For time indices

*k* = 1

to *k* = *nk*

,
the corresponding Measurement vector is denoted by
/ *z*(1) \
| *k* . |
*z* = | . |
*k* | . |
| *z*(*nz*) |
\ *k* /

where *nz*

is the number of elements in the measurement vector.
The Measurement function

*h*

maps a value for the
State vector
at time index *k*

to the mean value for the Measurement vector at time index *k*

.
The State vector must contain enough information to make this possible.
To be specific, for time indices *k* = 1

to *k* = *nk*

,
*z* = *h* (*xt* ) + *v*
*k* *k* *k* *k*

where *nk*

is the number of time points,
*xt*

is the true, but unknown, value for the state vector,
and *v*

is a random variable with mean zero.
For time indices

*k* = 1

to *k* = *nk*

,
the corresponding Measurement variance is denoted by
/ *E*[*v*(1) *v*(1)] ... *E*[*v*(1) *v*(*nz*)] \
| *k* *k* *k* *k* |
*T* | . . . |
*R* = *E*[ *v* *v* ] = | . . . |
*k* *k* *k* | . . . |
|*E*[*v*(*nz*) *v*(1)] ... *E*[*v*(*nz*) *v*(*nz*)] |
\ *k* *k* *k* *k* /

where *nz*

is the number of elements
in the Measurement vector *z*

,
and *E*

denotes expected value.
The filter in

/ *z*(1) \ /range measurement to station *P* at time index k\

*z* = | | = | |

*k* | *z*(2) | |range measurement to station *S* at time index k|

\ *k* / | /

Except for measurement error,
the range measurements are equal to the Euclidean distance
between the known shore stations and ship position.
The corresponding Measurement function is
/ _____________________________________ \
| \/ (*P*(1) - *x* (1))^2 + (*P*(2) - *x* (2))^2 |
*h* (*x* ) = | *k* *k* |
*k* *k* | \/ (*S*(1) - *x* (1))^2 + (*S*(2) - *x* (2))^2 |
\ *k* *k* /

Each measurement error for this example has a standard deviation
of `2`

and is not correlated with the other measurement.
The corresponding Measurement Variance is
/ 4 0 \
*R* = | |
*k* \ 0 4 /