperature combustions region generally become invisible, special and effective measurement techniques are required greatly to identify the combustion state [9].It can be concluded from the above examples that scientific research, economic benefits, ecological environment, and security requirements have aroused a strong need for improving thermometry techniques in industry fields.
Temperature measurement techniques of thermocouples and thermal resistances have been applied to industry for a long time.However, as intrusive methods, they require instruments to direct contact with measure medium, and the relatively slow time response in measurement always brings about inaccuracy; simultaneously, the temperature information gathered is insufficient owing to these methods' single-point measurement [2,[10][11][12].The infrared radiation technique is convenient but just for measuring surface temperature, and influences due to different emissivity and reflection of infrared radiations from other sources often result in deterioration of accuracy [11][12][13].Although methods using fiber optics become popular recently, they are not suitable for monitoring non-uniform temperature distribution either.For one thing, most of fiber optics also need direct contact with measure medium, which is similar to thermocouples and thermal resistances [13,14]; for another, this technique is always accompanied with a narrow operating range from 0 to 330 ∘ C, while temperature in applications is always beyond this range [15].Some studies of temperature distribution reconstruction have emerged in previous researches.The method based on analyzing colored radiation images captured by CCD cameras is working by a model relating the radiation image with temperature distribution [2,16].However, it is suitable only for visible luminous combustions or flames and not available for invisible situations.An inverse analysis has been presented for reconstruction of temperature profile in flames [17].But it is a pity that since the cross section of flame is divided into a series of concentric rings and temperature in each ring is expressed as a discrete value, it is just effective for axisymmetric free flames.A laser ultrasonic method for evaluation of temperature distribution is presented on the basis of temperature dependence of ultrasound [10,11].Nevertheless, it is practicable just for surface temperature distribution, as the surface acoustic waves used here propagate merely on the surface of medium.The CT method for calculating temperature distribution is also an effective reconstruction means of ultrasonic technique [18].Unfortunately, it requires large-scale machinery and a large number of transducers to collect enough projection data; thus it is bound to increase cost and running time.As another acoustic method, temperature distribution reconstruction using algorithm based on RBF neural network shows great advantages when compared to that using algorithm based on least square method [19].However, it requires a lot of training samples to calculate appropriate parameters of the neural network, while training samples are almost impossible to obtain in practice, even if obtained they are not accurate enough.
To overcome the shortcomings of conventional temperature measurement techniques and make up for the deficiencies of existing temperature distribution reconstruction methods, a nonintrusive ultrasonic technique of temperature distribution reconstruction is investigated in this paper, which is also presented based on the dependence of ultrasound velocity on temperature [3,[10][11][12][19][20][21][22][23].Temperature distribution reconstruction by this ultrasonic thermometry is an inversion problem, whose solution depends greatly on reconstruction algorithm.In practical application of this technique, reconstruction algorithm based on least square method has been commonly used, because it is simplest with high stability and can gain best match data by minimizing the error square sum.However, the temperature information it estimated is too limited to offer an accurate reflection of temperature distribution.Aiming at this shortage, an improved reconstruction algorithm based on least square method and multiquadric interpolation is presented, as it can deal with sparse data well while being accompanied with small error.
This paper is organized as follows.Section 2 describes the basic principle of ultrasonic thermometry.Section 3 presents a reconstruction algorithm based on least square method and multiquadric interpolation.Section 4 utilizes four two-dimensional temperature distribution models with different complexity in numerical studies to validate the reconstruction performance of our presented algorithm and gives experiment results and analysis.Conclusions and future research are presented in Section 5.

The Principle of Ultrasonic Thermometry
2.1.The Relationship between Ultrasound Velocity and Temperature.Ultrasonic thermometry is based on the dependence of ultrasound velocity on temperature, or that ultrasound velocity in any medium is a function of temperature.Such that, in ideal gases, the ultrasound velocity is directly proportional to the square root of temperature, and in most of liquids the dependence is linear, while in solid objects the ultrasound velocity generally decreases with the increment of temperature [22].Considering most widely situations in practice, principle based on ideal gas is given in detail.
The relationship between ultrasound velocity and temperature can be described as the following equation [3,[10][11][12][19][20][21][22]: where  is the ultrasound velocity,  is the gas absolute temperature,  is the universal gas constant, and  and  are the ratio and the average molecular weight of the gas mixture, respectively.As , , and  are fixed constants that can be measured for the specific gas, they may be replaced by a coefficient  which is also a constant.When the ultrasound velocity and gas properties are known, temperature  can be calculated by (2) derived from (1) as follows: A simplest ultrasonic thermometry instrument is composed by one ultrasonic transmitter mounted on one side and one ultrasonic receiver mounted on the other side along the same path.The transmitter is used to produce an ultrasonic signal at a specific time, and the receiver at a known distance is responsible for detecting the ultrasonic signal.As the distance between ultrasonic transmitter and ultrasonic receiver is fixed and known, ultrasound velocity can be calculated when travel time is measured.Thus (2) can be rewritten as where  is the distance between ultrasonic transmitter and ultrasonic receiver and   is the total travel time that the ultrasound is in transit.From (3), average temperature along the path is obtained.

Temperature Distribution Reconstruction.
Temperature distribution reconstruction will be accomplished using multiple ultrasonic transmitters and receivers (or collectively called ultrasonic transducers) installed over the measure area, as paths between them can be combined to compute temperature distribution with suitable algorithms [3,[19][20][21].With multiple transducers installed, the measure areas should be divided into several blocks according to specific situations.Figure 1 shows the typical situations in practical applications, which are square and circular measure areas, respectively.In the figures, transducers are represented by the black symbols, effective paths are represented by the solid lines within the areas, and measure areas are divided into blocks by the dashed lines.
The ultrasonic transducers used here may be called transceivers because of their ability to work as transmitters and receivers.That means that they can transmit and detect ultrasonic signals in different time.For example, as shown in Figure 1(a), when transceiver  1 acts as transmitter and radiates out ultrasonic signal, transceivers  2 ,  3 , . . .,  8 will play roles of receivers and detect the signal.And then, when transceiver  2 radiates out ultrasonic signal the other way round, transceivers  1 ,  3 , . . .,  8 will become receivers and detect the signal.The cycle repeats itself, all transceivers radiate out ultrasonic signal in rotation, and other transceivers catch the signal from the transceiver which serves as transmitter.
When all travel times of the paths between transceivers are obtained, they will be combined with information about both the arrangement of transceivers and the division of measure area, thus determining the temperature distribution.

Reconstruction Algorithm Based on Least Square Method and Multiquadric Interpolation
As inversion problem, an efficient algorithm is the core of temperature distribution reconstruction by ultrasonic thermometry.Only an excellent reconstruction algorithm can give an accurate reflection of temperature distribution and then help to evaluate the state of target area and develop control strategy.Reconstruction algorithm based on least square method is commonly used, since it is simplest with high stability and can gain best match data by minimizing the error square sum.However, there exists a limitation that the amount of divided blocks cannot exceed the amount of effective travel paths [10,19], which eventually leads to insufficient temperature information.To overcome this problem, a reconstruction algorithm based on least square method and multiquadric interpolation is presented, as it can deal with sparse data well while being accompanied with small error.

Estimating Temperature Array Using Least Square Method.
When transceivers are properly installed around the specific measure area and controlled to radiate out and detect ultrasonic signal, temperature array of the blocks will firstly be estimated using least square method.
In theory, travel time of a specific path can be described as the following equation of line integral: where   is the th travel path which the ultrasound travels through,    is the theoretical value of travel time, and  is the spatial character of ultrasound, that is, the reciprocal of ultrasound velocity.
Assuming that  is the amount of divided blocks, temperature of each block is uniform and   is the reciprocal of average ultrasound velocity in the specific block.Then ( 4) is turned to where  and , respectively, represent the number of path and the number of block and Δ  represents the length of the th path passing across the th block.
If   represents the actual measured value of travel time of the th path, then difference between   and the theoretical value    is expressed as Assigning that  is the amount of effective travel paths, then minimize the sum of the squares of these  differences, given by Then, canonical equation gained from (7) is where thus the reciprocal of average ultrasound velocity in each block can be calculated as When the geometric features of measure area as well as the arrangement of transceivers are known,  becomes a constant matrix which can be determined in advance.In addition, as  can be obtained by actual measurement techniques, average temperature of each block will be calculated from the following equation: where is the average temperature array of the blocks.
What must be noted is that, to ensure that the solution of this inversion problem is acceptable, it should be  ≤  in matrix .That is what leads to the limitation that the amount of divided blocks should not exceed the amount of effective travel paths.

Reconstructing Temperature Distribution Using Multiquadric Interpolation.
In Section 3.1, the average temperature array of blocks is calculated using least square method.It is clear that the amount of calculated temperatures is as same as the amount of divided block.However, as the amount of divided blocks cannot exceed the amount of effective travel paths, temperatures that can be estimated are too limited.Even using simple polynomial interpolation with these sparse temperatures cannot reconstruct temperature distribution effectively, as reconstructions will lose much information near the edge of measure area.In this subsection, a multiquadric interpolation is employed to reconstruct temperature distribution with these average temperatures, in consideration of its well capability of dealing with sparse data [24][25][26][27].
Set the average temperatures calculated from (11) as temperatures of the midpoints of the  blocks, and they will become sample points of the multiquadric interpolation model.Construct the multiquadric interpolation model of temperature distribution as where   (, ) is the nucleus function;   is the weight of the th nucleus function and is also the undetermined coefficient that will be solved afterward, and  is the amount of nucleus functions, which is also the amount of sample points and the amount of divided blocks mentioned before.Here, the nucleus function is chosen as multiquadric function: where ‖ ⋅ ‖ is the standard Euclidean norm, (, ) is the point whose function value needs to be approximated, (  ,   ) is the center coordinates of the nucleus function and is also the midpoint of the th block, parameter  is the smooth factor which is usually a small positive number or zero and can always be predetermined by research of practical data, and parameter  is also a constant which can be equal to 0.5, 1, or −0.5.
As described before, temperature of the midpoint of each block has been estimated.In other words, there are  sample points with known temperatures  1 ,  2 , . . .,   and known coordinates ( 1 ,  1 ), ( 2 ,  2 ), . . ., (  ,   ) conforming to (12).In this case, substitute these known quantities into (12), thus gaining the following: Equation ( 14) can then be rewritten as matrix form: where , and As  has been determined using numerical calculation, while  is also a fixed value, then all the  ×  elements of matrix Φ can be calculated from (13) in advance.Combining this predetermined constant matrix Φ with the average temperature matrix , then undetermined coefficient  will be gained from Finally, we can plug the solved  into (12) to get expression of the multiquadric interpolation model of temperature distribution, that is, (, ).

Simulation Experiments and Results Analysis
To validate the performance of temperature distribution reconstruction by ultrasonic thermometry based on the presented algorithm, numerical studies under atmosphere are designed.In this paper, numerical experiments and results analysis are both done on the MATLAB platform which is a powerful tool for numerical computation, simulation, and visualization.

Design of Experiment and Definitions of Evaluation.
As an improved algorithm presented on the basis of least square method, reconstruction results of the presented algorithm based on least square method and multiquadric interpolation are compared to that of algorithm based on least square method.For convenience, these two algorithms are, respectively, called LSM-MQ algorithm and LSM algorithm in the subsequent sections.
In order to analyze and evaluate the reconstruction results, effective contrast data is indispensable for comparison.However, due to constraints of practical environment and existing conventional techniques, it is difficult or even impossible to gain the accurate contrast data in realistic world.On the contrary, numerical studies can be a powerful means to investigate the reconstruction performance, since temperature distribution models artificially created in numerical environment can be used as standard of comparison.In this case, travel time used in reconstructions is gained via numerical computation from these models, and these models are what will be reconstructed afterwards in the experiments.The expressions of these four temperature distribution models created here are given as follows.
To better evaluate the reconstruction performance, not only qualitative analysis but also quantitative analysis is provided.The former is given as figures of isothermal contour displays.The latter includes two aspects, one is the overall temperature error analysis, which is given by calculating the mean relative error   and root-mean-square percent error   and the other is the real-time capability analysis that is given by calculating the mean reconstruction time, where reconstructions of each situation are repeatedly done 1000 times.Here,   and   are defined as ( 22) and ( 23), respectively, where  is the amount of calculating points of the area, TM mean is the mean temperature of the temperature distribution model, while TR  and TM  are temperatures of the point (  ,   ) of the reconstructed one and its model, respectively.

Experiment Results Analysis.
Considering that the edge of measure area is the installation of transceivers, so temperature distributions are reconstructed without the edge of 1 m, thus avoiding severe inaccuracy of reconstruction within this edge range.As a direct-viewing reflection of reconstruction performance, isothermal contour displays of temperature distribution models and the corresponding reconstructed ones, respectively, gained using LSM algorithm and LSM-MQ algorithm are shown in Figures 2-5.
It can be intuitively seen from the figure profiles that, within the range that can be reconstructed, temperature distributions reconstructed using these two algorithms are all in good agreement with the model ones.However, the reconstructed ones gained using LSM algorithm will lose much temperature information near the edge, while the reconstructed ones gained using LSM-MQ algorithm can offer all temperature information of the area.These qualitative figures indicate that reconstruction performance of our presented LSM-MQ algorithm is more excellent than that of LSM algorithm, as it can give a better reflection of temperature distribution of the whole area.
Quantitative analysis has more powerful persuasiveness with accurate data.Tables 1 and 2 show the mean relative error   and root-mean-square percent error   of reconstructions of the four temperature distribution models, of LSM algorithm and our presented LSM-MQ algorithm,  respectively.Furthermore, the mean reconstruction time of LSM algorithm and our presented LSM-MQ algorithm, respectively, is calculated and displayed in Table 3.
It can be seen from Tables 1 and 2 that, in situations of one-peak temperature distributions of both symmetrical and asymmetrical ones, the mean relative error   and rootmean-square percent error   of these two algorithms are all small enough, though the errors of LSM algorithm may be a little smaller than that of our presented LSM-MQ algorithm.However, in situations of two-and three-peak temperature distributions, all these errors of LSM algorithm are larger than that of our presented LSM-MQ algorithm, especially in the most complicated situation of the three-peak asymmetrical one.
Moreover, Table 3 indicates that the mean reconstruction time of both LSM algorithm and our presented LSM-MQ algorithm is very close, all of which is short and within 0.15 s.To be specific, in situations of one-peak symmetrical and three-peak asymmetrical ones, the reconstruction time of LSM algorithm is shorter than that of our presented LSM-MQ algorithm, while in situations of one-peak asymmetrical and two-peak asymmetrical ones, the reconstruction time of LSM algorithm is longer than that of our presented LSM-MQ algorithm.These data shows that our presented algorithm has well real-time capability and can meet the demand of online monitoring for most of industrial applications.It can be concluded from these quantitative data that, for the simple temperature distributions, the reconstruction performance of LSM algorithm and LSM-MQ algorithm is remarkable enough; however, for the complicated temperature distributions, our presented LSM-MQ algorithm is more excellent than the LSM algorithm.

Conclusions and Future Research
Temperature, especially temperature distribution is one of the most fundamental and vital parameters for theoretical study and control of various industrial applications, particularly of which related to burning and heating.In this paper, a nonintrusive ultrasonic technique to reconstruct temperature distribution is investigated, referring to the dependence of ultrasound velocity on temperature.Temperature distribution reconstruction by this ultrasonic thermometry is an inversion problem, and its reconstruction performance depends greatly on reconstruction algorithm.The commonly used algorithm based on least square method is simplest with high stability and can gain the best match data by minimizing the error square sum.However, due to the limitation that the amount of divided blocks of measure area cannot exceed the amount of effective travel paths, the temperature information it can provide is too limited to offer sufficient reference information for further control.To make up for this defect, an improved reconstruction algorithm based on least square method and multiquadric interpolation is presented in this paper, and its performance is validated via numerical studies using four artificial temperature distribution models.The reconstruction results of our presented algorithm are compared with that of algorithm based on least square method, under qualitative and quantitative analysis.Comparison and analysis indicate that the algorithm presented in this paper has more excellent reconstruction performance, as the temperature distributions it reconstructed will not lose information near the edge of area while being with small errors, and its mean reconstruction time is short enough within 0.15 s, thus meeting the real-time demand of most of practical applications.
The main contribution of this paper is to present a novel reconstruction algorithm based on least square method and multiquadric interpolation; thus the emphasis mostly focuses on studying the reconstruction performance of this algorithm via numerical studies.For better demonstrating the superiorities of our presented algorithm, experiments conducted in realistic world will be studied in the future.Thus, the major content of our future research will lie in the implementation of practical experimental system, the effectiveness of research of our presented algorithm in realistic environment, and the practical matters emerging during these processes and their solutions.

Figure 1 :
Figure 1: Schematic view of paths array and blocks: (a) square area and (b) circular area.

Table 1 :
Errors of LSM algorithm.