Meteorological Instrument Performance Characteristics

### Instrument Performance Characteristics » Static Calibration » Calibration Exercise

Use the interactive ShinyApp tool to complete the following questions. If you are viewing a downloaded version of this lesson, please see the "Printable Lesson" for the ShinyApp exercise information.

Our objective is to calibrate a sensor so that we can use the calibration to estimate the value x of a measurand from the output of the sensor M. We have a calibrator that produces values of a measurand with very small random error and no systematic error, so its contribution to error in the calibration can be neglected. Our sensor, on the other hand, makes individual measurements with a rather large random error, characterized by the precision σM = 1.5. The goal of calibration is to minimize systematic error in measurements using this sensor. We decide to make N = 20 measurements equally spaced over a measurand interval (x1 – x2), and as a result we get N measurements {Mi} that correspond to {xi}. Results are as listed in the following table.

x

M

x

M

x

M

x

M

1

8.9783

6

23.8731

11

49.9256

16

79.8253

2

12.4642

7

31.8451

12

58.1260

17

83.5705

3

14.7360

8

34.9857

13

61.4559

18

90.9026

4

17.9198

9

39.6738

14

67.3803

19

99.7811

5

23.4088

10

43.6210

15

71.3742

20

105.4500

Using these data, find a representation of this calibration in the form x = f(M), where f(M) can be a polynomial or other function, so that the output M from this sensor can be used to estimate the measurand x, under static conditions.

Procedure: Download the data from the table to an application or program (e.g., ShinyApp, MS Excel, MatLab, Python, etc.) that you can use to plot x = f(M) and generate a set of calibration curves.

#### Question

If the sensor output is 55 (M = 55), what would your estimate of the measurand x be using a linear model fit?

To estimate the measurand x for a sensor output of M = 55, we create a plot of x = f(M) and determine the linear fit from least-squares regression. The resulting linear model is x = 0.1929M + 0.6689. Using R2 = 0.9869 yields a value ofx = 11.28 for M = 55.

The residual standard deviation of the calibration values for x, given M, from that regression line is 0.6967. That is not very good, and the deviations are systematic, so it is justified to try a higher-order fit. Using a linear-model fit to determine coefficients for a relationship x = b1 + b2M + b3M2

#### Question

What is your estimate of x as you increase the degree of the polynomial used to fit the measurand data to a second degree polynomial? What is your estimate if fitting to a third degree polynomial?