A five-hole probe data processing method based on multiple linear regression and surface fitting

Abstract

The present invention discloses a five-hole probe data processing method based on multiple linear regression and surface fitting, which includes: obtaining the five-hole probe pressure value, the estimated value of the total pressure of the incoming flow, and the estimated value of the static pressure of the incoming flow; obtaining six dimensionless parameters and obtaining the multiple linear regression coefficients corresponding to each measuring point and each calibration point; fitting the corresponding regression coefficient surface and finding the maximum z value of the surface to determine the outflow angle; according to the airflow angle, calculating the total pressure value of the incoming flow and the static pressure value of the incoming flow at each measuring point by linear interpolation; judging whether the difference between the calculated static pressure value of the incoming flow and the estimated static pressure value of the incoming flow is less than a threshold value, and directly outputting the calculation result if it is. The present invention broadens the scope of use of the five-hole probe to a certain extent, reduces the processing accuracy of the five-hole probe, reduces the working environment requirements of the five-hole probe, and at the same time improves the service life of the five-hole probe.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-hole pneumatic probe data processing, and in particular to a five-hole probe data processing method based on multivariate linear regression and surface fitting. Background Art

[0002] Aircraft engines are not only the power behind human aircraft flight but also the thrust driving the development of aviation, often called the heart of the aircraft. Advances in aircraft engine technology have fueled every significant revolution in human aviation history. They are of great significance to my country's economy and national defense. With the increasing sophistication and diversity of flow field testing methods, many applications and experimental environments involve flow field measurement, requiring precise instrumentation and methods to measure flow field parameters. Compared to other testing methods, pneumatic probes offer advantages such as convenience, durability, and minimal testing environment requirements. Furthermore, and most importantly, they can measure the pressure at the test point.

[0003] Pneumatic probes can be divided into three-hole probes, five-hole probes, seven-hole probes and other types. Among them, the three-hole probe is used to measure flow parameters in two-dimensional space; the five-hole probe and the seven-hole probe can be used to measure airflow parameters in three-dimensional flow fields. However, due to the high manufacturing precision of the seven-hole probe, the large number of calibration data points and the large workload, the five-hole probe is used most frequently. The five-hole probe is often used as a simple and direct measurement tool that can accurately measure parameters such as airflow direction, velocity, total pressure, and static pressure in the flow field. However, it should be noted that the calculation accuracy of the five-hole probe depends on the interpolation algorithm, the calibration results and the machining accuracy of the probe.

[0004] For example, under the same processing accuracy, the interpolation algorithm of the five-hole probe has a great influence on the calculation accuracy. Generally speaking, there are three methods for measuring three-dimensional flow fields with a five-hole probe: the opposite measurement method, the semi-opposite measurement method, and the non-opposite measurement method; (1) The opposite measurement method: Although it is relatively intuitive, it takes a lot of time to find the pressure balance of holes 1, 3, 4, and 5, and the actual measurement is relatively complicated; (2) The semi-opposite measurement method: Although the operation is simple, it only requires finding the pressure balance of a pair of pressure holes, and the required data processing volume is small, but it is only suitable for uniform flow field measurement; (3) The non-opposite measurement method: By collecting the pressure values ​​of the five holes and comparing them with the calibration data of the probe, the pitch angle, yaw angle, total pressure, and static pressure of the measurement point can be interpolated and calculated. This method is the most commonly used, but its data processing volume is large and requires a more accurate interpolation method. The actual test system needs to use computer software programming to process data in order to have a faster test speed.

[0005] It should also be noted that during the manufacturing process of the probe, since the five holes on the head of each probe are not strictly symmetrical, the aerodynamic characteristics of each probe are different. In engineering, the five-hole probe is often required to be used at multiple Mach numbers. Each Mach number needs to be calibrated separately, and the calibration result can only be used within a certain range of Mach number changes.

[0006] The interpolation algorithm currently widely used in five-hole probes is the non-opposite bilinear interpolation method. The various calibration coefficients of this method - the angle coefficient K α , K β and the total static pressure calibration factor C Pt 、C ps , as shown in the following equations (2-1)-(2-5):

[0007]

[0008]

[0009]

[0010]

[0011]

[0012] When using the non-opposite bilinear interpolation method to calculate the five-hole probe, the calibration file will be imported first as the calculation standard. When calculating the flow field parameters of the measuring point, the pressure values ​​of each hole of the five-hole probe are used to calculate K_α and K_β. The α and β angles are interpolated in the calibration file through the bilinear interpolation method. Then, the total pressure and static pressure are interpolated from the total pressure coefficient and static pressure coefficient using the α and β angles. The total pressure and static pressure can be converted into Mach number, and the flow velocity can be calculated with the flow field temperature. The specific calculation process is as follows: Figure 1 In addition, there is a three-dimensional non-opposing linear interpolation algorithm with variable Mach number, as shown in Figure 2 This method adds the Mach number dimension to the bilinear interpolation calculation method. Since the bilinear interpolation calculation only uses the calibration file under one Mach number, it is difficult to accurately calculate the actual parameters when calculating the flow field with a large change in the incoming Mach number. This method first calibrates several equally spaced Mach numbers within a range, and then calculates the characteristic values ​​(K) of the same α and β positions of the calibrated Mach number characteristic curves. α , K β , C Pt , C Ps ) is fitted with the Mach number to obtain the relationship between the Mach number and K α , K β , C Pt , C PsThe relationship between them, that is, several one-variable multi-functions, can be used to obtain the characteristic curves at all Mach numbers in the calibration range by interpolation.

[0013] However, the above existing technologies all have the following disadvantages:

[0014] 1) The basis of linear interpolation is in K α , K β The interpolation process is carried out in the plane formed by Figure 3 This method has strict requirements for the interpolation cell, which needs to be kept as a convex quadrilateral. Once the interpolation cell becomes Figure 4 If a concave quadrilateral is present during calibration, the probe will be returned to the factory for reprocessing or used as a defective product for display purposes only.

[0015] 2) The multi-hole pneumatic probe usually adopts a pore size of 0.4mm, which is prone to clogging in some poor experimental environments, such as Figure 5 The figure shows a certain type of five-hole probe that is clogged. The current algorithm requires the pressure values ​​of the five holes of the probe to calculate each coefficient. Therefore, once the five-hole probe is clogged, it must either be scrapped directly or returned to the factory for repair and recalibration. Both methods are time-consuming and labor-intensive. Summary of the Invention

[0016] The present invention provides a five-hole probe data processing method based on multiple linear regression and surface fitting to overcome the above-mentioned technical problems.

[0017] The technical solution of the present invention is:

[0018] A five-hole probe data processing method based on multiple linear regression and surface fitting, characterized by comprising:

[0019] Step 1: Obtain the pressure value corresponding to each hole of each measuring point of the five-hole probe, namely, the pressure value of the five-hole probe, the estimated value of the total pressure of the incoming flow, and the estimated value of the static pressure of the incoming flow;

[0020] Step 2: Based on the calibration equation, six dimensionless parameters of the five-hole probe are obtained in the calibration wind tunnel as calibration coefficients for each calibration point;

[0021] Step 3: Obtain the multivariate linear regression coefficients corresponding to each measurement point and each calibration point;

[0022] Step 4: Fit the corresponding regression coefficient surface and find the maximum z value of the surface to determine the outflow angle. The regression coefficient surface is obtained by surface fitting using the angle data of the calibration point as the x and y variables and the multiple regression coefficients of the measurement point and each calibration point as the z variable.

[0023] Step 5: Calculate the total pressure value and static pressure value of the incoming flow at each measuring point by linear interpolation based on the airflow angle and the total static pressure calibration coefficient;

[0024] Step 6: Determine whether the difference between the calculated incoming flow static pressure value and the incoming flow static pressure estimate is less than the threshold value. If so, directly output the calculation result of step 5. Otherwise, use the calculation result of step 5 as the new incoming flow total pressure estimate and the incoming flow static pressure estimate, and return to step 2 for iterative calculation until the difference between the calculated incoming flow static pressure value and the current incoming flow static pressure estimate is less than the threshold value, and then end the iterative calculation.

[0025] Furthermore, step 2 includes the following calibration equations (1-1)-(1-6) to obtain six dimensionless parameters,

[0026]

[0027]

[0028]

[0029]

[0030]

[0031]

[0032] in P i (i=1, 2, 3, 4, 5) represents the pressure value of each hole of the five-hole probe, P t Indicates the total pressure of the incoming flow, P s Indicates the static pressure of the incoming flow.

[0033] Furthermore, in step 3, the pitch angle β is determined by Cp1 and Cp3, the deflection angle α is determined by Cp4 and Cp5, and the deflection angle α is determined by Cp2 and Cp ave Determine the total pressure and static pressure; and (Cp3+Cp5) 2 , (Cp1+Cp3) 2 , (Cp3+Cp2) 2 , (Cp2+Cp4) 2 , (Cp3+Cp4) 2 As the independent variable, Cp ave Multiple linear regression is performed with the above parameters of the calibration points as dependent variables to obtain the regression coefficients between the measurement points and each calibration point.

[0034] Beneficial effects of the present invention:

[0035] The present invention proposes a novel five-hole probe data processing method based on multivariate linear regression and surface fitting; it can not only solve the problem of the concave quadrilateral in the calibration curve of the current five-hole probe caused by processing, but also completely change the situation where the five-hole probe is directly scrapped when the measuring hole is blocked. In short, the present invention has broadened the scope of use of the five-hole probe to a certain extent, reduced the processing accuracy of the five-hole probe, reduced the working environment requirements of the five-hole probe, and increased the service life of the five-hole probe. Since the processing accuracy requirements for the five-hole probe are reduced, the qualified rate of the five-hole probe is better. In addition, the present invention can still be used when the measuring hole of the five-hole probe is blocked, so the use cost of the five-hole probe can be reduced to a certain extent. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following is a brief introduction to the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.

[0037] Figure 1 The figure is a calculation process step diagram of the existing non-opposing bilinear interpolation method;

[0038] Figure 2 The figure is a diagram of the existing three-dimensional non-opposing linear interpolation steps for variable Mach number;

[0039] Figure 3 This is a schematic diagram of the existing linear interpolation cell;

[0040] Figure 4 Schematic diagram of the concave quadrilateral (triangular area) appearing in the calibration curve;

[0041] Figure 5 This is a schematic diagram of a five-hole probe being blocked;

[0042] Figure 6 Schematic diagram of the five-hole probe hole sequence;

[0043] Figure 7 The present invention is a flowchart of the method steps. DETAILED DESCRIPTION

[0044] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0045] Currently, the existing technology for processing data from five-hole probes mostly uses non-opposed bilinear interpolation. This method can interpolate and calculate the total and static pressure values, Mach number, and airflow angle at the measurement point when the five-hole probe calibration curve is a convex quadrilateral. However, the calculation accuracy of the five-hole probe is significantly reduced when a concave quadrilateral exists. If the probe hole is clogged, the characteristic curve of the five-hole probe is completely destroyed, making the probe unusable and requiring only repair and recalibration or direct disposal.

[0046] To address these issues, the present invention proposes a novel five-hole probe calibration and calculation method. This method employs six dimensionless parameters as calibration coefficients and calculation criteria, employing an iterative calculation method based on multiple linear regression and surface fitting. By assigning a coefficient to each measuring hole, this method achieves accurate calculation results when the probe's measuring hole is blocked, while also enabling the use of five-hole probes with insufficient machining accuracy. To a certain extent, while maintaining the five-hole probe's calculation accuracy, this method broadens its scope of use and service life.

[0047] Therefore, if Figure 7 This embodiment provides a five-hole probe data processing method based on multiple linear regression and surface fitting, comprising the following steps:

[0048] Step 1: Obtain the pressure value corresponding to each hole of each measuring point of the five-hole probe, namely, the pressure value of the five-hole probe, the estimated value of the total pressure of the incoming flow, and the estimated value of the static pressure of the incoming flow;

[0049] Step 2: Based on the calibration equation, the six dimensionless parameters of the five-hole probe are obtained in the calibration wind tunnel as the calibration coefficients of each calibration point. Specifically, unlike the current interpolation algorithm, the present invention calculates the six dimensionless parameters by calculating the pressure value of each hole of the five-hole probe at the measuring point and preliminarily estimating the total static pressure value. That is, during calibration, the six dimensionless parameters shown in the following calibration equations (1-1) to (1-6) are used, where the hole number distribution of the five-hole probe is as follows: Figure 6 As shown;

[0050]

[0051]

[0052]

[0053]

[0054]

[0055]

[0056] in P i (i=1, 2, 3, 4, 5) represents the pressure value of each hole of the five-hole probe, P t Indicates the total pressure of the incoming flow, P s In practical applications, the dimensionless parameters shown in equations (1-1)-(1-6) can be recorded as calibration coefficients in a calibration file during calibration. This file includes the angle values ​​of the 169 calibration points and all the calibration coefficients for each point.

[0057] Step 3: Obtain the multivariate linear regression coefficients corresponding to each measurement point and each calibration point; Specifically, when processing the calibration data, the pitch angle β is determined by Cp1 and Cp3, the deflection angle α is determined by Cp4 and Cp5, and the deflection angle α is determined by Cp2 and Cp ave Determine the total pressure and static pressure; and (Cp3+Cp5) 2 , (Cp1+Cp3) 2 , (Cp3+Cp2) 2 , (Cp2+Cp4) 2 , (Cp3+Cp4) 2 As the independent variable, Cp ave As the dependent variable, multiple linear regression is performed with the above parameters of the calibration point to obtain the regression coefficients between the measurement point and each calibration point. Specifically, during calibration, different angles will have different Cp2 and Cp ave , then the corresponding Cp2 and Cp can be interpolated according to the determined α and β angles. ave , and using the pressure values ​​of the five holes at the same time, equation (1-5) can be used to inversely solve Pt (total pressure), and then equation (1-6) can be used to calculate Ps (static pressure); so that in the subsequent process, by comparing the calculated Ps (static pressure) with the Ps value used in the last iteration, it can be determined whether to continue the iterative calculation or output the results.

[0058] The preferred multiple linear regression method can be the least squares multiple linear regression method. Specifically, the independent variables and dependent variables required to determine the multiple linear regression equation are determined by the following method: First, the calibration file needs to be imported into the post-processing calculation software of the five-hole probe, and the calibration data Cp1, Cp2, Cp3, Cp4, Cp5, Cp6, Cp7, Cp8, Cp9, Cp10, Cp110, Cp12, Cp13, Cp14, Cp15, Cp16, Cp17, Cp18, Cp19, Cp20, Cp21, Cp22, Cp23, Cp24, Cp25, Cp26, Cp27, Cp28, Cp29, Cp30, Cp31, Cp32, Cp33, Cp34, Cp35, Cp36, Cp37, Cp38, Cp39, ave Calculate separately (Cp3+Cp5) 2 , (Cp1+Cp3) 2 , (Cp3+Cp2) 2 , (Cp2+Cp4) 2 , (Cp3+Cp4) 2And the angles are distinguished and placed in the corresponding calibration angles (i.e., deflection angle α, pitch angle β) in the software database; secondly, use the five-hole probe to start collecting data in the flow field. As the pressure values ​​of the pressure holes of the five-hole probe at the measuring point stabilize, the corresponding dimensionless parameters are calculated based on the equations (1-1)-(1-6) corresponding to each measuring point. Since the total pressure and static pressure of the measuring point cannot be determined during the initial calculation, the total pressure at the wind tunnel outlet and the static pressure at the wind tunnel outlet need to be used as initial estimates for the first calculation; after determining the dimensionless coefficients, the measuring point (Cp3+Cp5) 2 , (Cp1+Cp3) 2 , (Cp3+Cp2) 2 , (Cp2+Cp4) 2 , (Cp3+Cp4) 2 Calculate them separately; finally, the measured point and each calibration point (Cp3+Cp5) 2 , (Cp1+Cp3) 2 , (Cp3+Cp2) 2 , (Cp2+Cp4) 2 , (Cp3+Cp4) 2 As the independent variable x1-x11, the Cp of the measuring point and each calibration point ave As the dependent variable y, multiple linear regression is performed, and the regression coefficients of the measurement point and each calibration point are obtained.

[0059] Step 4, fit the corresponding regression coefficient surface and find the maximum z value of the surface to determine the airflow angle. The regression coefficient surface is obtained by using the angle data of the calibration point (i.e., the deflection angle α, the pitch angle β) as the x and y variables, and the multiple regression coefficients of the measuring point and each calibration point as the z variable for surface fitting. Specifically, the angles α and β of the calibration point and the regression coefficients corresponding to each measuring point are respectively used as x, y, and z coordinates to fit into the regression coefficient surface, and the maximum z value of the surface and its corresponding x and y coordinates are found to determine the α and β angles, that is, the angles α and β of the calibration point are used as x and y variables, and the regression coefficients of the measuring point and each calibration point are used as z variables for surface fitting to obtain a three-dimensional surface, and the point with the maximum z value (regression coefficient) is found on the surface, and the α and β angles corresponding to this point are used as the airflow angle of the measuring point for the first calculation.

[0060] Step 5: According to the airflow angle, the total static pressure calibration coefficient is used to calculate the total pressure value and static pressure value of the incoming flow at each measuring point through the linear stab value; then, Cp2 and Cp are calculated through the linear stab value. ave And inversely solve the total pressure and static pressure of the measuring point; use α and β angles in the calibration file Cp2, Cpave The total and static pressure values ​​of the measuring points calculated for the first time can be interpolated.

[0061] Step 6: Determine the difference between the calculated static pressure value of the incoming flow and the estimated static pressure value of the incoming flow (i.e., P Snew With P S The absolute value of the difference) is less than the threshold value. If so, the calculation result of step 5 is directly output. Otherwise, the calculation result of step 5 is used as the new estimated value of the total pressure of the incoming flow and the estimated value of the static pressure of the incoming flow, and the iterative calculation is returned to step 2 until the difference between the calculated static pressure value of the incoming flow and the current estimated static pressure value of the incoming flow is less than the threshold value. The iterative calculation ends. Preferably, the threshold value can be 0.001Pa. If the difference between the static pressure value and the initial estimated static pressure value is 0.001Pa, the calculation result is output. If the static pressure value differs from the initial estimated static pressure value by more than 0.001Pa, the total and static pressure values ​​of the measuring point calculated for the first time are used as the initial estimates to calibrate the measuring point coefficients Cp1, Cp2, Cp3, Cp4, Cp5, and Cp6 again. ave The angle, total pressure and static pressure values ​​of the measuring point are calculated directly in the same way until the difference between the newly calculated static pressure value and the last calculated static pressure value is less than 0.001Pa. The output result is the flow field parameters of the measuring point, that is, the final calculated total static pressure, as well as the deflection angle and pitch angle, and the necessary flow field parameters such as the Mach number, speed, and airflow angle of the measuring point are converted.

[0062] In addition, the calculation method proposed by the present invention can not only accurately calculate the flow field parameters of each measuring point, but also handle the process of calculating the measuring point parameters when one measuring hole of the five-hole probe is blocked. Since the calculation method proposed by the present invention does not calculate the pressure difference of the measuring holes of the deflection surface and the pitch surface when determining the airflow angle, but uses dimensionless calculation of each hole, when a measuring hole is found to be invalid, all the calculation coefficients related to the hole are directly eliminated, and the same calculation accuracy is achieved by increasing the number of iterations.

[0063] For example Figure 6 , it is determined that hole 1 is blocked, then the dimensionless parameters related to hole 1 will be eliminated during the calculation, that is, at this time, although there are Cp1, Cp2, Cp3, Cp4, Cp5, Cp ave However, the linear regression database of each measuring point will be updated as (Cp3+Cp5) 2 , (Cp3+Cp2) 2 , (Cp2+Cp4) 2 , (Cp3+Cp4) 2 , 9 independent variables, Cp1, (Cp1+Cp3) will also be calculated when measuring points 2The change from determining the linear regression coefficients for 11 independent variables to determining the multiple linear regression for 9 independent variables will inevitably lead to an increase in the number of iterations and calculation time. However, the calculation accuracy can still reach over 97% of the accuracy of the normal operation of the five holes.

[0064] In addition, the technical terms involved in this case are:

[0065] 1) Deflection Angle α and Pitch Angle β: In a three-dimensional flow field, the direction of airflow has a spatial angle. To facilitate research by researchers in this field, this spatial angle is usually decomposed into two angles: the deflection angle and the pitch angle. The deflection angle is the angle between the projection of the airflow spatial angle on the horizontal plane and the x-axis; the pitch angle is the angle between the airflow spatial angle and the vertical plane, that is, the complementary angle between the airflow angle and the z-axis.

[0066] 2) Five-hole probe calibration: The five-hole probe calibration mentioned in this invention refers to the probe sweeping the α and β angles in the calibration wind tunnel within a range of ±30°, with an angle interval of 5°, and a total of 169 collection points. That is, the angles taken are α = ±30°, ±25°, ±20°, ±15°, ±10°, ±5°, 0°, and β = ±30°, ±25°, ±20°, ±15°, ±10°, ±5°, 0°. First, the β angle is turned to -30°, and the α angle is swept from -30° to 30° at intervals of 5°. Then, the β angle is increased by 5°, and the α angle is repeated until all 13×13 points of data are collected.

[0067] 3) Calibration file: The pressure values ​​of each hole and the values ​​of each dimensionless parameter at different angles collected during the calibration of the five-hole probe are the standards for subsequent calculations, which are equivalent to the scale values ​​of a ruler.

[0068] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A five-hole probe data processing method based on multiple linear regression and surface fitting, characterized in that: include: Step 1: Obtain the pressure value corresponding to each hole of each measuring point of the five-hole probe, namely, the pressure value of the five-hole probe, the estimated value of the total pressure of the incoming flow, and the estimated value of the static pressure of the incoming flow; Step 2: Based on the calibration equation, obtain six dimensionless parameters of the five-hole probe in the calibration wind tunnel as calibration coefficients for each calibration point; Step 3: Obtain the multivariate linear regression coefficients corresponding to each measurement point and each calibration point; Step 4: Fit the corresponding regression coefficient surface and find the maximum z value of the surface to determine the outflow angle. The regression coefficient surface is obtained by surface fitting using the angle data of the calibration point as the x and y variables and the multiple regression coefficients of the measurement point and each calibration point as the z variable. Step 5: Calculate the total pressure value and static pressure value of the incoming flow at each measuring point by linear interpolation based on the airflow angle and the total static pressure calibration coefficient; Step 6: Determine whether the difference between the calculated inflow static pressure value and the inflow static pressure estimate is less than a threshold value. If so, directly output the calculation result of step 5. Otherwise, use the calculation result of step 5 as the new inflow total pressure estimate and inflow static pressure estimate, and return to step 2 for iterative calculation until the difference between the calculated inflow static pressure value and the current inflow static pressure estimate is less than the threshold value, and then end the iterative calculation. Step 2 uses the following calibration equations (1-1)-(1-6) to obtain six dimensionless parameters: in P i Indicates the pressure value of each hole of the five-hole probe, i = 1, 2, 3, 4, 5; P t Indicates the total pressure of the incoming flow, P s Indicates the static pressure of the incoming flow.

2. The five-hole probe data processing method based on multiple linear regression and surface fitting according to claim 1 is characterized in that: include: In step 3, the pitch angle β is determined by Cp1 and Cp3, the deflection angle α is determined by Cp4 and Cp5, and the rotation angle α is determined by Cp2 and Cp ave Determine the total pressure and static pressure; and (Cp3+Cp5) 2 , (Cp1+Cp3) 2 , (Cp3+Cp2) 2 , (Cp2+Cp4) 2 , (Cp3+Cp4) 2 As the independent variable, Cp ave Multiple linear regression is performed with the above parameters of the calibration points as dependent variables to obtain the regression coefficients between the measurement points and each calibration point.

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