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


Notation 
The following notation is used in this chapter in discussing models for
calibration curves.


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:


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.
 
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 xaxis should lead to a zero response on the yaxis. 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 higherorder polynomial). An example is a load cell where force exerted on the cell is a nonlinear function of load.  
Disadvantages of quadratic models 
Disadvantages of quadratic and higherorder polynomials are:


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 nonlinear 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 nonlinear 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 \(a\). 

Nonlinear models and their limitations 
Instruments whose responses are not linear in the coefficients can
sometimes be described by nonlinear 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 nonlinear functions that
have been shown to have practical value as calibration functions are:
Nonlinear models are an important class of calibration models, but they have several significant limitations.


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). $$ Y = \frac{a + bX + cX^2}{a_1 + b_1X + c_1X^2} + \epsilon $$ 