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8. PROPAGATION OF ERRORS: THE ERROR MATRIX

Consider the case in which a single physical quantity, y, is some function of the alpha's: y = y(alpha1, ..., alphaM). The "best" value for y is then y* = y(alphai*). For example y could be the path radius of an electron circling in a uniform magnetic field where the measured quantities are alpha1 = tau, the period of revolution, and alpha2 = v, the electron velocity. Our goal is to find the error in y given the errors in alpha. To first order in (alphai - alphai*) we have

Equation 12     (12)

A well-known special case of Eq. (12), which holds only when the variables are completely uncorrelated, is

Equation

In the example of orbit radius in terms of tau and v this becomes

Equation

in the case of uncorrelated errors. However, if img56 is non-zero as one might expect, then Eq. (12) gives

Equation

It is a common problem to be interested in M physical parameters, y1, ..., yM, which are known functions of the alphai. In fact the yi can be thought of as a new set of alphai or a change of basis from alphai to yi. If the error matrix of the alphai is known, then we have

Equation 13     (13)

In some such cases the ðyi / ðalphaa cannot be obtained directly, but the ðalphai / ðya are easily obtainable. Then

Equation

Example 3

Suppose one wishes to use radius and acceleration to specify the circular orbit of an electron in a uniform magnetic field; i.e., y1 = r and y2 = a. Suppose the original measured quantities are alpha1 = tau = (10 ± 1)µs and alpha2 = v = (100 ± 2) km/s. Also since the velocity measurement depended on the time measurement, there was a correlated error img56 = 1.5 × 10-3 m. Find r,Deltar, a, Deltaa.

Since r = vtau / 2pi = 0.159 m and a = 2piv / tau = 6.28 × 1010 m/s2 we have y1 = alpha1alpha2 / 2pi and y2 = 2pi alpha2 / alpha1. Then ðy1 / ðalpha1 = alpha2 / 2pi, ðy1 / ðalpha2 = alpha1 / 2pi, ðy2 / ðalpha1 = -2pialpha2 / alpha12, ðy2 / ðalpha2 = 2pi / alpha1 . The measurement errors specify the error matrix as

Equation

Eq. 13 gives

Equation

Thus r = (0.159 ± 0.184) m

For y2, Eq. 13 gives

Equation

Thus a = (6.28 ± 0.54) × 1010 m/s2.

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