Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 414

Innovation

e ( k ) c

z ( k )

u ( k )

( k )
c u

H k u ( k ) p

Fig. 6.5 The innovation is orthogonal to the past ( left ). The estimate error is orthogonal to the
estimate itself ( right )

This is the part of knowledge in the measure we could extract from the state at
the previous time step, or from the past . We do expect that z .k/ is adding new
information. The novel part of the information added by the measure is exactly the
innovation z .k/ H k u .k p .

Notice that

z .k/ H k u .k/ D H k u .k/ C .k/ H k u .k/ D .k/ H k e .k p ;

p p

consequently

z .k/

H k u .k p

e .k p

D E .k/ H k E

E D 0:

In addition, we compute the variance of the innovation

. z .k/

H k u .k p / T H k u .k p /. z .k/ E

. .k/

H k e .k p / T

H k e .k p /. .k/ D R k C H k ƒ .k p H k

D E

because the noise at k is not correlated to e .k/ D A k . u .k/

u .k 1/ / b .k 1/ .

p c

It is also possible to prove [ 68 ] that for j 1

. z .k/

H k u .k j p / T H k u .k p /. z .k j/

E D 0:

This means that the innovation at time k has no correlation with the innova-
tion at the previous time steps, so that we can conclude that the innovation is
a white process .