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If the bad measurement test detects bad measurements, these measurements should be identified. This is accomplished below.

Consider the nonlinear measurement model

(2.14)

where x^true is the unknown true state vector and e is the measurement error vector. Note that

(2.15)

which is a typical assumption in state estimation.

Consider the differential measurement equation:

(2.16)

If measurements are Gaussian distributed and independent, the least squares estimator can be obtained by minimizing the weighted sum of square deviations of the differential errors:

(2.17)

This minimization problem leads to the system of equations:

(2.18)

known as the system of normal equations [8]. Assuming that the gain matrix

(2.19)

has an inverse, the least squares estimates can be written explicitly as

(2.20)

from which we conclude that is a linear function of dz.

The residual and the differential residual are defined as

(2.21)

(2.22)

Thus

(2.23)

where P = HG⁻¹H^TW is known as hat or projection matrix.

Integrating (2.23), the linear transformation from z to r at the optimum is

(2.24)

where k is a constant vector and S is known as the residual sensitivity matrix.

If measurements z are Gaussian distributed and independent with zero mean and covariance matrix R (R = W⁻¹) and is an unbiased estimator of h(x^true), i.e. , the residual vector r provided by the linear transformation (2.24) is Gaussian distributed with parameters given by

(2.25)

(2.26)

and due to (2.14) and (2.25):

(2.27)

The normalized residual (N(0, 1²)) of measurement i is

(2.28)

The largest residual identifies a bad measurement with a 1 − α confidence level (e.g. 0.99) if . Further details can be found in [7].