MODELING OF WATER QUALITY - Water Quality Engineering in Natural Systems - page 404

Combining Equations (11.56) and (11.58) gives
the error in the calculated quantity, Δ Y , in terms of
the errors in the measured quantities, Δ X i = ( X i − 〈 X i 〉),
hence

Combining Equations (11.59) and (11.62) yields

N N

∑ ∑

σ

2

= φ φ ′ ′ ∆ ∆ X X

(11.63) Y X X i j

i j

i = 1 j = 1

N

∑ φ
1

where 〈Δ X i Δ X j 〉 is the covariance between the errors in
X i and X j . In many cases, the errors in different mea-
sured quantities are uncorrelated, leading to error
covariances of zero. Under these circumstances, Equa-
tion (11.63) becomes

∆ Y = ′ ∆ X (11.59)

X i

i

i =

where ϕ X i is given by

N

∂
∂

φ

∑ (

φ ′ = σ

2

= φ )

2

σ

2

(11.60) ′ (11.64)

X

Y X i

X X X

X

i

i

i =

i = 1

Equation (11.59) is a stochastic equation relating the
error in the derived quantity, Δ Y , to the errors in mea-
sured quantities, Δ X i , provided these latter errors are
small. Taking the ensemble average of Equation (11.59)
yields

This analysis is commonly referred to as first-order
second-moment (FOSM) analysis, and is limited to
simple models that are continuous with respect to model
parameters. Furthermore, FOSM approximation dete-
riorates when the coefficient of variation is greater than
10-20%, which is quite common.
First-order uncertainty analysis is sometimes referred
to as first-order error analysis (yan et al., 2006). The
major advantage of first-order uncertainty analysis
(FOUA) is its simplicity, requiring knowledge of only
the first two statistical moments of the basic variables
and a simple sensitivity analysis to calculate the incre-
mental change in model output corresponding to an
incremental change in model input. FOUA produces an
estimate of the combined effects of data and parameter
uncertainty and allows the partitioning of the model
output uncertainty into its sources. The applicability of
FOUA, however, is inherently restricted by the degree
of nonlinearity in the model. For nonlinear systems, this
analysis becomes increasingly inaccurate as the basic
variables depart from their central values (Melching
and yoon, 1997).

N

∑ ϕ
1

∆ Y = ′ ∆ X (11.61)

X i

i

i =

indicating that the average error in the estimated quan-
tity is the sum of the average errors in the measured
quantities, weighted by the sensitivities ϕ X i . Typically,
measurements are assumed to be without bias, in which
case the average errors, 〈Δ X i 〉, are equal to zero, and,
according to Equation (11.61), all calculated results are
also without bias. Another statistical parameter of inter-
est is the variance of the calculated quantity, Y , which is
denoted by σ 2 and defined by

σ Y

2

= ( ∆ Y )

2

(11.62)