2. Measurement Process Characterization
2.3. Calibration
2.3.6. Instrument calibration over a regime

## Models for instrument calibration

Notation The following notation is used in this chapter in discussing models for calibration curves.
• $$Y$$ denotes a measurement on a reference standard
• $$X$$ denotes the known value of a reference standard
• $$\epsilon$$ denotes measurement error
• $$a$$, $$b$$, and $$c$$ denote coefficients to bedetermined
Possible forms for calibration curves There are several models for calibration curves that can be considered for instrument calibration. They fall into the following classes:
• Linear: $$Y = a + bX + \epsilon$$
• Quadratic: $$Y = a + bX + cX^2 + \epsilon$$
• Power: $$Y = aX^b\epsilon$$
• Non-linear: $$Y = g(X) + \epsilon$$
Special case of linear model - no calibration required An instrument requires no calibration if $$a = 0 \mbox{ and } b = 1$$ i.e., if measurements on the reference standards agree with their known values given an allowance for measurement error, the instrument is already calibrated. Guidance on collecting data, estimating and testing the coefficients is given on other pages.
Advantages of the linear model The linear model ISO 11095 is widely applied to instrument calibration because it has several advantages over more complicated models.
• Computation of coefficients and standard deviations is easy.
• Correction for bias is easy.
• There is often a theoretical basis for the model.
• The analysis of uncertainty is tractable.
Warning on excluding the intercept term from the model It is often tempting to exclude the intercept, $$a$$, from the model because a zero stimulus on the x-axis should lead to a zero response on the y-axis. However, the correct procedure is to fit the full model and test for the significance of the intercept term.
Quadratic model and higher order polynomials Responses of instruments or measurement systems which cannot be linearized, and for which no theoretical model exists, can sometimes be described by a quadratic model (or higher-order polynomial). An example is a load cell where force exerted on the cell is a non-linear function of load.
• They may require more reference standards to capture the region of curvature.
• There is rarely a theoretical justification; however, the adequacy of the model can be tested statistically.
• The correction for bias is more complicated than for the linear model.
• The uncertainty analysis is difficult.
Warning A plot of the data, although always recommended, is not sufficient for identifying the correct model for the calibration curve. Instrument responses may not appear non-linear over a large interval. If the response and the known values are in the same units, differences from the known values should be plotted versus the known values.
Power model treated as a linear model The power model is appropriate when the measurement error is proportional to the response rather than being additive. It is frequently used for calibrating instruments that measure dosage levels of irradiated materials.

The power model is a special case of a non-linear model that can be linearized by a natural logarithm transformation to $$Y = \mbox{log}_e(a) + b \cdot \mbox{log}_e(X) + \mbox{log}_e(\epsilon)$$ so that the model to be fit to the data is of the familiar linear form $$W = a' + bZ + e$$ where $$W$$, $$Z$$, and $$e$$ are the transforms of the variables, $$Y$$, $$X$$ and the measurement error, respectively, and $$a'$$ is the natural logarithm of $$a$$.

Non-linear models and their limitations Instruments whose responses are not linear in the coefficients can sometimes be described by non-linear models. In some cases, there are theoretical foundations for the models; in other cases, the models are developed by trial and error. Two classes of non-linear functions that have been shown to have practical value as calibration functions are:
1. Exponential
2. Rational

Non-linear models are an important class of calibration models, but they have several significant limitations.

• The model itself may be difficult to ascertain and verify.
• There can be severe computational difficulties in estimating the coefficients.
• Correction for bias cannot be applied algebraically and can only be approximated by interpolation.
• Uncertainty analysis is very difficult.
Example of an exponential function An exponential function is shown in the equation below. Instruments for measuring the ultrasonic response of reference standards with various levels of defects (holes) that are submerged in a fluid are described by this function. $$Y = \frac{e^{-aX}}{b + cX} + \epsilon$$
Example of a rational function A rational function is shown in the equation below. Scanning electron microscope measurements of line widths on semiconductors are described by this function (Kirby). $$Y = \frac{a + bX + cX^2}{a_1 + b_1X + c_1X^2} + \epsilon$$