Analysis of Trends in Low-Flow Time Series of Canadian Rivers - Hydrologic Time Series Analysis: Theory and Practice - page 212

LTP in annual low flows at a single station (05BB001) and in winter low
flows at three stations (01AQ001, 02PJ007 and 05BB001). The DFA method
suggests the possibility of LTP in annual low flows at a single station
(02EC002), in winter low flows at three stations (01AQ001, 01EO001 and
02PJ007) and in summer low flows at stations 02EC002 and 02PJ007 (a
marginal case). A similar investigation using the MLE method for all the 49
stations suggests significant values of H for 12, 13 and 10 annual, winter and
summer low flow time series, respectively. The number of time series which
could possibly be assumed to exhibit LTP would increase in the case of
smaller than 90% confidence intervals. Differences between the results of
various methods for estimating Hurst exponent have also been noticed in
some earlier studies, e.g. Montanari et al. (1997), who suggested using FARIMA
modelling technique to estimate Hurst exponent.
10.3.4 Results of Trend Analysis
The analyses of STP and LTP diagnostics presented above suggest that the
independence assumption or the STP and LTP assumptions do not hold for all
of the low flow time series collectively. Therefore, to realize the influence of
each of these assumptions on trend significance, estimates of trend significance
for all of the low flow time series were obtained separately assuming
independence, STP and LTP. For the independence case, the original MK test
was applied without considering the effect of serial dependence. For the MMK1
test, first autocorrelation of ranks of data was considered, irrespective of it
being significant or non-significant. By doing so, even small departures from
independence would contribute in modifying the trend significance. Because
of the influence of trend on autocorrelations and vice versa, this test was
applied after removing an estimate of the linear trend obtained using the Sen's
slope estimation technique (Sen, 1968). However, a more reasonable alternative
would be a joint estimate of both first autocorrelation and linear trend following
the iterative procedure described in Khaliq and Gachon (2010) that is consistent
with the trend analysis procedure using time series modelling and simulation
approach introduced in the work of Cohn and Lins (2005). For the MK-BBS
test (i.e., when the MK test was combined with the BBS approach), the
number of contiguous significant autocorrelations, starting from the first one,
was determined and their effect was considered for estimating trend
significance. Thus, for those time series for which none of the first few
contiguous autocorrelations were found significant, the p -values for the MK
and MK-BBS tests should exactly match but they would differ slightly because
of the involvement of bootstrap resampling procedure for estimating trend
significance. For the case of LTP, both MKS and ALRT tests were used. Step
by step instructions for applying the MKS test are available in Hamed (2008)
and those of the ALRT in Cohn and Lins (2005). The MKS test was developed
on the basis of scaling approach and the ALRT on the basis of time series
modeling and simulation approach. Exactly the same procedure as described