Evaluating Variable Uncertainties

ERDC TN-DOER-I5

August 2000

The final uncertainty analysis expression for the solids content in the hopper is:

LMF -ρh I F Uw I 2 + F ρw Uρ I 2

H ρh - ρw JK GH Wh JK GH ρm bρm - ρw g JK

(14)

FG U p - U p IJ + FG U p IJ

2 12

H ρi - ρw ρh - ρw K H ρi - ρw K

Evaluating Variable Uncertainties. The equations are now in the form to insert the variable

values and their associated uncertainties for calculating the percent uncertainty in production. In

practice, to obtain a reliable estimate of production uncertainty, comprehensive data on the project

area sediment and water properties should be collected. If sediment and water samples are taken

over the project area, a statistical analysis can be performed to determine the uncertainty in the mean

value of the variables used in the uncertainty calculation. Assuming that the variation of the sample

values follow a normal (Gaussian) distribution, confidence intervals can be defined for the sample

population. Based on a 95 percent confidence interval, the precision limit can be calculated for the

sample population. The precision limit (PL) is defined as:

FG σ N -1 IJ

H NK

PL = t

(15)

with

t = the t distribution value

N = number of samples

σN-1 = the standard deviation

For example, 10 in situ density measurements are made (N=10). The mean value was 1.913 g/cm3,

and the standard deviation (σN-1) was calculated to be 0.068312 g/cm3. For N-1 degrees of freedom,

and a 95 percent confidence level, the t distribution value is 2.262 (Coleman and Steel 1989).

Therefore the precision limit value is calculated to be 0.04887 g/cm3, and the uncertainty for the

mean value of the in situ density measurements is 1.913 0.04887 g/cm3.

UNCERTAINTY ANALYSIS EXAMPLE CASES: To illustrate the utility of the uncertainty

analysis method, sample problems will be solved using production instrumentation and vessel

specifications from an example pipeline and hopper dredge. In the examples, it is assumed that an

adequate number of samples were taken to statistically define the variable uncertainties.

Example Pipeline Dredge Specifications. The production meter system on the example

pipeline dredge consists of a nuclear density gauge and a magnetic flow meter. The nominal dredge

pipe inside diameter (D) is 0.61 m and the average flow velocity in the pipe (V) is 4.57 m/sec. It is

assumed that the pipe diameter uncertainty is 0.00025 m (UD). For this example, it is assumed