control//state estimation//Kalman filter//measurement noise

How much to distrust a sensor, and the one number of the filter that is measured. The **measurement noise** \(v\) is the part of a reading that is not the truth, \(z=x+v\), and \(R\) is its variance: how much the filter distrusts the sensor. It is a property of the sensor, not of anyone's belief, and in a Kalman filter it is set by an experiment rather than guessed.

How much to distrust a sensor, and the one number of the filter that is measured. The measurement noise vvv is the part of a reading that is not the truth, z=x+vz=x+vz=x+v, and RRR is its variance: how much the filter distrusts the sensor. It is a property of the sensor, not of anyone's belief, and in a Kalman filter it is set by an experiment rather than guessed.

The measurement is literally a variance. Leave a GPS still on a point whose position is known to the centimetre, a geodetic vertex, take 10,000 readings and compute the errors ei=zi−xe_i=z_i-xei​=zi​−x. Their variance is RRR, say 4 m², so σ=2\sigma=2σ=2 m, and now you know how much to distrust that GPS. Their mean is the bias, which is subtracted before the sensor is ever used. The filter assumes the noise is centred, v∼N(0,R)v\sim\mathcal N(0,R)v∼N(0,R): the bell of possible readings sits on the truth. Obvious, and still the origin of a common confusion, because that σ2\sigma^2σ2 is RRR and not PPP. The sensor knows nothing about your PPP.

Why use a sensor that fails when a good reference exists? Because the reference does not travel. The geodetic vertex characterizes the GPS in the field or the lab; then the GPS rides in the car alone. Every real sensor has noise, bias, finite resolution, latency and drift, so the question is never whether it fails but how much and in which way (sensor calibration).

You think in σ\sigmaσ; the filter thinks in σ2\sigma^2σ2. PPP and RRR are both variances, so a GPS you picture as 2 m enters the filter as 4 m² (variance).

RRR may change from step to step. A GPS reports its own current precision from the geometry of the satellites, and a sensor whose noise grows with temperature can hand the filter a different RkR_kRk​ each time. One RRR for the whole day is too pessimistic when the sensor is good and too optimistic when it is bad (covariance).

10,000 consecutive readings at 10 Hz span sixteen minutes. If the GPS error changes slowly, and it does, they are not 10,000 independent opinions: the variance still comes out right, but something else breaks (autocorrelation).