**3. Uncertainty Analysis in the Planning of an Experiment**

# Introduction

Uncertainty analysis is a powerful tool for improving the value of
experimental work, and can be applied during all phases of an
experimental program. However, the greatest value of uncertainty
analysis is almost certainly obtained when it is used during the
planning of an experiment. In citing a dozen uses of uncertainty
analysis, Kline (1985a) highlighted five specific uses within the
planning phase

1.  Enforcing a complete examination of experimental procedures,

2.  Identifying situations in which improved instrumentation and/or
    procedures are needed to obtain desired accuracy of results,

3.  Minimizing instrumentation costs to obtain a given output accuracy,

4.  Identifying instruments and procedures that control accuracy
    (usually one or a few from the total set of those involved in a
    given experiment), and

5.  Informing beforehand when an experiment cannot meet desired accuracy
    requirements and is thus "hopeless". Such experiments can sometimes
    be redesigned, or if they must be abandoned, considerable time and
    money can be saved. *[(An example might be trying to measure the
    normal velocity through a fish screen with a 3-dimensional ADV when
    the sweeping velocity is extremely large compared to the normal
    component.)]{.mark}*

So important is uncertainty analysis in the planning phase of an
experiment that it was prominently featured in both of the primary
conclusions obtained from the landmark 1983 symposium on uncertainty
analysis sponsored by the ASME Journal of Fluids Engineering (JFE).
Those two conclusions were (Kline 1985b):

1.  Uncertainty analysis is an essential ingredient in planning,
    controlling, and reporting experiments. The important thing is that
    a reasonable uncertainty analysis be done. All differences of
    opinion about appropriate methods are subsidiary to this conclusion.

2.  It is particularly important to use an uncertainty analysis in the
    planning and checkout stages of an experiment!

In emphasizing the importance of uncertainty analysis during the
planning stages of an experiment, Hugh Coleman wrote to Kline (1985b)
"All experimentalists should be taught that an uncertainty analysis
PERFORMED IN THE PRELIMINARY DESIGN PHASE OF AN EXPERIMENT will often
yield results and insights far out of proportion to the relatively small
investment of time required for the analysis." (Coleman's emphasis).

So, uncertainty analysis in the planning phase of an experiment is a
worthy endeavor. This chapter attempts to highlight this fact with
discussion and examples of the use of uncertainty analysis techniques
that lend themselves to the planning phase. The material presented here
is drawn primarily from two sources, R.J. Moffat's 1985 paper from the
ASME/JFE symposium, titled "Using Uncertainty Analysis in the Planning
of an Experiment", and Coleman and Steele's 1999 book, *Experimentation
and Uncertainty Analysis for Engineers*, 2^nd^ ed.

# General Uncertainty Analysis

In all but the simplest of experiments, the end result of an
investigation is not measured directly, but rather is determined by
calculation from a data reduction equation. The end result, and the
uncertainty in it, are a product of the direct measurement of other
parameters and, in most cases, assumed values of material properties or
other physical "constants". In all the phases of experimentation we will
consider how the uncertainties in these individual variables propagate
through the data reduction equation into the end result.

In the planning phase of an experiment, we will consider the
uncertainties of individual variables and their propagation into the
result in the most basic ways possible. This level of analysis is often
called *general uncertainty analysis*, as opposed to detailed
uncertainty analysis, which is applied to later phases. In the planning
phase we are considering alternative methods for arriving at the
experimental result. We have not selected specific instruments or
equipment, and thus we are in no position to address the details of
systematic versus random errors, since any systematic errors in the
instruments we eventually choose are as likely to be positive and
negative. At this stage we consider all uncertainty to be caused by
random errors. Later, when we get into the detailed design of an
experiment, the debugging and data collection phases, and the data
analysis and reporting phases, we will be more interested in those
details, but for now our focus is on achieving the maximum amount of
early guidance for a small amount of expended effort. By using
uncertainty analysis at this stage to help select the correct
experimental approach, we can avoid expending unnecessary effort on
methods that might never achieve our objective.

# Uncertainty Propagation Equation

Recall from Chapter 2 that for the general case of an experimental
result, *r*, computed from *J* measured variables *X*~1*...J*~, the data
reduction equation is:

$r = r(X_{1},X_{2},\ldots,X_{J})$ ()

and the uncertainty in the experimental result is given by

$U_{r}^{2} = \left( \frac{\partial r}{\partial X_{1}} \right)^{2}U_{X_{1}}^{2} + \left( \frac{\partial r}{\partial X_{2}} \right)^{2}U_{X_{2}}^{2} + \cdots + \left( \frac{\partial r}{\partial X_{J}} \right)^{2}U_{X_{J}}^{2}$
()

where *U~r~* is the uncertainty in the result, *U~X1~* is the
uncertainty in the variable *X*~1~, etc. This is the most general form
of the uncertainty propagation equation (Coleman and Steele 1999). When
applying the uncertainty propagation equation, the individual
uncertainties should all be expressed with the same odds, e.g., at 95%
confidence. In the planning phase, this assumption is implicit. In
addition, the measured variables and their uncertainties are assumed to
be independent of one another.

## Nondimensional Forms

Two nondimensional forms of eq. (2) are useful in the planning phase.
Dividing each term by *r*^2^ and multiplying the terms on the right-hand
side by (*X~i~*/*X~i~*)^2^, we obtain

$\frac{U_{r}^{2}}{r^{2}} = \left( \frac{X_{1}}{r}\frac{\partial r}{\partial X_{1}} \right)^{2}\left( \frac{U_{X_{1}}}{X_{1}} \right)^{2} + \left( \frac{X_{2}}{r}\frac{\partial r}{\partial X_{2}} \right)^{2}\left( \frac{U_{X_{2}}}{X_{2}} \right)^{2} + \cdots + \left( \frac{X_{J}}{r}\frac{\partial r}{\partial X_{J}} \right)^{2}\left( \frac{U_{X_{J}}}{X_{J}} \right)^{2}$
()

In this equation, *U~r~*/*r* is the relative uncertainty in the result
and the factors *U~Xi~*/*X~i~* are the relative uncertainties of each
variable. The factors in parentheses that multiply the relative
uncertainties of the variables are called *uncertainty magnification
factors* (UMFs). They indicate the influence of uncertainty in a
particular variable on the uncertainty in the result. When the UMF is
greater than 1, uncertainty in a variable is magnified as it propagates
through the data reduction equation; if less than 1, the uncertainty in
the variable is reduced. The UMF depends on the value of a variable
relative to the result and the manner in which it is incorporated into
the data reduction equation, but it is independent of the actual
uncertainty in the variable, so it is useful before we know details
about measurement methods and their uncertainties. Since the UMFs are
always squared when inserted into eq. (3), only their absolute values
are important.

The second nondimensional form is obtained by dividing by *U~r~*^2^,
which produces

$1 = \left( \frac{\partial r}{\partial X_{1}} \right)^{2}\left( \frac{U_{X_{1}}}{U_{r}} \right)^{2} + \left( \frac{\partial r}{\partial X_{2}} \right)^{2}\left( \frac{U_{X_{2}}}{U_{r}} \right)^{2} + \text{.}\text{.}\text{.} + \left( \frac{\partial r}{\partial X_{J}} \right)^{2}\left( \frac{U_{X_{J}}}{U_{r}} \right)^{2}$
()

Each term on the right-hand side gives the fractional contribution of
the squared uncertainty in a given variable to the squared uncertainty
in the result. In percentage terms we can define *uncertainty percentage
contributions* (UPCs) as

$\text{UPC}_{i} = \left( \frac{\partial r}{\partial X_{i}} \right)^{2}\left( \frac{U_{X_{i}}}{U_{r}} \right)^{2}\text{100} = \left( \frac{X_{i}}{r}\frac{\partial r}{\partial X_{i}} \right)^{2}\left( \frac{U_{X_{i}}}{X_{i}} \right)^{2}\frac{1}{\left( U_{r}r \right)^{2}}\text{100}$
()

The UPCs include the effects of both the UMF and the uncertainty of the
variable, so they are useful in the late planning phase and early design
phase, when measurement equipment and methods are being selected and
measurement uncertainties can be estimated.

## Simplified Form

The most useful form of the uncertainty propagation equation for
planning purposes is probably eq. (3), in which the squares of the
relative uncertainties are related through the UMFs. In a great many
cases eq. (3) can be further simplified. When the data reduction
equation is of the form

$r = \text{kX}_{1}^{a}X_{2}^{b}X_{3}^{c}\cdots$ ()

with *a*, *b*, and *c*, and *k* being constants, applying eq. (3)
produces the simplified equation

$\frac{U_{r}^{2}}{r^{2}} = a^{2}\left( \frac{U_{X_{1}}}{X_{1}} \right)^{2} + b^{2}\left( \frac{U_{X_{2}}}{X_{2}} \right)^{2} + c^{2}\left( \frac{U_{X_{3}}}{X_{3}} \right)^{2} + \cdots$
()

In such a case, the UMFs are the exponents and the uncertainty
propagation equation can be written down by simple inspection. One must
keep several things in mind when considering the use of this simplified
form of the equation. First, one must solve for the experimental result
before applying the equation. Second, the *X~i~*s must be directly
measured variables, so an equation of the form *R*=*a*(cos(θ)) is not in
the proper form if θ is measured directly. Also, an equation of the form
*Q*=*C~d~A*(2*g*(*h*~2~-*h*~1~))^1/2^ is acceptable if *h*~2~-*h*~1~ is
measured directly, but not if *h*~2~ and *h*~1~ are measured separately.

# Examples -- Simple Cases

Consider the case of flow measurement over a fully contracted,
sharp-crested rectangular weir. We wish to determine the discharge
through the weir, using the equation

$Q = \text{CLh}_{1}^{1\text{.}5}$ ()

The variables that will be measured are the weir length, *L*, and the
upstream head, *h*~1~, and each will have an associated uncertainty. The
value of the discharge coefficient, *C*, is an empirical constant which
will also have some uncertainty associated with it.

Before determining how to measure the weir length and upstream head, a
general uncertainty analysis can be used to gain an understanding of the
relationship between the measurement uncertainties and the uncertainty
in the result.

Applying eq. (3), the general uncertainty expression is

$\left( \frac{U_{Q}}{Q} \right)^{2} = \left( \frac{C}{Q}\frac{\partial Q}{\partial C} \right)^{2}\left( \frac{U_{C}}{C} \right)^{2} + \left( \frac{L}{Q}\frac{\partial Q}{\partial L} \right)^{2}\left( \frac{U_{L}}{L} \right)^{2} + \left( \frac{h_{1}}{Q}\frac{\partial Q}{\partial h_{1}} \right)^{2}\left( \frac{U_{h_{1}}}{h_{1}} \right)^{2}$
()

And the UMFs are

$\text{UMF}_{C} = \frac{C}{Q}\frac{\partial Q}{\partial C} = \frac{C}{Q}\text{Lh}_{1}^{1\text{.}5} = \frac{\text{CLh}_{1}^{1\text{.}5}}{Q} = 1$
()

$\text{UMF}_{L} = \frac{L}{Q}\frac{\partial Q}{\partial L} = \frac{L}{Q}\text{Ch}_{1}^{1\text{.}5} = \frac{\text{CLh}_{1}^{1\text{.}5}}{Q} = 1$
()

$\text{UMF}_{h_{1}} = \frac{h_{1}}{Q}\frac{\partial Q}{\partial h_{1}} = \frac{h_{1}}{Q}\left( 1\text{.}5\text{CLh}_{1}^{0\text{.}5} \right) = \frac{1\text{.}5\text{CLh}_{1}^{1\text{.}5}}{Q} = 1\text{.}5$
()

Substituting the UMFs back into eq. (9), we obtain

$\left( \frac{U_{Q}}{Q} \right)^{2} = \left( \frac{U_{C}}{C} \right)^{2} + \left( \frac{U_{L}}{L} \right)^{2} + \left( 1\text{.}5 \right)^{2}\left( \frac{U_{h_{1}}}{h_{1}} \right)^{2}$
()

which we could have written down by inspection, since the data reduction
equation was in the simple form discussed previously. Notice that the
uncertainties in head measurement are magnified in the result due to the
exponent of 1.5. With this equation, we can now examine several
questions related to the uncertainty of the proposed discharge
measurement.

Suppose for example that the weir length is 2 m, the head is 0.3 m, and
the discharge coefficient is 1.71 m^0.5^/s. How accurately can the
discharge be measured if the relative uncertainty in *C* is 5%, the weir
length is known with an uncertainty of 2 mm, and the head is measured
with an uncertainty of 3 mm?

Inserting the values of the variables and their uncertainties, we obtain

$\begin{array}{r}
\left( \frac{U_{Q}}{Q} \right)^{2} = \left( 0\text{.}\text{05} \right)^{2} + \left( \frac{0\text{.}\text{002}}{2} \right)^{2} + \left( 1\text{.}5 \right)^{2}\left( \frac{U_{h_{1}}}{h_{1}} \right)^{2} \\
 = 0\text{.}\text{0025} + 0\text{.}\text{000001} + 0\text{.}\text{000225} \\
 = 0\text{.}\text{00273} \\
\frac{U_{Q}}{Q} = \sqrt{0\text{.}\text{00273}} = 0\text{.}\text{052}
\end{array}$

Thus, the relative uncertainty in discharge measurement is 5.2%, with
the primary sources of uncertainty being the discharge coefficient and
the measurement of the upstream head.

We can also answer "what if" questions related to design or selection of
measurement methods. For example, if the discharge measurement must be
made with an uncertainty of 6% or less and the length of the weir has an
uncertainty of 2 mm as before, what is the maximum allowable uncertainty
in the head measurement? Working through the equation, we find that

$\begin{array}{r}
\left( 0\text{.}\text{06} \right)^{2} = \left( 0\text{.}\text{05} \right)^{2} + \left( \frac{0\text{.}\text{002}}{2} \right)^{2} + \left( 1\text{.}5 \right)^{2}\left( \frac{U_{h_{1}}}{0\text{.}3} \right)^{2} \\
0\text{.}\text{0036} = 0\text{.}\text{0025} + 0\text{.}\text{000001} + 2\text{.}\text{25}\left( \frac{U_{h_{1}}}{0\text{.}3} \right)^{2} \\
U_{h_{1}} = 0\text{.}\text{00663}\text{ m} = 6\text{.}\text{63}\text{ mm}
\end{array}$

Similarly, if we were doing a laboratory calibration to determine the
discharge coefficient we might ask what uncertainty in head measurement
is needed to obtain a *C* value with relative uncertainty of 2%, if the
laboratory facilities are capable of independently measuring the
discharge with an uncertainty of 0.5%. Solving for *U~h~*~1~ in that
case, we obtain

$\left( 1\text{.}5 \right)^{2}\left( \frac{U_{h_{1}}}{h_{1}} \right)^{2} = \left( \frac{U_{Q}}{Q} \right)^{2} - \left( \frac{U_{C}}{C} \right)^{2} - \left( \frac{U_{L}}{L} \right)^{2}$
()

when we insert the values for the example we obtain

$\begin{array}{r}
\left( 1\text{.}5 \right)^{2}\left( \frac{U_{h_{1}}}{0\text{.}3} \right)^{2} = \left( 0\text{.}\text{005} \right)^{2} - \left( 0\text{.}\text{02} \right)^{2} - \left( \frac{0\text{.}\text{002}}{2} \right)^{2} \\
\left( \frac{U_{h_{1}}}{0\text{.}3} \right)^{2} = - 0\text{.}\text{00017}
\end{array}$

**We have obviously committed a serious error!** One cannot take the
square root of a negative number. The problem is that we forgot to
rearrange the data reduction equation to put the result on the left side
before applying the simplified form of the general uncertainty equation.
The correct data reduction equation and uncertainty expression are

$C = \frac{Q}{\text{Lh}_{1}^{1\text{.}5}} = \text{QL}^{- 1}h_{1}^{- 1\text{.}5}$
()

$\left( \frac{U_{C}}{C} \right)^{2} = \left( \frac{U_{Q}}{Q} \right)^{2} + ( - 1)^{2}\left( \frac{U_{L}}{L} \right)^{2} + \left( - 1\text{.}5 \right)^{2}\left( \frac{U_{h_{1}}}{h_{1}} \right)^{2}$
()

Inserting the appropriate numerical values for our example,

$$\begin{array}{r}
\left( 0\text{.}\text{02} \right)^{2} = \left( 0\text{.}\text{005} \right)^{2} + \left( \frac{0\text{.}\text{002}}{2} \right)^{2} + \left( 1\text{.}5 \right)^{2}\left( \frac{U_{h_{1}}}{0\text{.}3} \right)^{2} \\
0\text{.}\text{0004} = 0\text{.}\text{000025} + 0\text{.}\text{000001} + 2\text{.}\text{25}\left( \frac{U_{h_{1}}}{0\text{.}3} \right)^{2} \\
2\text{.}\text{25}\left( \frac{U_{h_{1}}}{0\text{.}3} \right)^{2} = 0\text{.}\text{000374} \\
U_{h_{1}} = 0\text{.}\text{0039}\text{ m} = 3\text{.}9\text{ mm}
\end{array}$$

This example shows how easy it is to make a mistake when writing down
the uncertainty equation by inspection. The requirements listed earlier
for use of the simplified form must be kept in mind when considering its
use.

# Using Uncertainty Analysis in the Planning of an Experiment

The examples given thus far are relatively trivial, simple cases. Before
proceeding on to a more detailed example, we should review the role of
uncertainty analysis in experimental planning.

Any experimental endeavor begins with a question whose answer is sought.
In order to be useful, the answer must be determined with some level of
certainty. It is important to establish this allowable level of
uncertainty in the planning phase, although it need not be a refined
value; an order of magnitude estimate (i.e., 0.1, 1, 10, or 50%) is
often sufficient.

In the planning phase, we usually have several options available to us
for arriving at the experimental result. These will often involve
completely different experimental methods and the measurement of
different parameters. For example, one experiment might be performed in
a steady-state condition while another is performed in a transient
condition. Different experiments may exploit different physical
properties of a material or different parameters of a process. For each
alternative, there may be a different data reduction equation with
unique error propagation characteristics. We want to determine which
approach is most likely to yield the desired result, and which
measurements may the most critical if a particular method is used. Once
we have selected an experimental method, then we may want to investigate
instrumentation issues more closely. As we do so, we gradually move from
the realm of general uncertainty analysis to more detailed uncertainty
analysis.

When applying general uncertainty analysis, it becomes necessary to
assign uncertainties to the variables that we intend to measure, even if
we do not yet know the exact methods of measurement or specific
instrumentation that will be used. Coleman and Steele (1999) observed
that there is a universal human reluctance to estimate uncertainties,
perhaps out of fear of using the wrong values. This can be overcome by
realizing that there is no "correct" value at this stage; we only seek
to gain understanding by assuming "reasonable" values. If one is still
unsure of what value to use, a parametric analysis can be made in which
a range of values are used. Even ridiculously large or small values can
be used, as they may help illustrate the sensitivity or insensitivity of
the uncertainty in the result to the uncertainties in the measured
variables.

We will now illustrate the more detailed application of general
uncertainty analysis to a nontrivial, real-world example. Unfortunately,
a single example cannot illustrate all of the nuances and possible
outcomes of a planning phase uncertainty analysis. For the reader
seeking additional examples, those given by Coleman and Steele (1999)
are highly recommended.

## Detailed Example -- Radial Gate Calibration Experiments

The calibration of radial (Tainter) gates for flow measurement is a
topic of current research interest to those who work with irrigation
canal systems, in which radial gates are often used at check structures
and canal bifurcations. Accurate calibration of such gates would allow
operators to measure flow rates through the gates without the need for
construction of dedicated flow measurement structures (e.g., flumes or
weirs) or the purchase of expensive measurement instruments such as
acoustic flow meters. Measurement at control structures also eliminates
some of the lag time involved in draining and filling canal reaches
upstream from flumes or weirs. Finally, accurate calibration equations
allow canal operators to more accurately set gates to deliver desired
flow rates.

In concept, the objective is to develop a calibration method that would
allow one to compute the flow rate through a gate from a measurement of
upstream and downstream water level and the gate position. A definition
sketch for a radial gate is provided in Fig. 1.

![](media/image1.wmf){width="3.501388888888889in"
height="1.9333333333333333in"}

Fig. 1. **Definition sketch for flow through a radial gate**

Clemmens et al. (2003) performed laboratory experiments on a model
radial gate and developed a new calibration method called the
Energy-Momentum method. Two key parameters in this method are an
upstream energy loss and velocity distribution factor, 1+ξ, and an
energy correction factor, *E*~corr~. To provide better definition of the
behavior of these parameters, a new series of experiments is now being
contemplated. We wish to use a general uncertainty analysis to explore
the possible experiments that might be performed. For this example we
will focus on just the 1+ξ factor.

The parameter 1+ξ appears in the energy equation applied to the flow
from the upstream pool through the gate to the vena contracta point.
When a gate is in a free flow condition, the calibration equation for
the gate (derived from the energy equation) is

$Q = \delta\text{wb}_{c}\sqrt{\frac{2g(H_{1} - \delta w)}{1 + \xi}}$ ()

where *Q* is the discharge, δ is the contraction coefficient, *w* is the
vertical gate opening, *b~c~* is the gate width, *g* is the acceleration
of gravity, *H*~1~ is the energy head at section 1, and ξ an energy loss
coefficient for losses that occur between sections 1 and 2. The velocity
distribution at the vena contracta (section 2 in Fig. 1) is assumed to
be uniform, hence the constant 1 in the denominator. If the velocity
distribution is actual nonuniform, that will be accounted for in the ξ
factor, so that 1+ξ accounts for the effects of both velocity
distribution and energy loss. We wish to perform experiments that will
yield the values of 1+ξ.

The first question that arises in considering this problem is what level
of uncertainty we should try to achieve in determining 1+ξ? To answer
this question we can first consider the final application of eq. (17)
for the purpose of flow measurement. We would like to be able to measure
*Q* in the field with a relative uncertainty of 2%, which is comparable
to the rating uncertainty of flumes and weirs. If we estimate the
uncertainty in field measurements or estimates of δ, *w*, *b~c~*,and
*H*~1~, we can solve for the allowable uncertainty in 1+ξ. We will
assume that the acceleration of gravity, *g*, is a constant known with
negligible uncertainty.

Eq. (17) is not in the form that allows the simplified development of
the uncertainty propagation equation, so we will need to work out the
partial derivatives. The UMFs are

$\text{UMF}_{\delta} = \frac{\delta}{Q}\frac{\partial Q}{\partial\delta} = \left( 1 - \frac{\delta w}{2\left( H_{1} - \delta w \right)} \right)$
()

$\text{UMF}_{w} = \frac{w}{Q}\frac{\partial Q}{\partial w} = \left( 1 - \frac{\delta w}{2\left( H_{1} - \delta w \right)} \right)$
()

$\text{UMF}_{b_{c}} = \frac{b_{c}}{Q}\frac{\partial Q}{\partial b_{c}} = 1$
()

$\text{UMF}_{H_{1}} = \frac{H_{1}}{Q}\frac{\partial Q}{\partial H_{1}} = \frac{H_{1}}{2\left( H_{1} - \delta w \right)}$
()

$\text{UMF}_{1 + \xi} = \frac{1 + \xi}{Q}\frac{\partial Q}{\partial(1 + \xi)} = \frac{1}{2}$
()

Note that we treat 1+ξ as a single parameter, since we expect to
determine it experimentally in that way. Determining the partial
derivatives is somewhat tedious, but many terms cancel to make the
resulting eqs. (18) to (22) reasonably compact. Substituting the UMFs
into the general uncertainty equation we obtain:

$\left( \frac{U_{Q}}{Q} \right)^{2} = \left( 1 - \frac{\delta w}{2\left( H_{1} - \delta w \right)} \right)^{2}\left\lbrack \left( \frac{U_{\delta}}{\delta} \right)^{2} + \left( \frac{U_{w}}{w} \right)^{2} \right\rbrack + \left( \frac{U_{b_{c}}}{b_{c}} \right)^{2} + \left( \frac{H_{1}}{2\left( H_{1} - \delta w \right)} \right)^{2}\left( \frac{U_{H_{1}}}{H_{1}} \right)^{2} + \frac{1}{4}\left( \frac{U_{1 + \xi}}{1 + \xi} \right)^{2}$\
()

We see that in addition to estimating relative uncertainties of the
measured variables, we will need to select appropriate values of δ, *w*,
and *H*~1~, since they are contained in the UMF's. The contraction
coefficient varies from about 0.6 to 0.8 at typical gate openings; a
representative value of 0.7 can be used for this analysis. The values of
*H*~1~ and *w* can vary widely during normal gate operations and for
gates of different sizes, so we will choose a range of typical values
and then solve for the relative uncertainty of 1+ξ for each condition
using a spreadsheet model. We stated earlier that the desired relative
uncertainty in *Q* is 2%. For the other variables we need to make
reasonable estimates of their uncertainties. The contraction
coefficient, δ, is usually determined from empirical equations that are
believed to have a relative uncertainty of about 1%. The estimated
relative uncertainty in gate opening, *w*, is also 1% for a field
application. The relative uncertainty in the gate width, *b~c~*, for a
typical prototype gate is estimated to be 0.25% (includes variation in
the width of a gate chamber over the height of a gate). Finally, the
uncertainty of upstream head measurement is estimated to be 6 mm (0.02
ft).

Inserting these values into eq. (23) we obtain the results shown in
Table 1. We limit the gate opening, *w*, to 0.66*H*~1~, since at larger
values the gate leaf would not control the flow (critical depth being
two-thirds of the upstream head).

  ------------- ---------- ---------- ---------- ---------- ---------- ----------
  **Table 1.                                                           
  Allowable                                                            
  relative                                                             
  uncertainty                                                          
  in 1+ξ**                                                             

                *w*,                                                   
                meters                                                 

  *H*~1\        0.1        0.2        0.33       1.33       3.33       5.33
  ~meters                                                              

  0.5           2.66%      2.79%      2.86%                            

  2             2.82%      2.87%      2.94%      3.59%                 

  5             2.80%      2.82%      2.85%      3.07%      3.63%      

  8             2.80%      2.81%      2.82%      2.96%      3.27%      3.63%
  ------------- ---------- ---------- ---------- ---------- ---------- ----------

The results in Table 1 show that for small gate openings we must have a
smaller uncertainty in 1+ξ to achieve our goal of a 2% uncertainty in
discharge measurement. We can conservatively conclude that we should try
to design our experiments so that we can determine 1+ξ with a relative
uncertainty of 2.5% or less.

Now we will consider the planned experiment. We rearrange the data
reduction eq. (17) to solve for the experimental result, 1+ξ.

$1 + \xi = 2g\left( H_{1} - \delta w \right)\left( \frac{\delta\text{wb}_{c}}{Q} \right)^{2}$
()

In the laboratory we will most likely measure δ*w* as a single quantity,
the jet thickness, *y~j~*, using either a point gage or a piezometer tap
located at the vena contracta. This simplifies the data reduction
equation to

$1 + \xi = 2g\left( H_{1} - y_{j} \right)\left( \frac{y_{j}b_{c}}{Q} \right)^{2}$
()

We will now determine the UMFs for this data reduction equation.

$\text{UMF}_{H_{1}} = \frac{H_{1}}{1 + \xi}\frac{\partial(1 + \xi)}{\partial H_{1}} = \frac{H_{1}}{H_{1} - y_{j}}$
()

$\text{UMF}_{y_{j}} = \frac{y_{j}}{1 + \xi}\frac{\partial(1 + \xi)}{\partial y_{j}} = \frac{2H_{1} - 3y_{j}}{H_{1} - y_{j}}$
()

$\text{UMF}_{b_{c}} = \frac{b_{c}}{1 + \xi}\frac{\partial(1 + \xi)}{\partial b_{c}} = 2$
()

$\text{UMF}_{Q} = \frac{Q}{1 + \xi}\frac{\partial(1 + \xi)}{\partial Q} = 2$
()

We see that, again, some of the UMFs contain the measured variables
themselves, so we will need to choose representative values of *H*~1~
and *y~j~* to perform the uncertainty analysis. Typically, *H*~1~ will
be significantly larger than *y~j~*, so we can quickly see that
UMF~*H*1~ will be on the order of 1, while UMF*~yj~* is on the order of
2. The complete uncertainty equation is

$\left( \frac{U_{1 + \xi}}{1 + \xi} \right)^{2} = \left( \frac{H_{1}}{H_{1} - y_{j}} \right)^{2}\left( \frac{U_{H_{1}}}{H_{1}} \right)^{2} + \left( \frac{2H_{1} - 3y_{j}}{H_{1} - y_{j}} \right)^{2}\left( \frac{U_{y_{j}}}{y_{j}} \right)^{2} + 4\left( \frac{U_{b_{c}}}{b_{c}} \right)^{2} + 4\left( \frac{U_{Q}}{Q} \right)^{2}$
()

In the facility that is available for the experiments, we have initially
planned to conduct tests on a model radial gate at heads ranging from
0.13 m up to 0.50 m, and gate openings ranging from 0.05 m to 0.20 m.
The jet thickness, *y~j~*, will be about two-thirds of the gate opening,
so *y~j~* will vary from about 0.033 m to 0.17 m. We believe that we can
construct and measure the width of the model gate chamber with a
relative uncertainty of 0.25%. The laboratory is equipped with a
weighing tank for determining discharge whose relative uncertainty is
0.1% or better. It is reasonable to assume that we can measure the
upstream head with an uncertainty of 0.5 mm with a point gage in a
stilling well. Knowing from the previous analysis that we need to
achieve a relative uncertainty of 2.5% or better in the experimental
result, 1+ξ, we can solve for the required uncertainty in the
measurement of *y~j~*. Table 2 shows the results. Note that we solve for
the absolute uncertainty, *U~yj~*, not the relative uncertainty, since
we expect variability in our measurements of the jet thickness to be
mostly independent of the actual jet thickness.

  ------------------ ----------------- ----------- ----------- -----------
  **Table 2.                                                   
  Allowable                                                    
  uncertainty in                                               
  measurement of                                               
  *y~j~*, meters.**                                            

                     *y~j~*, meters                            

  *H*~1~, meters     0.033             0.066       0.1         0.133

  0.13               0.00047           0.00158                 

  0.2                0.00044           0.00106                 

  0.3                0.00043           0.00093     0.00162     

  0.4                0.00042           0.00089     0.00146     0.00216

  0.5                0.00042           0.00087     0.00139     0.00198
  ------------------ ----------------- ----------- ----------- -----------

The results show that any tests carried out at very low gate openings
and jet thicknesses will require measurements of jet thickness that have
an uncertainty of about 0.4 mm.

We anticipate difficulty in reaching this objective, but believe that we
can probably achieve a measurement uncertainty for *y~j~* of 0.5 mm.
Using that value, we can compute the relative uncertainty in 1+ξ over
the potential range of test conditions (Table 3), and the uncertainty
percentage contributions (UPCs) associated with each measured variable
(Table 4).

  ------------------ ----------------- ----------- ----------- -----------
  **Table 3.                                                   
  Relative                                                     
  uncertainty in                                               
  1+ξ.**                                                       

                     *y~j~*, meters                            

  H1, meters         0.033             0.066       0.1         0.133

  0.13               2.62%             1.20%                   

  0.2                2.80%             1.32%                   

  0.3                2.90%             1.42%       0.96%       

  0.4                2.95%             1.48%       1.01%       0.80%

  0.5                2.97%             1.50%       1.04%       0.83%
  ------------------ ----------------- ----------- ----------- -----------

  --------------- ------------ ------------ ------------ ------------ -----------
  **Table 4. UPCs                                                     
  for the                                                             
  measured                                                            
  variables at                                                        
  different test                                                      
  conditions.**                                                       

  H1, meters      yj, meters   UPC~*H*1~    UPC*~yj~*    UPC*~bc~*    UPC*~Q~*

  0.13            0.033        4%           92%          4%           1%

  0.20            0.033        1%           95%          3%           1%

  0.30            0.033        0%           96%          3%           0%

  0.40            0.033        0%           96%          3%           0%

  0.50            0.033        0%           97%          3%           0%

  0.13            0.066        42%          37%          17%          3%

  0.20            0.066        8%           75%          14%          2%

  0.30            0.066        2%           83%          12%          2%

  0.40            0.066        1%           86%          11%          2%

  0.50            0.066        1%           87%          11%          2%

  0.30            0.10         7%           61%          27%          4%

  0.40            0.10         3%           69%          25%          4%

  0.50            0.10         1%           71%          23%          4%

  0.40            0.133        5%           50%          39%          6%

  0.50            0.133        3%           55%          36%          6%
  --------------- ------------ ------------ ------------ ------------ -----------

In almost all test cases the uncertainty in the measurement of *y~j~* is
responsible for the majority of the uncertainty in the result. In the
tests to be carried out at very small heads, the uncertainty in head
measurement becomes somewhat significant. Uncertainty in the gate width
is also a significant factor in some test conditions, but it is
primarily those in which the total uncertainty in the result is
relatively low, so this is probably not a serious concern.

The analysis performed here shows that uncertainty in the measurement of
*y~j~* may seriously affect the ability to achieve the test objective.
It may be necessary to focus effort on reducing the uncertainty of that
measurement through the selection of measurement methods and equipment.
One might also conclude that performing tests at very small heads and
gate openings is not worthwhile, although there might be other good
reasons for keeping such tests in the plan, such as changes in hydraulic
phenomena that only occur at those test conditions. The key point is
that the uncertainty analysis allows one to make informed decisions
about test design, rather than relying on intuition or discovering
problems by trial.

# Numerical Approximation of Partial Derivatives

Developing the general uncertainty equation for a given data reduction
equation often requires the determination of numerous partial
derivatives. When the data reduction equation is complex, this can be a
very tedious task. An alternative is to use numerical approximations of
the partial derivatives. They can be computed from

$$\frac{\partial r}{\partial X_{i}} \approx \frac{\Delta r}{{\Delta X}_{i}} = \frac{r\left( X_{i} + {\Delta X}_{i} \right) - r\left( X_{i} \right)}{{\Delta X}_{i}}$$

with the values of all other variables in the data reduction equation
held constant while varying *X~i~*. To implement this, one must choose a
value for ∆*X~i~*, such as 0.01*X~i~*, and compute the derivative, then
reduce ∆*X~i~* to one-half the starting value and compute the derivative
again. Continue reducing ∆*X~i~* until the estimated value of the
derivative converges. This method can be easily implemented in
commercial spreadsheet or mathematical analysis software.

# Summary

This chapter has demonstrated the use of general uncertainty analysis
for the planning of experiments. In the planning phase we are trying to
determine whether a proposed experiment can satisfy our objectives. If
alternative data reduction equations and experimental methods are
available, we consider the alternatives and whether one may be more
effective than another. General uncertainty analysis using UMFs also
gives us a first impression of which measurements may be most critical
for obtaining a desired level of uncertainty in the experimental result.
As we refine our experimental plan, we can analyze the UPCs to see which
measurements are the most critical, considering both the manner in which
they propagate into the end result (indicated by the UMFs) and the
actual values of the measurement uncertainties.

Several results are possible from an uncertainty analysis performed in
the planning phase. Infeasible approaches may be discarded and feasible
methods pursued, experimental methods may be altered, instrumentation
needs can be clearly identified, and in some cases experimental programs
that will never yield useful results may be abandoned. Regardless of the
outcome, the value obtained from the expenditure of time and money can
be increased, usually in an amount far greater than the cost of
performing the uncertainty analysis. In addition, time spent on
uncertainty analysis in the planning phase will lay the foundation for
the more detailed uncertainty analysis that follows once experiments are
underway.

# References

Clemmens, A.J., Strelkoff, T.S., and Replogle, J.A. (2003). "Calibration
of submerged radial gates." *Journal of Hydraulic Engineering*, Vol.
129, No.9, pp. 680-687.

Coleman, H.W., and Steele, W.G. (1999). *Experimentation and Uncertainty
Analysis for Engineers*, 2^nd^ Ed., John Wiley & Sons, New York, 275 pp.

Kline, S.J. (1985a). The purposes of uncertainty analysis. *Journal of
Fluids Engineering,* Transactions of the ASME, June 1985.

Kline, S.J., (1985b). Closure to 1983 Symposium on Uncertainty Analysis.
*Journal of Fluids Engineering*, Transactions of the ASME, June 1985.

Moffat, R.J. (1985). "Using uncertainty analysis in the planning of an
experiment." *Journal of Fluids Engineering*, Vol. 107, No. 6, pp.
173-178.