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to the error bars to judge how much importance to attach to any particular recon-
structed flow, and can use the error variance of the reconstructed annual flows to
derive appropriate error bars for relatively simple statistical summary statistics of
For more complicated statistical summary statistics, mathematical derivation
of the error bars may not be straightforward. An alternative approach is to dis-
pense with interpretation of the reconstruction itself and resort to probabilistic
analysis of a large number of plausible realizations of true flow derived from the
annual reconstructed values and their uncertainty. Such realizations have been called
equation:
u
i
= ˆ
ˆ
y
+
e
i
(8.1)
where
y
is a vector time series of reconstructed flows and
e
i
is a sample of random
noise of the same length drawn from a normal distribution with appropriate variance.
A large number (e.g., 1000) of such noise-added reconstructions constitutes a
plausible ensemble of 'true' flows, which can be analyzed probabilistically for
streamflow statistics. Noise-added reconstructions can be used to estimate prob-
abilities of past occurrence of any hydrologic 'event' that can be quantitatively
defined. For example, in the fourth year of a drought, we might be interested in the
probability that 4 consecutive low-flow years are followed by a fifth low-flow year.
The following example utilizes the updated reconstruction Lees-B of annual
to place the recent severe 5-year drought (2000-2004) in a long-term perspective.
The reconstruction was derived by multiple linear regression from a network of stan-
dard tree-ring chronologies, extends over the period 1490-1998, and is based on a
1906-1995 calibration period. The regression model explained 84% of the variance
of flows for the calibration period, verified well, and had residuals that conformed
Ferry observed flow for the period 1906-2004 has a mean of 18,540 million cubic
meters (mcm) and standard deviation of 5368.6 mcm.
Two essential steps in the analysis of the 2000-2004 drought are the definition
of the 'event' and the statement of a null hypothesis:
ˆ
1. Event: flow less than a specified drought threshold for at least 5 consecutive years
2. H0: at least one event occurred in the period 1490-1988
The 'event' defined for this example is the occurrence of 5 or more consecu-
tive years of flow below a drought threshold, which is arbitrarily specified as the
0.25 quantile of observed annual flows for 1906-2004. The 0.25 quantile adopted
as the drought threshold is equal to 14,365 mcm, or roughly 77% of the mean
annual flow. The event is of interest because five consecutive extremely low flows
indeed did occur on the Colorado River in water years 2000-2004, resulting in
