Atmospheric Data Assimilation and Quality Control
85
single large error can then be disastrous (Fig. 5.4 solid line). However if instead we
minimise the mean absolute deviation (We shall see that this is the correct norm if
the observational error probability distribution function is proportional to an exponential of the absolute deviation), the bad datum is ignored (Fig. 5.4 dotled line).
Analysis methods designed to ignore such outliers are also considered to be quality
control methods.
5.6.2 Discrete Bayes Theorem Applied to Gross Observational Errors
1 have two dice. One is weighted towards throwing sixes. 1 have performed some
experiments with them, and have the prior statistics that:
for the weighted (W) die, p(61 W)
= 8/60
for the good (G) die,
P(6 G)
= 10/60
1 choose one at random: P(W) = P( G) = 1/2
1 throw this die, and it shows a six. Now:
P(6) = p(61 W) P(W) + p(61 G) P(G)
= 58/60 1/2 + 10/60 1/2
= 34/60
We can now apply Bayes' Theorem:
P(G 16) = p(61 G) P(G) / P(6)
1
= 10/6
1
0 1/2 / 34/60 = 5/34
P(W 6) = P(6 W) P(W) / P(6)
= 58/60 1/2 /34/60 = 29/34
The information that 1 have thrown a six has added to my knowledge, so that the
posterior probability that the chosen die is weighted has increased. If 1 were to
throw again, and get another six, the probability would increase again.
5.6.3 Non-Gaussian Model for Observational Errors
The simplest model that allows for the observed fact that observational errors are
not in practice Gaussian, is to assume that a small fraction of the observations are
corrupted, and hence worthless. The others have Gaussian errors. For each observation we have:
p(/Ix) = p(/IG n x)P(G)p(/lc n x)P(C)
(31)
G is the event "there is a gross error" 2, and Ci means not G.
p(/Ic n x) = N(/IH(x), E + F)
p(/IG n x) = { ok
over the range ofplausible values
elsewhere
(32)
2. We assume that G is independent ofx, so that P(G Ix)=p(G).