CN108627841A - A kind of circle center locating method in shoal buried pipes supersonic sounding - Google Patents

Abstract

The invention discloses a method for locating the center of a circle in ultrasonic detection of buried pipelines in shoals. Based on the arrival time and the least square method, the method comprises the following steps: constructing a simplified bright spot model of buried pipelines; using MATLAB to design simulation parameters to obtain simulated echo signals; building A simple single-channel ultrasonic transmitting and receiving experimental platform in the laboratory environment; simulate the complex environment of the shoal, add Gaussian noise with a signal-to-noise ratio of a certain threshold to the original experimental echo signal, and obtain the echo signal; respectively use the maximum amplitude method , characteristic parameter correlation detection method, and fast energy center convergence method to estimate the echo TOA of the echo signal, and select the best estimation method; calculate the acoustic distance between the transducer and the theoretical mirror image point according to the simplified bright spot model and echo TOA estimation fit the objective function of the center of the pipe according to the sound path and calculate the approximate numerical solution, and use the approximate numerical solution as the initial coordinate to solve the exact solution of the center of the pipe through iterative operations.

Description

A Circle Center Location Method in Ultrasonic Detection of Shoal Buried Pipeline

technical field

The invention relates to the field of ultrasonic detection of buried pipelines in shoals, in particular to a method for locating the center of pipelines based on time of arrival and the least square method.

Background technique

Long oil pipelines buried in shoal areas (rivers, swamps, ponds, rice fields, etc.) are generally laid below 1 to 2 meters of soil. After a leakage accident, conventional pipeline repair cofferdam operations usually have the following three deficiencies:

(1) The scope of cofferdam construction is much larger than the actual required operation area;

(2) If the location of the cofferdam construction deviates from the leakage site, it must be rebuilt;

(3) There are safety risks of collapse and water seepage in traditional cofferdams.

Therefore, if there is an automatic and intelligent machine equipment in cofferdam repair work for oil pipeline leakage, it can accurately locate the position of the buried pipeline in the construction preparation stage, and monitor the edge position of the cross-section circle of the pipeline in real time during the construction process , to protect the pipeline from mechanical or physical damage that may be caused by construction operations, can greatly shorten the cycle of emergency repair operations, prevent delays in work or cause huge waste of funds. At present, there is no special research on this kind of equipment or system that can monitor the edge position of the buried pipeline target in real time during the construction and excavation process ^[1] .

The shoal environment is a heterogeneous medium in which sediment and water are mixed similar to the seabed. Numerous studies and mature applications have shown that ultrasonic waves are an effective short-distance detection and measurement method in this type of environment ^[2]-[4] . However, the application of the previous active sonar technology (for example: shallow formation profiler) in the pipeline measurement usually needs to transmit the sound wave signal at a long distance, so that the sound wave can penetrate the seawater-sediment medium to obtain the echo.

In the process of realizing the present invention, the inventor finds that at least the following disadvantages and deficiencies exist in the prior art:

The above methods can only qualitatively obtain the sonar image and depth orientation of the pipeline, but cannot accurately locate the radial cross section of the pipeline, let alone accurately determine the edge position information of the pipeline.

Therefore, in the actual engineering application of ultrasonic detection of shallow pipelines, it is of great significance to accurately locate and detect the position of pipelines in real time.

Contents of the invention

The invention provides a method for locating the center of a circle in ultrasonic detection of buried pipelines in shoals. Based on a simplified bright spot model, the invention proposes a method for locating the center of a pipeline based on arrival time and the least square method, which realizes efficient and accurate alignment The effect of positioning the center of the pipe circle, see the following description for details:

A method for locating the center of a circle in ultrasonic detection of buried pipelines in shoals, the method for locating the center of a circle is based on time of arrival and the least square method, and the method for locating the center of a circle comprises the following steps:

Construct a simplified bright spot model of buried pipelines; use MATLAB to design simulation parameters and obtain simulated echo signals;

Build a simple single-channel ultrasonic transmitting and receiving experimental platform in a laboratory environment;

Simulate the complex environment of the shoal, and add Gaussian noise with a signal-to-noise ratio of a certain threshold to the original experimental echo signal to obtain the echo signal;

Using the maximum amplitude method, the characteristic parameter correlation detection method, and the fast energy center convergence method to estimate the echo TOA of the echo signal, and choose the best estimation method;

Based on the simplified bright spot model and echo TOA estimation, the sound path between the transducer and the theoretical mirror bright spot is calculated;

Fit the objective function of the center of the pipe according to the sound path and calculate the approximate numerical solution. Using the approximate numerical solution as the initial coordinate, the exact solution of the center of the pipe is obtained through iterative operations.

Wherein, the simplified bright spot model is specifically:

Taking the position of the transducer as the origin O, and taking the radial cross section of the pipe as a two-dimensional plane, when n transducers with an interval of d form a transducer array, construct a positional coordinate system between each transducer and the pipe;

The working form of the transducer array is that a single transducer transmits and receives echoes sequentially, and correspondingly generates n bright spot positions at the edge of the pipeline to obtain corresponding received echo signals.

Further, the bright spot is the edge point of the cross-sectional circle of the pipe, and the bright spot is located on the line connecting the center of the transducer and the center of the cross-sectional circle of the pipe.

Wherein, the position of the cross-sectional circle of the pipeline is specifically:

Among them, (x _i , y _i ) are the coordinates of the bright spot; (x ₀ , y ₀ ) are the coordinates of the center of the circle; (x′ _i , y′ _i ) are the coordinates of the transducer; l _i is the distance between the transducer and the bright spot sound process.

Further, the brief single-channel ultrasonic transmitting and receiving experimental platform set up under the laboratory environment is specifically:

Select the starting coordinate directly above the edge of the pipe, and the vertical distance from the center of the pipe to the transducer is a preset distance;

Starting from the starting coordinates, set a number of transducers from left to right to receive the echoes to be measured at intervals of a certain preset value;

Trigger an acoustic wave emission and reception at each point to be measured, and perform a data acquisition; obtain the echo signal obtained from a certain group of water tank experiments.

During specific implementation, the fitting of the pipeline center objective function according to the sound path and the calculation of the approximate numerical solution are specifically as follows:

Fit the center and radius of the circle according to the least square method; use the coordinates of the transducer and its relationship with the distance between the sound path and the discrete point of the arc to obtain the corrected objective function;

The modified objective function is written in an approximate form; the multivariate function is used to find the extremum method to solve it, and the equations are obtained, and then the numerical solution is obtained by calculation.

Further, the exact solution for calculating the center of the pipeline through iterative calculation using the approximate numerical solution as the initial coordinates is specifically:

Establish the matrix P _i according to the initial coordinates of the center of the circle and the initial step size, bring the matrix P _i into the revised objective function, obtain another matrix F _i , and find the minimum element F(x _j ,y _j in the other matrix F _i );

If i≠j, update the coordinates of the center of the circle as x _i =x _j , y _i =y _j ;

If i=j, then update the step size S=S/2, repeat the iterative steps;

If the step size S<S _min , the iteration stops, and S _min is the set minimum iteration step size.

The beneficial effects of the technical solution provided by the invention are:

1. This method is based on the simplified bright spot model, based on the idea of least squares, first establishes the corrected objective function, then deduces the numerical solution of the center of the circle, and then uses the numerical solution as the initial coordinate, and iteratively calculates the final precise solution of the center of the circle;

2. This method has important engineering significance for the precise positioning of the pipeline and the real-time monitoring of the edge position in the emergency repair operation of the shoal buried oil pipeline leakage, and is a key step to realize the automation and intelligence of detection;

3. This method verifies the calculation performance and accuracy of the positioning method through design simulation and experiments. The results show that the positioning method meets the accuracy requirements of actual engineering in water, and proves its feasibility in real environment applications.

Description of drawings

Fig. 1 is a flow chart of a method for locating the center of a circle in the ultrasonic detection of a shoal buried pipeline;

Fig. 2 is a schematic diagram of a bright spot model of a single transducer pipeline;

Figure 3 is a bright spot model of the transducer array pipeline;

Fig. 4 is a water tank experiment system diagram;

Fig. 5 is the echo signal chart that water tank experiment obtains;

Figure 6 is a simulated tidal flat echo signal diagram containing Gaussian noise;

Figure 7 is a schematic diagram of the comparison of the results of three commonly used TOA estimation methods.

Detailed ways

In order to make the purpose, technical solution and advantages of the present invention clearer, the implementation manners of the present invention will be further described in detail below.

Example 1

A method for locating the center of a circle in ultrasonic detection of shallow buried pipelines, the method for locating the center of a circle is based on the time of arrival and the least square method, see Figure 1, the method for locating the center of a circle includes the following steps:

101: Construct a simplified bright spot model of buried pipelines; use MATLAB to design simulation parameters and obtain simulated echo signals;

102: Build a simple single-channel ultrasonic transmitting and receiving experimental platform in the laboratory environment;

103: Simulate the complex environment of the shoal, and add Gaussian noise with a signal-to-noise ratio of a certain threshold to the original experimental echo signal to obtain the echo signal;

104: Estimate the echo TOA of the echo signal by using the maximum amplitude method, the characteristic parameter correlation detection method, and the fast energy center convergence method, and select the best estimation method among them;

105: Calculate the sound path between the transducer and the theoretical image bright spot according to the simplified bright spot model and echo TOA estimation;

106: Fit the objective function of the center of the pipe according to the sound path and calculate the approximate numerical solution, use the approximate numerical solution as the initial coordinate, and solve the exact solution of the center of the pipe through iterative operations.

Wherein, the simplified bright spot model in step 101 is specifically:

Taking the position of the transducer as the origin O, and taking the radial cross section of the pipe as a two-dimensional plane, when n transducers with an interval of d form a transducer array, construct a positional coordinate system between each transducer and the pipe;

The working form of the transducer array is that a single transducer transmits and receives echoes sequentially, and correspondingly generates n bright spot positions at the edge of the pipeline to obtain corresponding received echo signals.

Further, the above-mentioned bright spot is the edge point of the cross-sectional circle of the pipe, and the bright spot is located on the line connecting the center of the transducer and the center of the cross-sectional circle of the pipe.

Further, in step 102, the brief single-channel ultrasonic transmitting and receiving experimental platform in the laboratory environment is specifically as follows:

Select the starting coordinate directly above the edge of the pipe, and the vertical distance from the center of the pipe to the transducer is a preset distance;

Starting from the starting coordinates, set a number of transducers from left to right to receive the echoes to be measured at intervals of a certain preset value;

Trigger an acoustic wave emission and reception at each point to be measured, and perform a data acquisition; obtain the echo signal obtained from a certain group of water tank experiments.

During specific implementation, in step 106, the objective function of fitting the center of the pipeline according to the sound path and calculating the approximate numerical solution are specifically as follows:

Fit the center and radius of the circle according to the least square method; use the coordinates of the transducer and its relationship with the distance between the sound path and the discrete point of the arc to obtain the corrected objective function;

The modified objective function is written in an approximate form; the multivariate function is used to find the extremum method to solve it, and the equations are obtained, and then the numerical solution is obtained by calculation.

Further, in step 106, the approximate numerical solution is used as the initial coordinate, and the exact solution of the pipeline center is obtained through iterative calculation as follows:

Establish the matrix P _i according to the initial coordinates of the center of the circle and the initial step size, bring the matrix P _i into the revised objective function, obtain another matrix F _i , and find the minimum element F(x _j ,y _j in the other matrix F _i );

If i≠j, update the coordinates of the center of the circle as x _i =x _j , y _i =y _j ;

If i=j, then update the step size S=S/2, repeat the iterative steps;

If the step size S<S _min , the iteration stops, and S _min is the set minimum iteration step size.

To sum up, the embodiment of the present invention has important engineering significance for solving the precise positioning of the pipeline and the real-time monitoring of the edge position in the emergency repair operation of the oil pipeline buried in the shallows, and is a key step to realize the automation and intelligence of the detection.

Example 2

The scheme in Embodiment 1 is further introduced below in conjunction with specific calculation formulas and Figures 1-7, see the description below for details:

201: Construct a simplified bright spot model of buried pipelines;

The detailed operation of this step 201 is:

1) As shown in Figure 2, take the position of the transducer as the origin O, and take the radial cross section of the pipe as the two-dimensional plane (XOY plane), construct a single underwater acoustic transducer and the position coordinate system of the pipe. Assuming that the cross-section of the pipe, that is, the circle O', is within the -3dB beam width angle of the transducer as a whole, the edge AB of the arc in Figure 2 is the illuminated area of the sound beam, and the other arc areas are the sound shadow area.

It can be obtained from the geometrical acoustic theory that the point L is the intersection point of OO' and AB, which is also a theoretical mirror reflection point, and its reflection contributes the most to the echo. Other illuminated areas on arc AB except point L will produce scattered echoes equivalent to bright spots. It can be seen intuitively from the positional relationship in Fig. 1 that the mirror reflected echo at point L will arrive at point O before other scattered echoes.

2) When n transducers with an interval of d form a transducer array (T1, T2...Tn), a coordinate system for the positional relationship between each transducer and the pipeline is constructed, as shown in Fig. 3 . Assuming that the working form of the transducer array is that a single transducer transmits and receives echoes sequentially, then n bright spot positions L1, L2, ..., Ln will be correspondingly generated at the edge of the pipeline, and the corresponding received echo signals y _i ( t) (i=1,2,...,n).

3) It can be concluded from Figure 3 of the model that after the model is simplified, the bright spot is the edge point of the pipe cross-section circle, and the bright spot is located on the line connecting the center of the transducer and the pipe cross-section circle. If the coordinates of the transducer are (x′ _i , y′ _i ), the coordinates of the bright spot are ( _xi , y _i ), and l _i is the sound path between the transducer and the bright spot, then they have the following relationship:

Among them, (x ₀ , y ₀ ) are the coordinates of the center of the circle.

202: Use MATLAB to design simulation parameters to obtain simulated echo signals;

The details are as follows: take the transducer T1 as the origin of the coordinate system, the distance between the transducers d=0.01m, the center frequency of the transducer is 100kHz, the -3dB beamwidth is 17°, and the sound source level per volt is 164dB. The simulated oil pipeline is made of steel, the coordinates of the center O' of the pipeline are (0.08, 0.53) (unit: m), and the outer radius of the pipeline is 0.08m. Excite single-frequency rectangular pulse signal frequency f ₀ =100KHz, pulse width T=40us, amplitude 20V, sound velocity in water c=1500m/s, system sampling rate f _s =2MHz, single sampling depth N=2000.

203: Build a simple single-channel ultrasonic transmitting and receiving experimental platform in the laboratory environment;

The detailed operation of this step is:

1) As shown in Figure 4, use the signal generator as the excitation source to directly connect to a single transducer, and receive data through the data acquisition card to form a simple single-channel sound wave receiving and transmitting experimental platform in the laboratory environment;

2) Select the starting coordinate directly above the edge of the pipeline, the vertical distance between the center of the pipeline and the transducer is 0.53m, and the parameters such as pipeline material, pipeline radius, and excitation frequency are consistent with those under the simulation conditions in step 202;

3) Starting from the starting coordinates, set 8 transducers from left to right at an interval of 0.01m to receive the echoes to be measured;

4) Trigger an acoustic wave transmission and reception at each point to be measured, and perform a data collection;

Wherein, the parameter setting of the data acquisition card is consistent with that in step 202 .

5) Obtain the echo signals obtained from a certain group of water tank experiments, as shown in Figure 5.

The embodiments of the present invention do not limit the models of the signal generator, the transducer, the data acquisition card, etc., as long as they can complete the above functions.

204: Simulate the complex environment of the shoal, add Gaussian noise with a signal-to-noise ratio of 10dB to the original experimental echo signal, and obtain the echo signal as shown in Figure 6;

205: Estimate the echo TOA of the echo signal obtained in step 204 by using the maximum amplitude method, the characteristic parameter correlation detection method, and the fast energy center convergence method, and select the best estimation method;

Wherein, the above-mentioned methods are all methods known to those skilled in the art, which will not be described in detail in this embodiment of the present invention.

From the comparison results shown in Figure 7, it can be seen that in the case of adding Gaussian noise, the fast energy center convergence method can achieve better estimation results.

206: Calculate the sound path l _i between the transducer and the theoretical mirror bright spot according to the echo TOA estimation:

Among them, t _i is the echo arrival time, c is the speed of sound.

207: Fit the objective function of the pipeline center and calculate an approximate numerical solution;

The detailed operation of this step is:

1) Fit the center and radius of the circle according to the method of least squares;

Assuming that the radius of the oil pipeline is R, in practical engineering applications, R is usually known in advance, then the objective function of the least squares fitting circle can be expressed as:

2) The corrected objective function is obtained by using the transducer coordinates and the sound path distance relationship with the discrete points of the arc;

Since the arc discrete points (x _i , y _i ) are unknown parameters, the objective function cannot be directly calculated, and the modified objective function can be obtained by substituting the formula (1) in step 201 into formula (3):

3) Write the modified objective function of formula (4) as an approximate form:

4) Utilize the multivariate function to find the extremum method to solve, and obtain the equation system:

Since y′ _i =0, the numerical solution can be calculated after substitution:

in, Indicates the mean value of each l _i , and other similar forms are the same.

208: Using the approximate numerical solution as the initial coordinates, the exact solution of the pipeline center is obtained through iterative operations.

The detailed operation of this step is:

1) Assuming that the initial coordinates of the center of the circle are ( _xi ,y _i ), a 3*3 matrix P _i is established:

The steps between elements are equal, and the initial step size is S, that is, x _i = _xi-1 +S, y _i =y _i-1 +S. Matrix P _i is substituted into the formula (4) in step 207 to obtain F _i as:

2) Find the minimum element F(x _j ,y _j ) in the matrix F _i , if i≠j, update the coordinates of the center of the circle as x _i =x _j , y _i =y _j ; if i=j, update the step size S=S/2, repeat iteration step 1).

3) If the step size S<S _min , the iteration stops, and S _min is the set minimum iteration step size.

In summary, the embodiment of the present invention achieves the effect of efficiently and accurately locating the center of the pipeline; it has important engineering significance for solving the precise positioning of the pipeline and the real-time monitoring of the edge position in the emergency repair operation of the shoal buried oil pipeline. It is a key step to realize the automation and intelligence of detection.

Example 3

Below in conjunction with concrete example, table 1-table 4, carry out feasibility verification to the scheme in embodiment 1 and 2, see the following description for details:

According to the center fitting method given in the above-mentioned examples 1 and 2, the experimental signal is used for calculation to verify the accuracy of the proposed method for locating the center of the circle by ultrasonic detection of buried pipelines. The detailed operation is as follows:

1) Use the experimental platform and relevant experimental parameters in step 201 to carry out 7 repeated tests;

2) Using the fast energy center convergence method to estimate and compare the TOA of 7 groups of water tank experimental echo signals and simulated echo signals, the results are shown in Table 1;

3) use the numerical solution of the method fitting circle center in step 202;

Use |δ _x |, |δ _y | to represent the error between the fitted value and the real value in the x direction and y direction respectively, and use Represents the degree to which the fitted circle center deviates from the true circle center, and the results are shown in Table 2.

4) use the iterative algorithm in step 203 to solve the center of circle;

The initial step size S is 0.1m, the minimum iterative step size S _min is 0.000001m, and the initial coordinates of the iteration are the numerical solutions obtained in Table 2. The iterative calculation results are shown in Table 3.

5) Compare the performance of the iterative algorithm under different initial coordinates.

The numerical solution of the origin of the coordinates and the center of the circle obtained in Table 2 is used as the initial coordinates respectively, and the comparison of the number of iterations is shown in Table 4.

Table 1 Echo signal TOA estimation

Table 2 Numerical solution of the coordinates of the center of the circle

Table 3 Iterative calculation results

Table 4 Comparison of the number of iterations

6) Based on the above experimental results, it can be seen that the error of the circle center coordinates obtained after iterative calculation is relatively small.

The positioning accuracy requirements in practical engineering applications are generally less than 0.05m. Except for the measurement error of group 1 of the water tank experiment reaching 0.02m, the measurement errors of groups 2-7 are all below 0.01m, which meets the engineering accuracy requirements. In addition, after the iterative calculation is optimized, the number of iterations is significantly reduced, which improves the calculation speed.

references

[1] Wang Changwen. Application Research on Deepwater Submarine Pipeline Maintenance System Engineering [D]. Tianjin University: School of Architectural Engineering, 2010.

[2] Zhang Quanxing. Research on propagation and attenuation characteristics of ultrasonic inhomogeneous media [D]. Shenyang University of Technology: School of Information Science and Engineering, 2015.

[3] Nie Shijun. Research on the Transmission Law of Ultrasound in Mud [D]. China University of Petroleum: Oil and Gas Well Engineering, 2007.

[4] Cao Maoyong, Zhang Yifang. Error Analysis and Correction of Ultrasonic Ranging in Mud [J]. Metrology Technology, 1996, (10): 23-24.

Those skilled in the art can understand that the accompanying drawing is only a schematic diagram of a preferred embodiment, and the serial numbers of the above-mentioned embodiments of the present invention are for description only, and do not represent the advantages and disadvantages of the embodiments.

The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included in the protection of the present invention. within range.

Claims (7)

1. A circle center positioning method in ultrasonic detection of a shallow buried pipeline is characterized in that the circle center positioning method is based on arrival time and a least square method, and comprises the following steps:

constructing a simplified bright spot model of the buried pipeline; using MATLAB to design simulation parameters to obtain simulation echo signals;

a simple single-channel ultrasonic transmitting and receiving experiment platform under a laboratory environment is built;

simulating a complex shoal environment, and adding Gaussian noise with a signal-to-noise ratio of a certain threshold value into an original experiment echo signal to obtain an echo signal;

respectively adopting a maximum amplitude method, a characteristic parameter correlation detection method and a rapid energy center convergence method to carry out echo TOA estimation on echo signals, and selecting an optimal estimation method;

estimating and calculating a sound path between the transducer and a theoretical mirror image bright spot according to the simplified bright spot model and the echo TOA;

fitting a pipeline circle center objective function according to the sound path, calculating an approximate numerical solution, taking the approximate numerical solution as an initial coordinate, and solving an accurate solution of the pipeline circle center through iterative operation.

2. The method of claim 1, wherein the simplified bright spot model is specifically:

taking the position of the transducer as an origin O, taking the radial cross section of the pipeline as a two-dimensional plane, and constructing a coordinate system of the position relation between each transducer and the pipeline when n transducers with the interval of d form a transducer array;

the working form of the transducer array is that a single transducer sequentially transmits and receives echoes, and n bright spot positions are correspondingly generated at the edge of a pipeline to obtain corresponding received echo signals.

3. The method as claimed in claim 2, wherein the bright spots are the edge points of the pipe cross-section circle, and the bright spots are located on the line connecting the center of the transducer and the center of the pipe cross-section circle.

4. The method as claimed in claim 3, wherein the location of the circle center in the ultrasonic detection of the shallow buried pipeline is specifically as follows:

wherein (x)_i，y_i) Coordinates of the bright spots; (x)₀，y₀) As the coordinates of the circle center; (x'_i，y′_i) Are the coordinates of the transducer; l_iThe acoustic path between the transducer and the bright spot.

5. The method as claimed in claim 2, wherein the simple single-channel ultrasonic transmitting and receiving experiment platform under the laboratory environment is specifically:

selecting an initial coordinate right above the edge of the pipeline, wherein the vertical distance between the center of the pipeline and the transducer is a certain preset distance;

starting from the initial coordinate, setting a plurality of transducers from left to right at intervals of a certain preset value to receive the echo to be detected;

triggering once sound wave transmission and reception at each point to be measured, and performing once data acquisition; and obtaining echo signals obtained by a certain group of water tank experiments.

6. The method of claim 1, wherein the fitting of the pipe circle center objective function according to the acoustic path and the calculation of the approximate numerical solution are specifically as follows:

fitting the circle center and the radius according to a least square method; obtaining a corrected target function by using the coordinate of the transducer and the sound path distance relation between the transducer coordinate and the arc discrete point;

writing the modified objective function into an approximate form; and solving by utilizing a multivariate function extremum method to obtain an equation set, and further calculating to obtain a numerical solution.

7. The method as claimed in claim 6, wherein the approximate numerical solution is used as the initial coordinate, and the exact solution for solving the center of the pipeline by iterative operation is specifically:

establishing a matrix P according to the initial coordinates and initial step length of the circle center_iWill matrix P_iAfter correctionAn objective function, obtaining another matrix F_iFind another matrix F_iThe smallest element F (x) in_j，y_j)；

If i is not equal to j, updating the coordinates of the circle center as x_i＝x_j，y_i＝y_j；

If i is j, updating the step length S to be S/2, and repeating the iteration step;

if the step length S is less than S_minIteration stops, S_minIs the set minimum iteration step size.