A hysteresis loop fitting method

Abstract

The application discloses a hysteresis loop fitting method, and relates to the technical field of hysteresis loop fitting, and comprises the following steps: using an alternating gradient magnetic measurement device, sequentially performing sweep frequency testing and sweep field testing on a test sample, and obtaining a raw data pair of the vibration speed and the magnetic field intensity of the test sample; preprocessing the magnetic field intensity signal of the test sample, and piecewise fitting magnetization curves and demagnetization curves; using a complex nonlinear function comprising a polynomial function term and an exponential function term to fit the hysteresis loop of the test sample; solving the related parameters of the complex nonlinear function; and establishing the fitted hysteresis loop of the vibration speed and the magnetic field intensity of the test sample according to the solved related parameters. The application can ensure the accuracy of the fitted hysteresis loop result, has sufficient model convergence speed, and has good noise resistance effect.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of hysteresis loop fitting, and more particularly to a hysteresis loop fitting method. BACKGROUND

[0002] The hysteresis loop refers to a relationship curve between vibration speed and magnetic field strength of a test sample under alternating magnetic field magnetization. In the test process of an alternating gradient magnetometer, some abnormal data with large fluctuations exist due to the influence of hardware devices, user behavior, environmental noise and the like, which has a significant negative impact on the accurate characterization of the magnetism of the sample. Therefore, a method for fitting and smoothing the test data is needed to improve the signal-to-noise ratio, and it is very important to finally fit an accurate hysteresis loop.

[0003] At present, most of the related technologies for hysteresis loop regression analysis use neural networks for approximate fitting. However, training a neural network requires a large number of parameters, such as the initial values of network topology structure, weight values and threshold values. Moreover, the learning process between observations cannot be observed, the output result has poor interpretability, there is no corresponding mathematical expression to express the relationship, which affects the credibility and acceptability of the result. The iteration speed is slow, the learning time is too long, and hundreds or thousands of iterations may not even achieve the purpose of learning. There is no actual physical constraint in the fitting process, which makes the corresponding hysteresis loop not conform to the actual loop model.

[0004] Therefore, how to make up for the deficiencies in the prior art, ensure the accuracy of the fitted hysteresis loop result, and improve the model convergence speed are technical problems that need to be solved by those skilled in the art. SUMMARY

[0005] Therefore, the present application provides a hysteresis loop fitting method, which can ensure the accuracy of the fitted hysteresis loop result while having a fast enough model convergence speed, and has good noise resistance effect.

[0006] In order to achieve the above purpose, the present application provides the following technical scheme:

[0007] A hysteresis loop fitting method, comprising the following steps:

[0008] An alternating gradient magnetic measurement device is used to sequentially perform sweep frequency testing and sweep field testing on a test sample to obtain a pair of original data of hysteresis characteristics of the test sample vibration speed and magnetic field strength;

[0009] The magnetic field strength signal of the test sample is preprocessed, and the magnetization curve and demagnetization curve are fitted in sections;

[0010] The complex nonlinear function is composed of a polynomial function term and an exponential function term, and the expression of the complex nonlinear function is designed as follows:

[0011] y1=(a1*(Gauss+x1)+b1);

[0012] y2=(a2*(Gauss+x2)+b2);

[0013] y3=(a3*(Gauss+x3)+b3);

[0014] y4=(a4*(Gauss+x4)+b4);

[0015]

[0016] In the formula, v represents the amplitude of the vibration speed of the sample, and the unit is mm / s; Gauss represents the magnetic field strength, and the unit is Oe; x1, x2, x3, x4, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, h1, and z1 are to-be-optimized fitting parameters;

[0017] Solving the related parameters of the complex nonlinear function;

[0018] According to the solved related parameters, a fitting hysteresis loop of the vibration speed of the test sample and the magnetic field strength is established.

[0019] The technical effects achieved by the above technical solution are as follows: according to the characteristic that the relationship between the vibration speed of the test sample and the applied magnetic field is an S-shaped hysteresis loop, based on the idea of hysteresis loop fitting, a complex nonlinear function composed of a polynomial function term and an exponential function term is designed to fit the related data of the hysteresis loop, so as to reduce the data test error and improve the accuracy of the hysteresis loop.

[0020] Optionally, obtaining a pair of original data of the hysteresis characteristic of the vibration speed of the test sample and the magnetic field strength specifically includes the following steps:

[0021] Turning on the power switches of all instruments of the alternating gradient magnetic measurement device, starting the control program on the industrial computer, and fixing the test sample at the sample seat of the sample rod;

[0022] Setting the corresponding parameters in the control program, performing a sweep frequency test on the test sample, and obtaining the vibration speed of the sample rod at different driving frequencies through the Doppler laser vibration measurement method;

[0023] Taking the frequency corresponding to the maximum vibration speed as the resonance frequency, performing a sweep field test on the test sample, obtaining the vibration information under different magnetic fields, and obtaining a pair of original data of the hysteresis characteristic of the vibration speed of the test sample and the magnetic field strength.

[0024] The technical effects achieved by the above technical solutions are that the alternating gradient magnetic measurement device adopts the Doppler laser vibration measurement technology to measure the vibration of the test sample, has the advantages of high measurement accuracy and fast response speed, and can quickly reach the signal-to-noise ratio threshold under a changing magnetic field and perform field scanning measurement.

[0025] Optionally, the magnetic field strength signal of the test sample is preprocessed, specifically including the following steps:

[0026] The single magnetic field strength signal data corresponding to the test sample at the resonance frequency is filtered and denoised by using a band-pass filtering method, and only the single signal at the resonance frequency is filtered out;

[0027] The hysteresis characteristic raw data of the vibration speed and the magnetic field strength of the test sample are divided into two groups of data according to the gradually increasing and gradually decreasing magnetic field strength, and the piecewise fitting of the magnetization curve and the demagnetization curve is performed.

[0028] Optionally, the related parameters of the complex nonlinear function are solved, specifically including the following steps:

[0029] The initial values of the parameters are set, wherein x1, x2, x3, x4 are constrained by the coercive force, used to control the left and right translation of the curve fitting result; a1, a2, a3, a4 are constrained by the magnetic field strength range, used to control the curve slope when the magnetic field is equal to 0; the preset values of b1, b2, b3, b4 are 0, used to assist the bias of the slope; the preset values of c1, c2, c3, c4, h1 are 1; z1 is constrained by the magnetization saturation value, used to control the saturation value / convergence value of the curve fitting result, i.e. the maximum value of the y-axis;

[0030] The nonlinear least squares method is used for fitting, a loss function is constructed through the error sum of squares between the real data and the complex nonlinear function containing the to-be-optimized fitting parameters, and the global minimum value of the loss function is obtained through the optimization algorithm, i.e. finding the parameter vector β in the model curve f(x, β) to minimize the residual sum of squares S(β):

[0031]

[0032] Wherein, β T =(x1, x2, x3, x4, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, h1, z1), i=1, 2, 3, 4.

[0033] The technical effects achieved by the above technical solutions are that the fitting method of the nonlinear least squares can reflect good noise resistance effect, and the test sample can also maintain a sufficient accurate hysteresis loop fitting result even in an environment with interference.

[0034] Via the technical solutions described above, compared with the prior art, the application provides a hysteresis loop fitting method, which has the following beneficial effects:

[0035] (1) According to the characteristics of the S-shaped hysteresis loop of the relationship between the vibration speed of the test sample and the applied magnetic field, based on the idea of hysteresis loop fitting, a complex nonlinear function composed of a polynomial function term and an exponential function term is designed to fit the hysteresis loop related data, which reduces the data test error and improves the accuracy of the hysteresis loop, and has practical application value in engineering.

[0036] (2) The monotonicity of the fitting function designed by the application can ensure the physical characteristics of the hysteresis loop fitting result, and the appropriate number of fitting parameters can ensure the accuracy of the hysteresis loop fitting result while having a fast model convergence speed; through the fitting method of nonlinear least squares, good noise resistance effect can be achieved, and even in the presence of interference, the test sample can also maintain a sufficiently accurate hysteresis loop fitting result.

[0037] (3) The application uses an alternating gradient magnetic measurement device to perform magnetic characterization on the test sample, which has the advantages of high measurement accuracy and fast response speed, and can quickly reach the signal-to-noise ratio threshold under varying magnetic fields and perform field scanning measurement. BRIEF DESCRIPTION OF DRAWINGS

[0038] In order to more clearly illustrate the technical solutions in the embodiments of the application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiment or prior art description. Obviously, the drawings in the following description are only embodiments of the application, and for those skilled in the art, other drawings can also be obtained without creative labor based on the provided drawings.

[0039] Figure 1 It is a flowchart of the hysteresis loop fitting method.

[0040] Figure 2 It is a structural schematic diagram of the alternating gradient magnetic measurement device.

[0041] Figure 3 It is a flowchart of the Levenberg-Marquardt algorithm for solving parameters in the optimization parameter iteration process.

[0042] Figure 4 It is a schematic diagram of the hysteresis loop fitting result of sample one.

[0043] Figure 5 It is a schematic diagram of the hysteresis loop fitting result of sample two.

[0044] Figure 6 It is a schematic diagram of the hysteresis loop fitting result of sample three.

[0045] Figure 7 a schematic diagram of the hysteresis loop fitting result of sample four;

[0046] Figure 8 a schematic diagram of the hysteresis loop fitting result of sample five;

[0047] Figure 9 a schematic diagram of the hysteresis loop fitting result of sample six;

[0048] Label: 1-test sample, 2-sample rod. DETAILED DESCRIPTION

[0049] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative work fall within the scope of protection of the present application.

[0050] To solve the problems involved in the background art and make up for the deficiencies in the prior art, the embodiments of the present application disclose a hysteresis loop fitting method, as shown in Figure 1 , comprising the following steps:

[0051] An alternating gradient magnetic measurement device is used to sequentially perform frequency sweep testing and field sweep testing on the test sample, so as to obtain a pair of original data of the hysteresis characteristics of the vibration speed and the magnetic field intensity of the test sample;

[0052] The magnetic field intensity signal of the test sample is preprocessed, and the magnetization curve and the demagnetization curve are fitted in sections;

[0053] A complex nonlinear function containing a polynomial function term and an exponential function term is used to fit the associated hysteresis loop of the vibration speed and the magnetic field intensity of the test sample;

[0054] The related parameters of the complex nonlinear function are solved;

[0055] According to the solved related parameters, the fitting hysteresis loop of the vibration speed and the magnetic field intensity of the test sample is established.

[0056] Next, the flow of the hysteresis loop fitting shown in Figure 1 will be specifically described to further understand the technical solutions of the present embodiment.

[0057] First, about the preparation of sample testing. Turn on the power switch of each instrument of the Alternating Gradient Magnetometer (AGM) device, start the control program on the industrial computer, fix the test sample 1 at the sample seat of the sample rod 2, see Figure 2 ;

[0058] Set the corresponding parameters in the control program, perform a sweep test on the test sample 1, that is, change the frequency of the gradient field in a certain frequency range with a fixed step under the condition that the bias field is unchanged, and obtain the vibration speed of the sample rod 2 under different driving frequencies by means of Doppler laser vibration measurement;

[0059] Take the frequency corresponding to the maximum vibration speed as the resonance frequency, and then perform a sweep field test on the test sample 1, that is, modify the magnetic field size in a certain range under the condition that the amplitude and frequency of the gradient field are unchanged. First, a sweep field is performed on the sample with a larger step under a large field, for example, the sweep field range is 1T (10 4 Oe) and the step is 50Oe. According to the results, the approximate position of the saturation field of the sample can be seen. Then, the sweep field range is reduced to slightly larger than the saturation field, and the sweep field step is also reduced to perform a fine sweep on the sample. The vibration information under different magnetic fields is obtained, that is, the original data pair of the vibration speed v of the test sample and the magnetic field strength Gauss.

[0060] Based on the above process, it can be understood that the embodiment adopts an Alternating Gradient Magnetometer device integrating sample testing, data acquisition and hysteresis loop fitting. The Alternating Gradient Magnetic Measurement technology is a magnetic characterization method developed based on the magnetic balance method. The technology measures the vibration of the magnetized sample in the alternating gradient field through double piezoelectric crystals, and then characterizes the sample. The AGM uses the Doppler laser vibration measurement technology to measure the vibration of the sample. The vibration unit and the vibration measurement unit of the traditional alternating gradient magnetic measurement device are separated, which can overcome the technical bottleneck of the traditional alternating gradient magnetic measurement and further improve the test precision. The AGM has the advantages of high measurement precision and fast response speed. It can quickly reach the signal-to-noise ratio threshold under a changing magnetic field and perform a sweep field measurement. The data of the test sample under the corresponding magnetic field strength is collected to fit the hysteresis loop of the corresponding sample characteristics, and the fitting result of the hysteresis loop is output.

[0061] Further, the magnetic field strength signal of the test sample is preprocessed, specifically including the following steps:

[0062] A band-pass filtering method is used to filter and reduce noise of the single magnetic field strength signal data of the test sample at the resonance frequency, and only the single signal at the resonance frequency is filtered out;

[0063] According to the step-by-step increase and step-by-step decrease of the magnetic field intensity, the hysteresis characteristic original data of the test sample vibration speed and the magnetic field intensity are divided into two groups of data, and the magnetization curve and the demagnetization curve are segmented and fitted.

[0064] Further, in order to fit the hysteresis loop of the test sample vibration speed and the magnetic field intensity, a complex nonlinear function as shown below is designed:

[0065] y1=(a1*(Gauss+x1)+b1);

[0066] y2=(a2*(Gauss+x2)+b2);

[0067] y3=(a3*(Gauss+x3)+b3);

[0068] y4=(a4*(Gauss+x4)+b4);

[0069]

[0070] In the formula, v represents the sample vibration speed amplitude, the unit is mm / s; Gauss represents the magnetic field intensity, the unit is Oe; x1, x2, x3, x4, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, h1, z1 are to-be-optimized fitting parameters. In addition, the monotonicity of the fitting function can guarantee the physical characteristics of the hysteresis loop fitting result, and an appropriate number of fitting parameters can guarantee the accuracy of the hysteresis loop fitting result while having a sufficient model convergence speed.

[0071] Further, the related parameters of the complex nonlinear function are solved, and the steps include the following steps:

[0072] The initial values of the parameters are set, wherein x1, x2, x3, x4 are constrained by the coercive force, for controlling the left and right translation of the curve fitting result; a1, a2, a3, a4 are constrained by the magnetic field intensity range, for controlling the curve slope when the magnetic field is equal to 0; the preset values of b1, b2, b3, b4 are 0, for assisting the bias of the slope; the preset values of c1, c2, c3, c4, h1 are 1, which is also an auxiliary role, and is automatically adjusted according to the optimization algorithm, increasing the degree of freedom of the fitting result; z1 is constrained by the magnetization saturation value, for controlling the saturation value / convergence value of the curve fitting result, that is, the maximum value of the y-axis;

[0073] The nonlinear least square method is used for fitting, a loss function is constructed through the error square sum term between the real data and the complex nonlinear function containing the to-be-optimized fitting parameters, and the global minimum value of the loss function is obtained through the optimization algorithm.

[0074] In the nonlinear least squares fitting problem, the goal is to find the parameter vector β in the model curve f(x, β) that minimizes the sum of squared residuals S(β):

[0075]

[0076] where β T = (x1, x2, x3, x4, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, h1, z1), i = 1, 2, 3, 4.

[0077] Further, for the parameter optimization problem of complex nonlinear function, the embodiment adopts Levenberg-Marquardt method, which is also called Trust Region Reflective algorithm when solving the nonlinear optimization problem. The Trust Region Reflective algorithm is a numerical iterative algorithm for solving the nonlinear optimization problem, that is, starting from the initial value of the set parameter, through step-by-step iteration and continuous improvement, until the optimal solution of the to-be-optimized fitting parameter is obtained. The basic idea of the Trust Region Reflective algorithm is to convert the optimization problem into a series of simple local optimization problems. If you want to start minimizing, you need to provide an initial guess of the parameter vector β. When there is only one minimum value, an unknowing standard guess such as β T = (1, 1, …, 1) algorithm will work normally; when there are local minimum values, only when the initial guess is close to the final solution, the algorithm will converge to the global minimum value.

[0078] In the iteration process, the parameter variable β will be replaced by the new β+δ. To determine δ, the fitting function f(x i , β) is linearly approximated: f(x i , β+δ)≈f(x i , β)+J i δ, where: represents the gradient of the fitting function f(x i , β) with respect to the parameter variable β;

[0079] The necessary condition for the residual sum of squares S(β) to reach the minimum value is Combined with the first-order approximation of the fitting function f(x i , β), the residual sum of squares

[0080] Add a range to δ as the Trust Region, which is considered to be effective within the Trust Region. The algorithm converges until the residual sum of squares is obtained, and the optimal fitting parameter of the complex nonlinear function is obtained.

[0081] Specifically, an index p is defined to characterize the degree of approximation, which is determined according to the difference between the approximate model and the actual function: if p is close to 1, it means that the approximation effect is good; if p is too small, it means that the actual decline value is much smaller than the approximate decline value, and it is considered that the approximation effect is poor, and the approximate range needs to be reduced; if p is too large, it means that the actual decline value is much larger than the approximate decline value, and it is considered that the approximation effect is poor, and the approximate range needs to be expanded.

[0082] Referring to Figure 3 The flow chart of the Levenberg-Marquardt algorithm for solving parameters is used to further understand the specific process, and the steps are as follows:

[0083] S1, presetting the initial value of the model parameter β and the initial optimization trust radius μ;

[0084] S2, for the kth iteration, solving:

[0085]

[0086] S3, calculating p k :

[0087]

[0088] S4, if then μ k+1 = 2μ k ;

[0089] S5, if then μ k+1 = 0.5μ k ;

[0090] S6, if p k is greater than a threshold value η, it is considered that the approximation is feasible, and β k+1 = β k + δ k ;

[0091] S7, judging whether the algorithm converges or not, if not, returning to S2, if yes, ending.

[0092] In the technical solution, the increment δ is limited in a sphere with a radius of μ, and it is considered that only in this sphere is effective; after D is added, this sphere can be regarded as an ellipsoid. In the optimization method proposed by Levenberg, D is taken as the unit matrix I, which is equivalent to directly restricting δ in a sphere; subsequently, Margaurdt proposed to take D as a non-negative diagonal matrix, and in practice, the square root of the diagonal element of J T is usually used, so that the range is larger in the dimension with small gradient.

[0093] Finally, according to the solved related parameters, a fitting hysteresis loop of the vibration speed of the test sample and the magnetic field strength can be established, as shown in the following table. Figures 4-9 FIG. 1 shows the fitting results of the hysteresis loop of the related test samples, wherein: sample one is a 16nm-thick FGT film, with a size of 5mm*10mm, in-plane test, a driving frequency of 49.7Hz, a sweep field step of 50e, and a sweep field range of 300e; sample two is a 16nm-thick FGT film, with a size of 5mm*10mm, in-plane test, a driving frequency of 49.7Hz, a sweep field step of 20e, and a sweep field range of 300e; sample three is a 16nm-thick FGT film, with a size of 5mm*10mm, in-plane test, a driving frequency of 49.7Hz, a sweep field step of 20e, and a sweep field range of 300e; sample four is a 20nm-thick FGT film, with a size of 5mm*10mm, in-plane test, a driving frequency of 51.5Hz, a sweep field step of 20e, and a sweep field range of 300e; sample five is a 20nm-thick CoFe film, with a size of 10mm*8mm, in-plane test, a driving frequency of 43Hz, a sweep field step of 10e, and a sweep field range of 60e; and sample six is a 20nm-thick CoFe film, with a size of 10mm*8mm, in-plane test, a driving frequency of 211.3Hz, a sweep field step of 10e, and a sweep field range of 60e.

[0094] It can be seen that the technical scheme is designed based on the idea of hysteresis loop fitting, and a complex nonlinear function composed of a polynomial function term and an exponential function term is used to fit the related data of the hysteresis loop. The monotonicity of the fitting function ensures the physical characteristics of the hysteresis loop fitting result. The appropriate number of fitting parameters ensures the accuracy of the fitting hysteresis loop result and has a fast model convergence speed, which makes up for the shortcomings of the prior art.

[0095] The embodiments in the specification are described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts of each embodiment can be referred to each other. The above description of the disclosed embodiments enables a person skilled in the art to implement or use the present application. Various modifications of the embodiments will be apparent to those skilled in the art, and the general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, the present application will not be limited to the embodiments shown herein, but will conform to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A hysteresis loop fitting method characterized by, The method comprises the following steps: The alternating gradient magnetic measurement device is used to sequentially perform sweep frequency testing and sweep field testing on the test sample, and obtain a pair of original data of the hysteresis characteristic of the vibration speed and the magnetic field strength of the test sample; The magnetic field strength signal of the test sample is preprocessed, and the magnetization curve and the demagnetization curve are segmented and fitted; A complex nonlinear function containing a polynomial function term and an exponential function term is used to fit the associated hysteresis loop of the vibration speed and the magnetic field strength of the test sample; wherein the expression of the complex nonlinear function is designed as: y1=(a1*(Gauss+x1)+b1); y2=(a2*(Gauss+x2)+b2); y3=(a3*(Gauss+x3)+b3); y4=(a4*(Gauss+x4)+b4); In the formula, v represents the sample vibration speed amplitude, and the unit is mm / s; Gauss represents the magnetic field strength, and the unit is Oe; x1, x2, x3, x4, a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, h1, z1 are to-be-optimized fitting parameters; The related parameters of the complex nonlinear function are solved; According to the solved related parameters, the fitting hysteresis loop of the vibration speed and the magnetic field strength of the test sample is established.

2. A hysteresis loop fitting method according to claim 1, characterized in that, The pair of original data of the hysteresis characteristic of the vibration speed and the magnetic field strength of the test sample is obtained, and the method comprises the following steps: The power switches of all instruments of the alternating gradient magnetic measurement device are turned on, the control program on the industrial computer is started, and the test sample is fixed at the sample seat of the sample rod; The corresponding parameters in the control program are set, the test sample is subjected to sweep frequency testing, and the vibration speed of the sample rod under different driving frequencies is obtained by means of Doppler laser vibration measurement; The frequency corresponding to the maximum vibration speed is taken as the resonance frequency, the test sample is subjected to sweep field testing, and the vibration information under different magnetic fields is obtained, that is, the pair of original data of the hysteresis characteristic of the vibration speed and the magnetic field strength of the test sample.

3. A hysteresis loop fitting method according to claim 2, characterized in that, The magnetic field strength signal of the test sample is preprocessed, and the method comprises the following steps: The single magnetic field strength signal data corresponding to the test sample at the resonance frequency is filtered and denoised by using a band-pass filtering method, and only the single signal at the resonance frequency is filtered out; According to the step-by-step increase and step-by-step decrease of the magnetic field strength, the pair of original data of the hysteresis characteristic of the vibration speed and the magnetic field strength of the test sample is divided into two groups of data, and the magnetization curve and the demagnetization curve are segmented and fitted.

4. The hysteresis loop fitting method of claim 1, wherein, The related parameters of the complex nonlinear function are solved, and the method comprises the following steps: The initial values of the parameters are set, wherein x1, x2, x3, x4 are constrained by the coercive force, which is used to control the left and right translation of the curve fitting result; a1, a2, a3, a4 are constrained by the magnetic field strength range, which is used to control the curve slope when the magnetic field is equal to 0; the preset value of b1, b2, b3, b4 is 0, which is used to assist the bias of the slope; the preset value of c1, c2, c3, c4, h1 is 1; z1 is constrained by the magnetization saturation value, which is used to control the saturation value / convergence value of the curve fitting result, that is, the maximum value of the y-axis; The nonlinear least squares method is used for fitting, a loss function is constructed through a sum of squares of errors between real data and a complex nonlinear function containing to-be-optimized fitting parameters, and a global minimum value of the loss function is obtained through an optimization algorithm, that is, a parameter vector β in a model curve f(x, β) is found to minimize a residual sum of squares S(β): where β T = (xl, x2, x3, x4, al, a2, a3, a4, bl, b2, b3, b4, cl, c2, c3, c4, hi, zi), i = 1, 2, 3, 4.

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