Page 161 - Modern Spatiotemporal Geostatistics

142 Modern Spatiotemporal Geostatistics — Chapter 7

Physical laws—Hard and soft data

As we discussed in Chapter 3 ("General knowledge in terms of physical laws,"
p. 76), an important general knowledge base is expressed in terms of laws
of nature (physical, biological, etc.). In this section, therefore, we present
some analytical and numerical results in which the BME estimates obtained
incorporate this sort of physical knowledge. The first example is concerned
with a situation in which the physical law has the form of a stochastic ordinary
differential equation.

EXAMPLE 7.4: Consider the general knowledge base in the form of the physical
law (Eq. 3.16, p. 79). As we saw in Example 5.4 (p. 112-114), the correspond-
ing 9§-operator is given by

where t map = (ti,... ,t m,t k) and iJ. a,\(t a) (a = 1,... ,m,fc and A = 1, 2)
are the solutions of Equations 5.20 and 5.21 (p. 113). Also, assume that there
are hard data at times t haTd = (ti,... ,t mh) and interval (soft) data at times
tsoft = (t mh+i, • • • ,t m)- Based on this knowledge, an estimate is sought at
the future time tk- The BMEmode estimate at time tk is simply the solution
of Equation 7.6 above, where 9£ is given by Equation 7.16. For illustration
purposes, let us consider the simple case in which a = k, so that Xa — Xk- 'n
this case, as we saw in Example 5.4, the 9« is given by

The moments involved in the prior stage were implied by the physical law and
did not need to be calculated experimentally from the data. The BMEmode
estimate (which, due to the symmetry of the pdf, is the same as the BME
conditional mean estimate) is the solution of the following equation:

which, in view of Equation 7.17, gives

As should be expected, Equation 7.19 is in agreement with the mean solution
of the physical law in Equation 3.16 (p. 79).
Furthermore, some numerical results involving Darcy's law of groundwater
flow are discussed in the following example.