- Email: [email protected]
Step-functional analysis of long records of streamflow
CATENA
Vol. 9,379-396
Braunschweig 1982
STEP-FUNCTIONAL ANALYSIS OF LONG RECORDS OF STREAMFLOW G.H. Dury, Cambridge
SUMMARY Stream discharge records published by UNESCO include long series for thirteen stations. Step-functional testing resolves the series into sequences of square waves, which in some cases are superimposed on secular trends. Analysis deals throughout in terms of arithmetic values: the patterns of variation are such as to make the use of transform values unnecessary. The frequencies of block length in the square-wave sequences fall into a Poisson distribution. The percentage departure of block mean from series mean is linearly related to log of block length. Results of the analysis of random series of normally distributed random digits simulate all the results obtained from the analysis of streamflow records, from the patterns of cumulative deviations from series means, through square-wave sequences, secular trends, patterns of frequency distribution of block length, and patterns of departure of block means from series means, to the contrast between high-flow and low-flow regime. The unusually wide spacing of regime change on one river, and the unusually close spacing on three others, can also be matched from parts of the randomly generated sequences, but if the observed spacing of change in streamflow regime is truly representative of long-term behaviour, then in these four cases it should probably be regarded as different from what would be expected from purely random variation.
ZUSAMMENFASSUNG Die yon der UNESCO publizierten Datenserien des Wasserzulaufes von Str6men schliegen dreizehn Beobachtungsorte mit langen Datenserien ein. Stufen-Funktionenuntersuchungen Ibsen diese Serien in eine Reihenfolge von rechteckigen Wellen auf, in einigen F/illen sind diese Wellen auf siikularen Tendenzen iiberlagert. Die Analyse behandelt arithmetische Gr6fJen: eine Transformation der Gr66en war nicht erforderlich. Die H/iufigkeit der L/inge der rechteckigen Wellenblbcke folgt der Poisson Streuung. Der Prozentsatz der Abweichungen des Blockdurchschnittes yon Seriendurchschnitten zeigt eine gradlinige Beziehung zu dem Logarithmus der Blockl/inge auf. Die Resultate einer Analyse von zufallsverteilten Serien normiert gestreuter zufallsverteilter Ziffem simulieren alle Resultate der Analyse der Wasserauslaufsbeobachtungen, yon den kumulativen Abweichungen vom Seriendurchschnitt bis zu den rechteckigen Wellensequenzen, s/ikularen Tendenzen, H/iufigkeitsstreuungen der Blockl~ngen, Abweichungen der Blockdurchschnitte von den Seriendurchschnitten bis zum Gegensatz zwischen hohem und niedrigem Wasserstand. Die ungewbhnlich weiten Abst/inde der Wasserauslaufszustand/inderungen fiir einen der Rtisse kbnnen Teile.n.einer zufallsverteilten Serie angepafJt werden, abet sollten die beobachteten AbstS_nde der Anderungen im Wasserauslaufszustand wirklich langzeitige Verhaltungsweisen repr/isentieren, dann sollten diese vier F/ille als unterschiedlich yon einer reinen zufallsverteilten Variation betrachtet werden.
380 1.
DURY
INTRODUCTION
The stream discharge records published by UNESCO (1971) include series for 13 stations with records ranging in length from 89 to 162 years (Table 1). The records vary somewhat in character, five including m a x i m u m daily flow, seven m a x i m u m mean monthly flow, and one mean annual flow only. There can be little objection, however, to subjecting all the series to the same form of analysis. Where m a x i m u m daily flow and m a x i m u m mean monthly flow are both listed, the two sets of values can be closely similar, as for instance with the Neva and the V'~inern-G6ta; the record of annual mean discharge on the Loire at Blois appears somewhat more sensitive to the form of analysis employed than does the record of m a x i m u m mean monthly discharges on the same river at Montjean.
Tab. 1: RECORDS ANALYZED River
Station
Columbia Danube Labe ( = Elbe) Loire Loire Neman New~ Niagara St Lawrence St Mary's Severnaya Dvina (Northern Dvina) V:,inern-G6ta Vuoksi * missing ** 1942 missing
The Dalles, Ore. Orsova De,in Blois Montjean Smalininkai Novosaratovka Queenston Oldenburg Sault Ste Marie Ust-Pinega
Nature of discharge record 1879 - 1967 Maximum monthly 1840 -1968 Maximum daily 1851 -1968 Maximum monthly 1863 - 1969 Mean annual 1863 - 1969 Maximum monthly 1812 - 1969" Maximum daily 1858- 1969"* Maximum daily 1860 - 1967 Maximum monthly 1860 - 1964 Maximum monthly 1860 - 1969 Maximum monthly 1842 - 1969 Maximum monthly
V~anesborg Imatra
1807 - 1968 1847 - 1968
Period
Maximum daily Maximum daily
Drainage area km 2 614,000 645,000 (approx.) 51,104 38,180 110,000 812,000 281,000 665,000 764,000 20,700 348,000 46,830 61,280
The analysis tests for step-functional variation in the regime of maximal flow, as this is given in the available records. As will be seen, it breaks down the various sequences into trains of square waves. In certain cases, part or all of a square-wave sequence appears to be superimposed on a secular trend, which may affect high flows and low flows alike, or high flows only.
2.
ANALYTICAL METHOD
The analytical method, described elsewhere in connection with the analysis of precipitation series (DURY 1980), is extremely simple. It involves the reduction of a given time series to stationary form, by calculation of the series mean, and then the calculation of cumulative deviations from that mean. What happens next depends on the range of data-processing equipment available. Computer programmes exist which will handle the whole operation, including calculation of series means and of cumulative deviations, but the
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
381
method is most easily explained with reference to partly manual working. If for a given series the cumulative deviations are graphed, the graph will typically display rising, falling, and roughly horizontal segments, these last being absent in some instances (Fig. 1). Subject to difference-of-means tests, each segment identifies a block - here a block of years - during which observed values run generally above the series mean (rising segment), generally below that mean (falling segment), or generally close to that mean (roughly horizontal segment).
.?"
15'
.%
,
e
E
I
10-
o 0
.:,¢
x
•
°
.;" r
60
u
~
1900
I
1950
Fig. 1: Graph of cumulative deviations for maximal streamflow on the Niagara River at Queenston: straight lines have been fitted by regression. Difference-of-means tests on successive pairs of block means have taken standard form. The cutoffP = 0.05 has been applied thoughout in the F test for difference of variance. The same cutoff has been applied in the ttest for difference of means, except where N1, N2 or both ~< 3, when a difference at P ~< 0.10 has been accepted. The reason for this slight relaxation is that the graph of cumulative deviations tends to be somewhat more sensitive in revealing trends in the mn of observed values than does the difference-of-means test in identifying significant changes between pairs of blocks. If one wishes, the values of cumulative deviations may be regressed, segment by segment, on their serial positions. The results of such regression are shown in Fig. 1. It is usual to obtain high correlations with rising and falling segments, while those associated with narhorizontal segments are typically low, and in some cases are not statistically significant. While this regression is not an essential part of the analysis, it does serve to confirm the identification of the various blocks, and also to confirm that the trends of the segments are linear through the time. The entire exercise can be conveniently handled by an electronic calculator and a small programmable desk computer, used in tandem. Possible observed values in the series here considered range upward from zero: the same is true of precipitation series. Although the difference-of-means test assumes, or at least implies, that the observed values are normally distributed about their means, in particular the means of blocks, this may not be so: hence the practice of using transform values. I have myself drawn attention to the advantage gained, in the analysis of residuals in a precipitation series, from the use of cube-root transforms (DURY, op.cit. 1980). Should a series contain a
382
DURY
~ o b s e r v e d
30
% ,-- - -
--]~fitted
I 20
10
o ---
5---i
250
_1 i
J
150
50
"3
i
--
0
At-
50
150
250
Fig. 2: Distribution of residuals from their block means, V'tinern-G6ta river, 1807-1968, with approximately fitted normal curve. Regression of fitted on observed values gives_r = + 0.954, r9 = 0.910, 0.01 > P.
noticeable frequency of zero observations, still further advantage might result from use of the transform 3 vr~ + 1, wherey is an observed value. In the present exercise, however, it seems unlikely that any useful purpose would be served by dealing in transform as opposed to arithmetic values. Series containing occasional very high and numerous rather low values produce distinctly skewed distributions of residuals from means, with thin but long positive tails and with a peak frequency in the bracket next below zero. Fig. 2 shows the distribution of arithmetic residuals from their block means, for the streamflow series of the V'~inern-G6ta. A slight skew does exist; but, as indicated, a normal curve can be successfully fitted to the observed distribution. A potentially more serious matter is that the difference-of-means test makes no provision for single aberrant items - a very high value in a generally low series, or a very low value in a generally high series. In the records considered, however, such items are rare to absent.
3.
ANALYTICAL RESULTS
The combination of the breakdown of deviations graphs into segments with the testing for differences of roans between the blocks represented by those segments permits the observed records ofpak streamflow to be resolved into sequences of square waves (Figs. 3-7, Table 2). Each horizontal tread in a wave sequence represents a block length and a block mean. This mean differs significantly from the means of immediately preceding and succeeding blocks, - or, at one end or other of the sequence, differs either from the succeeding or the
STREAMFLOW: S T E P - F U N C T I O N A L ANALYSIS
383
4 I
St M A R Y ' S
R.
NIAGARA
0O9
St
LAWRENCE
15
~
__
In is'so
lgoo
-
F
~-n- -~,.r~R--- I]_- - rrt ~
l~so
Fig. 3: Histograms of peak flow, with square-wave sequences superimposed, for the St Mary's, Niagara, St Lawrence, and Columbia Rivers: trend lines are discussed in the text.
E r9
[1
L O I R E / MONTJ EAN
]l
20t
n r-I ,_nlln:_4Ll~L! 1o ~~1 LrUIdUll~ uI..,-,,~ ~yu ''~i
Fig. 4:
1T009
U,
H I~VH ~ 1
I
Histograms and square waves for the Loire at Blois and at Montjean.
19'50
384
DURY
1000 VANERN-GOTA E
50O 1850
1900
"'""" °°'" ...."
ooo'i •
F
1950
100C 80C
1850 Fig. 5:
i15~f
1950
1900 Histograms and square waves for the V'~inern-G6ta and the Vuoksi.
DANUBE
o~5_J ~l 1o_
n .,-.-,_ t~ °°Hn U UUL,-Hu U U u Lr
15
~ r,~.-.-, Yi ._,~lm, A-in rl N ILrlnn _,~,-, ~r~
LJ!_J LP.Juu u L-El Lid
LABEIELBE)
UL'-]'~rr'
~LH u L
~
{ ~8SO
El U u L " / h J ~ -
r~
o-"-Lllr n 19'00
-,'-,.1~50
Fig. 6: Histograms and square waves for the Danube and the Labe (Elbe).
preceding mean. Each vertical riser marks both the incidence and the magnitude of an abrupt change from one block, with its particular mean, to the next block with its different mean. The abrupt changes constitute step functions. Most of the square-wave sequences suggest little or nothing byway of secular trend. The high-flow and low-flow regimes alternate, with occasional interpolations of blocks with nearaverage flow, without any sign that either the high-flow or the low-flow regime is trending either upward or downward. Something could of course be made of the comparative distribution of blocks with high or low means, but the necessary allowance is already made by the graphs of cumulative deviations. In any event, a concentration of long blocks with high or low means in a particular part of a sequence does not necessarily imply a secular trend of increase or decrease: it could easily be a simple accident of distribution. Possible exceptions, to be referred to again, occur in the square-wave sequences for the Columbia, St Lawrence, and St Mary's Rivers (Fig. 3). Regression of mean values for the low-flow blocks on the St
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
385
NEVA
I
T
r
~
1860
,
1 g 00
O
i
,
195O
i
i
i°] o
t
0
0 ~
o° ® E
n -,11
~ I[IL. I1J ~ ] i Lr-,-,~u ~ --'~-u-"z:r7" L~
[
1
T
1
1810 5
NEMA
"
i
-
, -
-
1850
cont'd
"
Lr,.-rh
~
~ r
-
-
l
~
OO
II--
=
0
0 "L'~---t
1880
Fig. 7:
r
~
1900
r
-r
"r
r
"r
r
1950
Histograms and square waves for the Neva, Scvernaya (Northern) Dvina, and the Neman.
Lawrence, 1872-81 to 1962-64, on the locations of their midpoints on the timescale produces r = --0.869, r 2 = 0.755, and 0.05 > P. The regression equation indicates a reduction of the mean value of the low-flow blocks, during an interval of about 90 years, by some 20 per cent of its starting value. For part of this interval, the means of high-flow blocks sustained an almost parallel descent. Regression, as above, gives_r = --0.989 and_r2 = 0.978, although the inclusion of only three pairs of items in the calculation ensures that 0.10 > P > 0.05. The regression equation indicates a reduction through 50 years of 9 per cent of starting value. Although the square-wave pattern for the Niagara River at Queenston, a station upstream of Oldenburg on the St Lawrence, appears at first sight generally similar to the pattern drawn for the St Lawrence itself, no statistically definable trend, comparable to that identified for the St Lawrence, seems to be present. At the same time, the two records in combination could be taken as indicating a general decline in peak flow from 1860 to 1942, except for the sharp rise in 1929-1930. In direct contrast, the St Mary's River, another member of the Great Lakes system, combined an increase in variability of peak flow, during the approximate period 1880-1954, with an increase in the average magnitude of high flows. Selection of values for entry into regression is complicated by the occurrence of block means close to the series mean. If however such blocks are considered, along with high-flow blocks proper, then regression of block means on the serial locations of their midpoints produces r = + 0.919, r_2 = 0.845, and 0.05 > P, the regression equation indicating an increase through some 30 per cent of starting value in an interval of about 75 years. No accompanying trend can be identified for the low-flow regime. Equally for the record of the Columbia river, analysis fails to demonstrate a significant
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
387
falling trend for the low-flow blocks, even though in the second part of the sequence these seem to descend in parallel with the high-flow blocks. For these latter, regression as above produces r = --0.871,~ = 0.759, and 0.05 > P. The regression equation indicates a reduction through some 20 per cent of starting value during an interval of some 90 years, a result identical to that obtained for the reduction in the means of low-flow blocks, and closely similar to that for high-flow blocks on the St Lawrence. For the time being, no explanation of the resemblances will be attempted. Particularly perhaps with respect to the Columbia, one might ask if artificial works may not have seriously affected the regime of maximal flows. The major dams, however, the Grand Coulee and the Bonneville, date only from the 1930s, while the major irrigation project began abstracting water only in 1952. Irrespective of the location of works in relation to the gauging station, it seems reasonable therefore to suppose that the secular trend recognised for the high-flow regime on the Columbia is essentially natural. A parallel comment can be assumed to apply to the secular trends identified for the St Lawrence and St Mary's Rivers.
4. CUMULATIVE DEVIATIONS, SQUARE-WAVE SEQUENCES, AND RANDOM VARIATION
Graphs of cumulative deviations, similar to those obtained for streamfiow, can readily be produced by random operations. A particularly noteworthy experiment is that of JACCHIA (1975); he used a computer to generate a large random number, to select the final digit, to add this digit and subtract 4.5, and so on through 200,000 steps. The machine printed out a point for the cumulative deviation at every 200th step, providing a 1,000-point graph. Parts of this graph can be closely matched to deviation graphs for climatic and hydrologic series (cf. DURY, op.cit. 1980), the only difference being that the climatic and hydrologic graphs are drawn for stationary series, whereas JACCHIA's graph belongs to a non-stationary series. Now in JACCHIA's experiment, a constant probability of 10 per cent applied to the occurrence of all the digits 0 through 9 - although, as the result demonstrates, 200,000 passes produced a series mean sufficiently different from 4.5 to' ensure a large final cumulative deviation. The digits which may be read, or extracted by sampling, from the usual published table of random numbers are also regularly distributed, with an equal probability of occurrence attached to each. Series constructed from them can be shown to produce deviation graphs similar to those obtained for streams, and to break down into square-wave sequences resembling those drawn for the variation of peak streamflow. Stream discharges in general, and peak discharges in particular, are not however regularly distributed about their means, but clustered, as is illustrated by Fig. 2 herewith. I have accordingly undertaken a series of trials with normally distributed random digits, wherein for instance the probability of the occurrence of the digit 4 or 5 is 40 times as great as the occurrence of 0 or 9. The tables, which include 918 entries each, arranged in 27 columns and 34 rows, exist in two versions. Read down the columns, the first version provides six almost stationary series of 153 bits each, and three stationary series of 306 bits each. The second version is a randomized redistribution of the entries in the first version. Read serially either across the rows or down the columns, it provides a 918-bit stationary series, as the first version does when read serially across the rows. Either version can be used to generate non-stationary series of any desired
388
DURY
length. For purposes of comparison with the results of the analysis of streamflow, both tables have been read across the rows, the two 918-bit stationary series being analyzed for cumulative deviations, tested for step-functional breakdown, and resolved into sequences of square waves.
1000
"
25
o
+
E
0
•
random
o
Vuoksi
data o
ooO'°O
~. 15
o
•
20
o•
o
o o
o o
o
o
o
o
•
o
•
•
o o
•
•
o
o
• o
•
•
o °
•
o o
""'i i j-" o o.
1000
o o
o o
o
o
670 i
10
•
o
°
o
700 I
I
l
~
i
900
i
i
i
I
1950
Fig. 8: Cumulative residuals for the Vuoksi river, 1903-54, and similar residuals for a sequence of 52 normally distributed random digits. Regression of observed on generated values given r = + 0.792, r2 = 0.627, 0.01 > P.
/
400"
+ 0
400
800
1200
O0 I
i
5
random
10
digits
1 '5
2'0
25
Fig. 9: Plot of cumulative residuals for the Vuoksi river against randomly generated residuals: regression line shown.
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
389
8
654
1820 t
1850 i
i
1
i
188(3 i
i
i
25O
32
Fig. 10: Part of the square-wave sequence for the Vzinern-G6ta compared to part of a randomly generated sequence. A sample of the results is given in Fig. 8, where part of the deviation graph for the Vuoksi River is compared to part of the deviation graph drawn for a sequence of normally distributed random digits. There is an obvious family resemblance between the two plots. Regression of the observed on the generated values produces a highly significant correlation (fig. 9). Many other examples of the same kind could be offered. Similarly, the square-wave sequences into which peak streamflow has been resolved can be matched from the sequences obtained from normally distributed random digits. Fig. 10 compares part of the square-wave train for the V'~inern-G6ta with part of a randomly generated train. Once again, many additional examples could be offered.
5.
DISTRIBUTIONS OF BLOCK LENGTH
Comparisons of block length, which amount to comparisons of the frequency of stepfunctional change, are rendered somewhat uncertain by the presence in the streamflow series of uncompleted blocks. Such blocks, of 56, 47, 36, and 35 years in length, appear in the square-wave trains for the Danube, the Loire at Blois, and the Niagara. The longest completed blocks are recorded for the Neman (37 years) and the Loire at Montjean (36 years). If all blocks, completed or not, are taken into analysis, and a frequency distribution is constructed by 5-year intervals, then that distribution is closely matched by the distribution, arranged by five-bit intervals, obtained from the two 918-bit series of normally distributed random digits (Fig. 11). The correlation between the observed and the generated distribution is close, and statistically highly significant. The same Poisson distribution can be fitted to both arrays, as shown in fig. 10, where, solely for the sake of convenience, the frequencies are grouped by 10-year and 10-bit ranges. The closeness of fit between the observed and the Poisson distributions is perhaps especially satisfying; the fractional difference in closeness of fit between the observed and the generated distribution has no importance. Mean block length varies considerably from one stream to another. Difference-of means tests, between mean block length for streams and the mean of some 11.58 bits for the randomly generated data, indicate no significant difference in nine of the 13 cases. The mean
390
DURY 40 % 30 " ~ ' 1 - - --lr"~/~ r andom
20-
_~rivers
__
10-
-l_ =
:
block
©
P-- - - 1 - - - - ' I
;
~
leng ~"h ~
~
oJ
o~
r-- --i
~
d"
~
=
~"
m
7o
® =
,,o
Fig. 11: Frequency distributions of block length, for rivers and for randomly generated sequences. Totals are 149 blocks for rivers and 158 blocks for random data. Regression of observed on generated distributions gives r ~ +0.984, [2 = 0.968, 0.01 > P.
70 ~ ~rivers
6'O
50-
40Poisson
/
30-
20-
/
)--
10~
/ random
0
i
2--11 12--21 block l e n g t h
g
t-
-
-f-
22-- 31 32-- 41
-
42-51
-'
52-- 61 62--71
Fig. 12: Block frequencies for river data and for random data compared to a Poisson distribution. Regression on values in the Poisson distribution gives_r = + 0.999, r~ = 0.998 for the river data,_r + 0.999, [2 = 0.980 for the radom data, 0.01 > P in both cases.
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
391
block length of 32 years for the Danube, reckoned in any event from largely uncompleted blocks, is significantly greater than the mean for the random data, while the means for the St Mary's River (5.5 years), V'~inern-G6ta (8.53 years), and Vuoksi (7.12 years) are significantly less. What should be made of the differences must however be regarded as to some extent a matter of opinion. The mean block length on the Columbia River, 8.08 years, is less than hat on the V'~inern-G6ta, but a greater variance causes the t-test for difference of means between observed and randomly generated distributions to fail. Again, one sequence of 111 bits in the random data resolves into four blocks of average length 27.75 bits: compare the 129 years and four blocks on the Danube. Another sequence of 102 bits resolves into 16 blocks of average length about 5.02: compare the 110 years and 20 blocks on the St Mary's River. That is to say, even the sets of the longest and the shortest block means can be matched from the sequences obtained from the normally distributed random digits. Again, if the percentage frequency of block length is plotted against a logarithmic scale, the frequency distribution can be described by a normal curve (Fig. 13).
40%
observed 30-
20-
10/
i
log
0"54 0"74 block length
0"94
1 . 14
1-34
1,54
Fig. 13: Logarithmic frequency of mean block length, river data: the x scale is subdivided so as to include a mode at about 11.0, and to provide for seven columns. Corresponding arithmetic cutoffs are approximately 3.47, 5.50, 6.92, 13.80, 21.88, and 34.67._r = + 0.974,_r2 ~ 0.949, 0.01 > P.
The total sequence of random digits analyzed for step-functional change amounts to 1830 bits, the difference from 912 x 2 = 1836 being accounted for by short loose ends. As a simulation model of the variation of peak streamflow, the sequence corresponds to a run of observations nearly two millenia in length. As has just been pointed out, parts of the sequence match observed sequences where block means are unusually long or unusually short: only if the unusually long and unusually short observed means are truly representative of long-term behaviour should their differences from randomly generated effects be accepted. It remains true that some square-wave sequences for rivers are distinctly more, or less, variable than some others. Among the possible physical causes that could be investigated are size of catchment, regime of precipitation, and regime of runoff. Investigation of the two
392
DURY
types of regime is outside the scope of the present paper. There is no significant correlation, for the rivers considered, between size of catchment on the one hand, and mean block length or standard deviation of block length on the other. The data do not provide the means to scrutinize individual systems, except that the modest downstream increase in mean block length on the Loire is not statistically significant, and that no significant relationship between block length and drainage area appears on the Great Lakes - St Lawrence system. The tally of three stations, needless to say, is quite inadequate for the study of a whole large system; it can nevertheless be said that, whereas mean block length for the St Mary's River is significantly less than the mean for either the Niagara or the St Lawrence, mean length actually decreases (although not significantly so) between this last pair.
6.
BLOCK LENGTH AND DEVIATIONS FROM SERIES MEANS
One would expect on general grounds that the departures of block means from series means would increase with decreasing length of block. For a given variance, difference of means must increase as block length decreases, in order for the criterion of difference to be satisfied. Conversely, with variance again assumed to be constant, increasing block length, and especially the increase in length of two successive blocks, reduces the value of the statistic v,, ,~, and thus increases the value oft obtained in the difference-of-means test. Because block lengths themselves assume a Poisson distribution, one might look to a similar distribution to fit the distribution of the ratio between block length and deviation of block mean from series mean. In actuality, when departures of block means are expressed as percentages, averaged for each block length, they indicate a normal/log relationship between departure and block length (Fig. 14). While the points and the regression line for the randomly generated data indicate a somewhat greater departure than do those for rivers, especially in the lower part of the range of block length, the two data sets belong to a single order. We may conclude, at least provisionally, that the degree of percentage departure of block mean values from series mean values, as observed for rivers, can be simulated by random operations.
40
%
• ~
O
•
regression
line, random
data
(0)
,20
10
regression 0
line,
rivers
•
(0)
• •
[
,
2
block
i
i 5
i
i
)
,
i 10
iQ
20
•
o-Jo •
o )
n
~
•
-~o~
~
O~ I
o
~ i
50
length
Fig. 14: Mean percentage deviation from series mean, by block length, for river data and for randomly generated data.
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
7. RANDOMLY STREAMFLOW
GENERATED
SEQUENCES
AND
SECULAR
393
TRENDS
IN
Even if we accept that the secular trends in peak flow identified for the St Lawrence, St Mary's, and Columbia river, and possibly also present in the record for the Niagara, are both real and natural, this does not necessarily mean that they are not random. Fig. 15 shows parts of the square-wave sequence obtained by analysis of normally distributed random digits. The upper part of the diagram closely simulates the falling trends on the St Lawrence and the Columbia, while the lower part closely simulates the rising trend observed for the high-flow regime on the St Mary's River. Degrees of change are comparable between the two sets of trend. The falling trend for the high blocks in the upper part of the diagram corresponds to a loss of 17% of starting value in the space of 86 bits, that for low blocks to a loss of 50% in 80 bits. The rising trend in the lower part of the diagram corresponds to a gain of 38% of starting value in the space of 55 bits. AZ 2 test, using these loss and gain rates, the gain rate for the St Mary's River, the loss rate for the high-flow regime on the St Lawrence, and the mean loss rate for the low-flow regimes on the St Lawrence and the Columbia, shows no significant difference, with P - 0.5. Like the square-wave sequences, the observed secular trends and trend rates can be simulated by random operation. 7
j
4510
Fig. 15: Seculartrends in square wavesdrawn for random digit sequences: compare the trends shown for certain rivers in Fig. 3. Regressiongives r = --0.978,0.05 > P for the top trend,_r ~ -0.941, 0.10 > P for the middle one, and r = + 0.933, 0.05 > P for the bottom one.
8.
HIGH-FLOW AND LOW-FLOW REGIME
In considerable part, the square-wave sequences drawn for peak streamflow, and the randomly generated sequences, alike consist of blocks with means distinctly above or below the series means. For some sequences it is possible to separate high, low, and intermediate regimes. Thorough analysis of the cases concerned would presumably demand a criterion of distinction between the intermediate regimes on the one hand, and the high and low regimes on the other. Such a criterion could obviously relate to the presence or absence ofa signifi-
394
DURY
1000¢n
high-flow
regime
E 50 0
o OOO;::oo ......
.
[
,,~O
O
~
I.
~
random
I
. . .
Iregime
digits
3" 2-
-o /
o 1-
1.01
1 1
1.58
2
recurrence
5
interval
10
20
50
Fig. 16: Magnitude-frequency graphs for high and low regimes, V'~inern-G6ta(146 items in all) and for a sequence of random digits (129 items in all).
cant difference between a block mean and a series mean. For immediate purposes it will suffice to consider the high-flow and low-flow regimes obviously identifiable in the record of the V'~inern-G6ta, 1807-1969, with the block 1951-64 alone excluded as averaging very close indeed to the series mean. This exclusion apart, there remain a total of 72 years included in low-flow blocks, and 74 years in high-flow blocks. Magnitude-frequency analysis for the high-flow and the low-flow regimes produces the results plotted in the upper part of Fig. 16. The interpretation of the uppermost end of the graph for the high-flow regime is a matter of some doubt, but between the recurrence intervals of 1.58 and 10 years the values for the high-flow regime run some 30 per cent greater than those for the low-flow regime. The lower part of the Figure shows outline magnitudefrequency graphs for the high and low regimes in a sequence of 129 normally distributed random digits, of which 69 occur in blocks of the high regime and 50 in blocks of the low. The points are plotted for the first appearance of a given digit in a reading up the scale of recurrence interval. Parallel results would be achieved by the plotting of last appearances, while a complete plot, needless to say, would rise in steps. The graphs obtained are similar to those drawn for the high-flow and low-flow regimes of the V'~inern-G6ta, the mean difference between high and low values, in the range of recurrence interval from 1.58 to 10, being only slightly less, in percentage terms, than that for the river. The outcome of this part of the analysis is, in view of the results of the analysis of departures in relation to block length, not at all surprising. It does however serve to show that the contrast between high-flow and low-flow regimes, as these are observed on rivers, can be replicated by random operation.
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS 9.
395
CONCLUSIONS
Twenty years ago, CURRY (1962) advanced theoretical reasons for regarding climatic change as a random series. His comments, even though he was mainly considering major climatic variations, are certainly transferable to streamflow series. The present paper makes an empirical comparison between step-functioning streamflow series and the step-functional sequences obtained from normally distributed random digits. Peak streamflow on rivers with long records can, by reference to cumulative deviations from series means, and by the application of difference-of-means testing, be broken down into sequences of square waves. These sequences consist of blocks of years during which observed values averaged higher than, lower than, or occasionally close to a series mean. Successive blocks are separated by step-functional changes in regime. The real square-wave sequences, the real distribution of block length, the real distribution of the ratio between block length and percentage departure from series mean, the real secular trends, and the real distinction between high-flow and lowflow regimes, can all be simulated by random methods. The one uncertainty concerns the spacing of regime change on the Danube significantly greater than the average obtained from the random digits - and on the St Mary's, V'~inern-G6ta, and Vuoksi Rivers - significantly less. The observed spacing on these rivers can be matched from parts of the randomly generated sequences; but if the actual records are accurate representations of long-term behaviour, then the frequency of regime change on these four rivers should probably not be regarded as truly random through time. Contributions to distributional analysis include the identification of the frequency of block length as according with a Poisson distribution, and the provisional identification of the relationship of percentage departure from series mean to block length as forming a normal-log distribution.
10.
END-NOTE
The research project, the results of which are presented here, forms one of a series of analyses of the step-functional change of climatic and hydrologic variables. A parallel analysis of precipitation at Sydney, NSW, has already been cited (DURY, op.cit. 1980). Anlysis of the early precipitation records for Sydney and for Adelaide, S.A., is in hand. I have shown elsewhere that step-functional analysis can also be applied to tree-ring series and to the documentary records of harvest quality and of farming weather (DURY, 1981). The research was in part supported by a grant from the Graduate School of The University of Wisconsin-Madison (Project No. 181441). Professor Barbara Bell (Harvard) originally drew my attention to the usefulness of cumulative deviations, with which her own published work in part deals. Professors Waltraud Brinkmann and Grace Wahba (Madison) discussed, respectively, the use of transforms and the desirability of using normally, as opposed to regularly, distributed random numbers.
BIBLIOGRAPHY
CURRY, L. (1962): Climaticchange as a random series. Annals, AssociationofAmerican Geographers, 52, 21-31. DURY, G.H. (1980): Step-functionalchange in precipitation at Sydney. Australian Geographical Studies, 18, 62-78.
396
DURY
DURY, G.H. (1981): Climate and settlement in late-medieval central England. In: SMITH & PARRY (eds.): Consequences of Climatic Change, 40-53, University of Nottingham, Department of Geography. JACCHIA, L.C. (1975): Some thoughts about randomness. Sky and Telescope, 50, 371-375.
Anschrift des Autors: George H. Dury 46 Woodland Close, Risby, Bury St Edmunds IP 28 6QN, England, United Kingdom
Vol. 9,379-396
Braunschweig 1982
STEP-FUNCTIONAL ANALYSIS OF LONG RECORDS OF STREAMFLOW G.H. Dury, Cambridge
SUMMARY Stream discharge records published by UNESCO include long series for thirteen stations. Step-functional testing resolves the series into sequences of square waves, which in some cases are superimposed on secular trends. Analysis deals throughout in terms of arithmetic values: the patterns of variation are such as to make the use of transform values unnecessary. The frequencies of block length in the square-wave sequences fall into a Poisson distribution. The percentage departure of block mean from series mean is linearly related to log of block length. Results of the analysis of random series of normally distributed random digits simulate all the results obtained from the analysis of streamflow records, from the patterns of cumulative deviations from series means, through square-wave sequences, secular trends, patterns of frequency distribution of block length, and patterns of departure of block means from series means, to the contrast between high-flow and low-flow regime. The unusually wide spacing of regime change on one river, and the unusually close spacing on three others, can also be matched from parts of the randomly generated sequences, but if the observed spacing of change in streamflow regime is truly representative of long-term behaviour, then in these four cases it should probably be regarded as different from what would be expected from purely random variation.
ZUSAMMENFASSUNG Die yon der UNESCO publizierten Datenserien des Wasserzulaufes von Str6men schliegen dreizehn Beobachtungsorte mit langen Datenserien ein. Stufen-Funktionenuntersuchungen Ibsen diese Serien in eine Reihenfolge von rechteckigen Wellen auf, in einigen F/illen sind diese Wellen auf siikularen Tendenzen iiberlagert. Die Analyse behandelt arithmetische Gr6fJen: eine Transformation der Gr66en war nicht erforderlich. Die H/iufigkeit der L/inge der rechteckigen Wellenblbcke folgt der Poisson Streuung. Der Prozentsatz der Abweichungen des Blockdurchschnittes yon Seriendurchschnitten zeigt eine gradlinige Beziehung zu dem Logarithmus der Blockl/inge auf. Die Resultate einer Analyse von zufallsverteilten Serien normiert gestreuter zufallsverteilter Ziffem simulieren alle Resultate der Analyse der Wasserauslaufsbeobachtungen, yon den kumulativen Abweichungen vom Seriendurchschnitt bis zu den rechteckigen Wellensequenzen, s/ikularen Tendenzen, H/iufigkeitsstreuungen der Blockl~ngen, Abweichungen der Blockdurchschnitte von den Seriendurchschnitten bis zum Gegensatz zwischen hohem und niedrigem Wasserstand. Die ungewbhnlich weiten Abst/inde der Wasserauslaufszustand/inderungen fiir einen der Rtisse kbnnen Teile.n.einer zufallsverteilten Serie angepafJt werden, abet sollten die beobachteten AbstS_nde der Anderungen im Wasserauslaufszustand wirklich langzeitige Verhaltungsweisen repr/isentieren, dann sollten diese vier F/ille als unterschiedlich yon einer reinen zufallsverteilten Variation betrachtet werden.
380 1.
DURY
INTRODUCTION
The stream discharge records published by UNESCO (1971) include series for 13 stations with records ranging in length from 89 to 162 years (Table 1). The records vary somewhat in character, five including m a x i m u m daily flow, seven m a x i m u m mean monthly flow, and one mean annual flow only. There can be little objection, however, to subjecting all the series to the same form of analysis. Where m a x i m u m daily flow and m a x i m u m mean monthly flow are both listed, the two sets of values can be closely similar, as for instance with the Neva and the V'~inern-G6ta; the record of annual mean discharge on the Loire at Blois appears somewhat more sensitive to the form of analysis employed than does the record of m a x i m u m mean monthly discharges on the same river at Montjean.
Tab. 1: RECORDS ANALYZED River
Station
Columbia Danube Labe ( = Elbe) Loire Loire Neman New~ Niagara St Lawrence St Mary's Severnaya Dvina (Northern Dvina) V:,inern-G6ta Vuoksi * missing ** 1942 missing
The Dalles, Ore. Orsova De,in Blois Montjean Smalininkai Novosaratovka Queenston Oldenburg Sault Ste Marie Ust-Pinega
Nature of discharge record 1879 - 1967 Maximum monthly 1840 -1968 Maximum daily 1851 -1968 Maximum monthly 1863 - 1969 Mean annual 1863 - 1969 Maximum monthly 1812 - 1969" Maximum daily 1858- 1969"* Maximum daily 1860 - 1967 Maximum monthly 1860 - 1964 Maximum monthly 1860 - 1969 Maximum monthly 1842 - 1969 Maximum monthly
V~anesborg Imatra
1807 - 1968 1847 - 1968
Period
Maximum daily Maximum daily
Drainage area km 2 614,000 645,000 (approx.) 51,104 38,180 110,000 812,000 281,000 665,000 764,000 20,700 348,000 46,830 61,280
The analysis tests for step-functional variation in the regime of maximal flow, as this is given in the available records. As will be seen, it breaks down the various sequences into trains of square waves. In certain cases, part or all of a square-wave sequence appears to be superimposed on a secular trend, which may affect high flows and low flows alike, or high flows only.
2.
ANALYTICAL METHOD
The analytical method, described elsewhere in connection with the analysis of precipitation series (DURY 1980), is extremely simple. It involves the reduction of a given time series to stationary form, by calculation of the series mean, and then the calculation of cumulative deviations from that mean. What happens next depends on the range of data-processing equipment available. Computer programmes exist which will handle the whole operation, including calculation of series means and of cumulative deviations, but the
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
381
method is most easily explained with reference to partly manual working. If for a given series the cumulative deviations are graphed, the graph will typically display rising, falling, and roughly horizontal segments, these last being absent in some instances (Fig. 1). Subject to difference-of-means tests, each segment identifies a block - here a block of years - during which observed values run generally above the series mean (rising segment), generally below that mean (falling segment), or generally close to that mean (roughly horizontal segment).
.?"
15'
.%
,
e
E
I
10-
o 0
.:,¢
x
•
°
.;" r
60
u
~
1900
I
1950
Fig. 1: Graph of cumulative deviations for maximal streamflow on the Niagara River at Queenston: straight lines have been fitted by regression. Difference-of-means tests on successive pairs of block means have taken standard form. The cutoffP = 0.05 has been applied thoughout in the F test for difference of variance. The same cutoff has been applied in the ttest for difference of means, except where N1, N2 or both ~< 3, when a difference at P ~< 0.10 has been accepted. The reason for this slight relaxation is that the graph of cumulative deviations tends to be somewhat more sensitive in revealing trends in the mn of observed values than does the difference-of-means test in identifying significant changes between pairs of blocks. If one wishes, the values of cumulative deviations may be regressed, segment by segment, on their serial positions. The results of such regression are shown in Fig. 1. It is usual to obtain high correlations with rising and falling segments, while those associated with narhorizontal segments are typically low, and in some cases are not statistically significant. While this regression is not an essential part of the analysis, it does serve to confirm the identification of the various blocks, and also to confirm that the trends of the segments are linear through the time. The entire exercise can be conveniently handled by an electronic calculator and a small programmable desk computer, used in tandem. Possible observed values in the series here considered range upward from zero: the same is true of precipitation series. Although the difference-of-means test assumes, or at least implies, that the observed values are normally distributed about their means, in particular the means of blocks, this may not be so: hence the practice of using transform values. I have myself drawn attention to the advantage gained, in the analysis of residuals in a precipitation series, from the use of cube-root transforms (DURY, op.cit. 1980). Should a series contain a
382
DURY
~ o b s e r v e d
30
% ,-- - -
--]~fitted
I 20
10
o ---
5---i
250
_1 i
J
150
50
"3
i
--
0
At-
50
150
250
Fig. 2: Distribution of residuals from their block means, V'tinern-G6ta river, 1807-1968, with approximately fitted normal curve. Regression of fitted on observed values gives_r = + 0.954, r9 = 0.910, 0.01 > P.
noticeable frequency of zero observations, still further advantage might result from use of the transform 3 vr~ + 1, wherey is an observed value. In the present exercise, however, it seems unlikely that any useful purpose would be served by dealing in transform as opposed to arithmetic values. Series containing occasional very high and numerous rather low values produce distinctly skewed distributions of residuals from means, with thin but long positive tails and with a peak frequency in the bracket next below zero. Fig. 2 shows the distribution of arithmetic residuals from their block means, for the streamflow series of the V'~inern-G6ta. A slight skew does exist; but, as indicated, a normal curve can be successfully fitted to the observed distribution. A potentially more serious matter is that the difference-of-means test makes no provision for single aberrant items - a very high value in a generally low series, or a very low value in a generally high series. In the records considered, however, such items are rare to absent.
3.
ANALYTICAL RESULTS
The combination of the breakdown of deviations graphs into segments with the testing for differences of roans between the blocks represented by those segments permits the observed records ofpak streamflow to be resolved into sequences of square waves (Figs. 3-7, Table 2). Each horizontal tread in a wave sequence represents a block length and a block mean. This mean differs significantly from the means of immediately preceding and succeeding blocks, - or, at one end or other of the sequence, differs either from the succeeding or the
STREAMFLOW: S T E P - F U N C T I O N A L ANALYSIS
383
4 I
St M A R Y ' S
R.
NIAGARA
0O9
St
LAWRENCE
15
~
__
In is'so
lgoo
-
F
~-n- -~,.r~R--- I]_- - rrt ~
l~so
Fig. 3: Histograms of peak flow, with square-wave sequences superimposed, for the St Mary's, Niagara, St Lawrence, and Columbia Rivers: trend lines are discussed in the text.
E r9
[1
L O I R E / MONTJ EAN
]l
20t
n r-I ,_nlln:_4Ll~L! 1o ~~1 LrUIdUll~ uI..,-,,~ ~yu ''~i
Fig. 4:
1T009
U,
H I~VH ~ 1
I
Histograms and square waves for the Loire at Blois and at Montjean.
19'50
384
DURY
1000 VANERN-GOTA E
50O 1850
1900
"'""" °°'" ...."
ooo'i •
F
1950
100C 80C
1850 Fig. 5:
i15~f
1950
1900 Histograms and square waves for the V'~inern-G6ta and the Vuoksi.
DANUBE
o~5_J ~l 1o_
n .,-.-,_ t~ °°Hn U UUL,-Hu U U u Lr
15
~ r,~.-.-, Yi ._,~lm, A-in rl N ILrlnn _,~,-, ~r~
LJ!_J LP.Juu u L-El Lid
LABEIELBE)
UL'-]'~rr'
~LH u L
~
{ ~8SO
El U u L " / h J ~ -
r~
o-"-Lllr n 19'00
-,'-,.1~50
Fig. 6: Histograms and square waves for the Danube and the Labe (Elbe).
preceding mean. Each vertical riser marks both the incidence and the magnitude of an abrupt change from one block, with its particular mean, to the next block with its different mean. The abrupt changes constitute step functions. Most of the square-wave sequences suggest little or nothing byway of secular trend. The high-flow and low-flow regimes alternate, with occasional interpolations of blocks with nearaverage flow, without any sign that either the high-flow or the low-flow regime is trending either upward or downward. Something could of course be made of the comparative distribution of blocks with high or low means, but the necessary allowance is already made by the graphs of cumulative deviations. In any event, a concentration of long blocks with high or low means in a particular part of a sequence does not necessarily imply a secular trend of increase or decrease: it could easily be a simple accident of distribution. Possible exceptions, to be referred to again, occur in the square-wave sequences for the Columbia, St Lawrence, and St Mary's Rivers (Fig. 3). Regression of mean values for the low-flow blocks on the St
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
385
NEVA
I
T
r
~
1860
,
1 g 00
O
i
,
195O
i
i
i°] o
t
0
0 ~
o° ® E
n -,11
~ I[IL. I1J ~ ] i Lr-,-,~u ~ --'~-u-"z:r7" L~
[
1
T
1
1810 5
NEMA
"
i
-
, -
-
1850
cont'd
"
Lr,.-rh
~
~ r
-
-
l
~
OO
II--
=
0
0 "L'~---t
1880
Fig. 7:
r
~
1900
r
-r
"r
r
"r
r
1950
Histograms and square waves for the Neva, Scvernaya (Northern) Dvina, and the Neman.
Lawrence, 1872-81 to 1962-64, on the locations of their midpoints on the timescale produces r = --0.869, r 2 = 0.755, and 0.05 > P. The regression equation indicates a reduction of the mean value of the low-flow blocks, during an interval of about 90 years, by some 20 per cent of its starting value. For part of this interval, the means of high-flow blocks sustained an almost parallel descent. Regression, as above, gives_r = --0.989 and_r2 = 0.978, although the inclusion of only three pairs of items in the calculation ensures that 0.10 > P > 0.05. The regression equation indicates a reduction through 50 years of 9 per cent of starting value. Although the square-wave pattern for the Niagara River at Queenston, a station upstream of Oldenburg on the St Lawrence, appears at first sight generally similar to the pattern drawn for the St Lawrence itself, no statistically definable trend, comparable to that identified for the St Lawrence, seems to be present. At the same time, the two records in combination could be taken as indicating a general decline in peak flow from 1860 to 1942, except for the sharp rise in 1929-1930. In direct contrast, the St Mary's River, another member of the Great Lakes system, combined an increase in variability of peak flow, during the approximate period 1880-1954, with an increase in the average magnitude of high flows. Selection of values for entry into regression is complicated by the occurrence of block means close to the series mean. If however such blocks are considered, along with high-flow blocks proper, then regression of block means on the serial locations of their midpoints produces r = + 0.919, r_2 = 0.845, and 0.05 > P, the regression equation indicating an increase through some 30 per cent of starting value in an interval of about 75 years. No accompanying trend can be identified for the low-flow regime. Equally for the record of the Columbia river, analysis fails to demonstrate a significant
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
387
falling trend for the low-flow blocks, even though in the second part of the sequence these seem to descend in parallel with the high-flow blocks. For these latter, regression as above produces r = --0.871,~ = 0.759, and 0.05 > P. The regression equation indicates a reduction through some 20 per cent of starting value during an interval of some 90 years, a result identical to that obtained for the reduction in the means of low-flow blocks, and closely similar to that for high-flow blocks on the St Lawrence. For the time being, no explanation of the resemblances will be attempted. Particularly perhaps with respect to the Columbia, one might ask if artificial works may not have seriously affected the regime of maximal flows. The major dams, however, the Grand Coulee and the Bonneville, date only from the 1930s, while the major irrigation project began abstracting water only in 1952. Irrespective of the location of works in relation to the gauging station, it seems reasonable therefore to suppose that the secular trend recognised for the high-flow regime on the Columbia is essentially natural. A parallel comment can be assumed to apply to the secular trends identified for the St Lawrence and St Mary's Rivers.
4. CUMULATIVE DEVIATIONS, SQUARE-WAVE SEQUENCES, AND RANDOM VARIATION
Graphs of cumulative deviations, similar to those obtained for streamfiow, can readily be produced by random operations. A particularly noteworthy experiment is that of JACCHIA (1975); he used a computer to generate a large random number, to select the final digit, to add this digit and subtract 4.5, and so on through 200,000 steps. The machine printed out a point for the cumulative deviation at every 200th step, providing a 1,000-point graph. Parts of this graph can be closely matched to deviation graphs for climatic and hydrologic series (cf. DURY, op.cit. 1980), the only difference being that the climatic and hydrologic graphs are drawn for stationary series, whereas JACCHIA's graph belongs to a non-stationary series. Now in JACCHIA's experiment, a constant probability of 10 per cent applied to the occurrence of all the digits 0 through 9 - although, as the result demonstrates, 200,000 passes produced a series mean sufficiently different from 4.5 to' ensure a large final cumulative deviation. The digits which may be read, or extracted by sampling, from the usual published table of random numbers are also regularly distributed, with an equal probability of occurrence attached to each. Series constructed from them can be shown to produce deviation graphs similar to those obtained for streams, and to break down into square-wave sequences resembling those drawn for the variation of peak streamflow. Stream discharges in general, and peak discharges in particular, are not however regularly distributed about their means, but clustered, as is illustrated by Fig. 2 herewith. I have accordingly undertaken a series of trials with normally distributed random digits, wherein for instance the probability of the occurrence of the digit 4 or 5 is 40 times as great as the occurrence of 0 or 9. The tables, which include 918 entries each, arranged in 27 columns and 34 rows, exist in two versions. Read down the columns, the first version provides six almost stationary series of 153 bits each, and three stationary series of 306 bits each. The second version is a randomized redistribution of the entries in the first version. Read serially either across the rows or down the columns, it provides a 918-bit stationary series, as the first version does when read serially across the rows. Either version can be used to generate non-stationary series of any desired
388
DURY
length. For purposes of comparison with the results of the analysis of streamflow, both tables have been read across the rows, the two 918-bit stationary series being analyzed for cumulative deviations, tested for step-functional breakdown, and resolved into sequences of square waves.
1000
"
25
o
+
E
0
•
random
o
Vuoksi
data o
ooO'°O
~. 15
o
•
20
o•
o
o o
o o
o
o
o
o
•
o
•
•
o o
•
•
o
o
• o
•
•
o °
•
o o
""'i i j-" o o.
1000
o o
o o
o
o
670 i
10
•
o
°
o
700 I
I
l
~
i
900
i
i
i
I
1950
Fig. 8: Cumulative residuals for the Vuoksi river, 1903-54, and similar residuals for a sequence of 52 normally distributed random digits. Regression of observed on generated values given r = + 0.792, r2 = 0.627, 0.01 > P.
/
400"
+ 0
400
800
1200
O0 I
i
5
random
10
digits
1 '5
2'0
25
Fig. 9: Plot of cumulative residuals for the Vuoksi river against randomly generated residuals: regression line shown.
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
389
8
654
1820 t
1850 i
i
1
i
188(3 i
i
i
25O
32
Fig. 10: Part of the square-wave sequence for the Vzinern-G6ta compared to part of a randomly generated sequence. A sample of the results is given in Fig. 8, where part of the deviation graph for the Vuoksi River is compared to part of the deviation graph drawn for a sequence of normally distributed random digits. There is an obvious family resemblance between the two plots. Regression of the observed on the generated values produces a highly significant correlation (fig. 9). Many other examples of the same kind could be offered. Similarly, the square-wave sequences into which peak streamflow has been resolved can be matched from the sequences obtained from normally distributed random digits. Fig. 10 compares part of the square-wave train for the V'~inern-G6ta with part of a randomly generated train. Once again, many additional examples could be offered.
5.
DISTRIBUTIONS OF BLOCK LENGTH
Comparisons of block length, which amount to comparisons of the frequency of stepfunctional change, are rendered somewhat uncertain by the presence in the streamflow series of uncompleted blocks. Such blocks, of 56, 47, 36, and 35 years in length, appear in the square-wave trains for the Danube, the Loire at Blois, and the Niagara. The longest completed blocks are recorded for the Neman (37 years) and the Loire at Montjean (36 years). If all blocks, completed or not, are taken into analysis, and a frequency distribution is constructed by 5-year intervals, then that distribution is closely matched by the distribution, arranged by five-bit intervals, obtained from the two 918-bit series of normally distributed random digits (Fig. 11). The correlation between the observed and the generated distribution is close, and statistically highly significant. The same Poisson distribution can be fitted to both arrays, as shown in fig. 10, where, solely for the sake of convenience, the frequencies are grouped by 10-year and 10-bit ranges. The closeness of fit between the observed and the Poisson distributions is perhaps especially satisfying; the fractional difference in closeness of fit between the observed and the generated distribution has no importance. Mean block length varies considerably from one stream to another. Difference-of means tests, between mean block length for streams and the mean of some 11.58 bits for the randomly generated data, indicate no significant difference in nine of the 13 cases. The mean
390
DURY 40 % 30 " ~ ' 1 - - --lr"~/~ r andom
20-
_~rivers
__
10-
-l_ =
:
block
©
P-- - - 1 - - - - ' I
;
~
leng ~"h ~
~
oJ
o~
r-- --i
~
d"
~
=
~"
m
7o
® =
,,o
Fig. 11: Frequency distributions of block length, for rivers and for randomly generated sequences. Totals are 149 blocks for rivers and 158 blocks for random data. Regression of observed on generated distributions gives r ~ +0.984, [2 = 0.968, 0.01 > P.
70 ~ ~rivers
6'O
50-
40Poisson
/
30-
20-
/
)--
10~
/ random
0
i
2--11 12--21 block l e n g t h
g
t-
-
-f-
22-- 31 32-- 41
-
42-51
-'
52-- 61 62--71
Fig. 12: Block frequencies for river data and for random data compared to a Poisson distribution. Regression on values in the Poisson distribution gives_r = + 0.999, r~ = 0.998 for the river data,_r + 0.999, [2 = 0.980 for the radom data, 0.01 > P in both cases.
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
391
block length of 32 years for the Danube, reckoned in any event from largely uncompleted blocks, is significantly greater than the mean for the random data, while the means for the St Mary's River (5.5 years), V'~inern-G6ta (8.53 years), and Vuoksi (7.12 years) are significantly less. What should be made of the differences must however be regarded as to some extent a matter of opinion. The mean block length on the Columbia River, 8.08 years, is less than hat on the V'~inern-G6ta, but a greater variance causes the t-test for difference of means between observed and randomly generated distributions to fail. Again, one sequence of 111 bits in the random data resolves into four blocks of average length 27.75 bits: compare the 129 years and four blocks on the Danube. Another sequence of 102 bits resolves into 16 blocks of average length about 5.02: compare the 110 years and 20 blocks on the St Mary's River. That is to say, even the sets of the longest and the shortest block means can be matched from the sequences obtained from the normally distributed random digits. Again, if the percentage frequency of block length is plotted against a logarithmic scale, the frequency distribution can be described by a normal curve (Fig. 13).
40%
observed 30-
20-
10/
i
log
0"54 0"74 block length
0"94
1 . 14
1-34
1,54
Fig. 13: Logarithmic frequency of mean block length, river data: the x scale is subdivided so as to include a mode at about 11.0, and to provide for seven columns. Corresponding arithmetic cutoffs are approximately 3.47, 5.50, 6.92, 13.80, 21.88, and 34.67._r = + 0.974,_r2 ~ 0.949, 0.01 > P.
The total sequence of random digits analyzed for step-functional change amounts to 1830 bits, the difference from 912 x 2 = 1836 being accounted for by short loose ends. As a simulation model of the variation of peak streamflow, the sequence corresponds to a run of observations nearly two millenia in length. As has just been pointed out, parts of the sequence match observed sequences where block means are unusually long or unusually short: only if the unusually long and unusually short observed means are truly representative of long-term behaviour should their differences from randomly generated effects be accepted. It remains true that some square-wave sequences for rivers are distinctly more, or less, variable than some others. Among the possible physical causes that could be investigated are size of catchment, regime of precipitation, and regime of runoff. Investigation of the two
392
DURY
types of regime is outside the scope of the present paper. There is no significant correlation, for the rivers considered, between size of catchment on the one hand, and mean block length or standard deviation of block length on the other. The data do not provide the means to scrutinize individual systems, except that the modest downstream increase in mean block length on the Loire is not statistically significant, and that no significant relationship between block length and drainage area appears on the Great Lakes - St Lawrence system. The tally of three stations, needless to say, is quite inadequate for the study of a whole large system; it can nevertheless be said that, whereas mean block length for the St Mary's River is significantly less than the mean for either the Niagara or the St Lawrence, mean length actually decreases (although not significantly so) between this last pair.
6.
BLOCK LENGTH AND DEVIATIONS FROM SERIES MEANS
One would expect on general grounds that the departures of block means from series means would increase with decreasing length of block. For a given variance, difference of means must increase as block length decreases, in order for the criterion of difference to be satisfied. Conversely, with variance again assumed to be constant, increasing block length, and especially the increase in length of two successive blocks, reduces the value of the statistic v,, ,~, and thus increases the value oft obtained in the difference-of-means test. Because block lengths themselves assume a Poisson distribution, one might look to a similar distribution to fit the distribution of the ratio between block length and deviation of block mean from series mean. In actuality, when departures of block means are expressed as percentages, averaged for each block length, they indicate a normal/log relationship between departure and block length (Fig. 14). While the points and the regression line for the randomly generated data indicate a somewhat greater departure than do those for rivers, especially in the lower part of the range of block length, the two data sets belong to a single order. We may conclude, at least provisionally, that the degree of percentage departure of block mean values from series mean values, as observed for rivers, can be simulated by random operations.
40
%
• ~
O
•
regression
line, random
data
(0)
,20
10
regression 0
line,
rivers
•
(0)
• •
[
,
2
block
i
i 5
i
i
)
,
i 10
iQ
20
•
o-Jo •
o )
n
~
•
-~o~
~
O~ I
o
~ i
50
length
Fig. 14: Mean percentage deviation from series mean, by block length, for river data and for randomly generated data.
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS
7. RANDOMLY STREAMFLOW
GENERATED
SEQUENCES
AND
SECULAR
393
TRENDS
IN
Even if we accept that the secular trends in peak flow identified for the St Lawrence, St Mary's, and Columbia river, and possibly also present in the record for the Niagara, are both real and natural, this does not necessarily mean that they are not random. Fig. 15 shows parts of the square-wave sequence obtained by analysis of normally distributed random digits. The upper part of the diagram closely simulates the falling trends on the St Lawrence and the Columbia, while the lower part closely simulates the rising trend observed for the high-flow regime on the St Mary's River. Degrees of change are comparable between the two sets of trend. The falling trend for the high blocks in the upper part of the diagram corresponds to a loss of 17% of starting value in the space of 86 bits, that for low blocks to a loss of 50% in 80 bits. The rising trend in the lower part of the diagram corresponds to a gain of 38% of starting value in the space of 55 bits. AZ 2 test, using these loss and gain rates, the gain rate for the St Mary's River, the loss rate for the high-flow regime on the St Lawrence, and the mean loss rate for the low-flow regimes on the St Lawrence and the Columbia, shows no significant difference, with P - 0.5. Like the square-wave sequences, the observed secular trends and trend rates can be simulated by random operation. 7
j
4510
Fig. 15: Seculartrends in square wavesdrawn for random digit sequences: compare the trends shown for certain rivers in Fig. 3. Regressiongives r = --0.978,0.05 > P for the top trend,_r ~ -0.941, 0.10 > P for the middle one, and r = + 0.933, 0.05 > P for the bottom one.
8.
HIGH-FLOW AND LOW-FLOW REGIME
In considerable part, the square-wave sequences drawn for peak streamflow, and the randomly generated sequences, alike consist of blocks with means distinctly above or below the series means. For some sequences it is possible to separate high, low, and intermediate regimes. Thorough analysis of the cases concerned would presumably demand a criterion of distinction between the intermediate regimes on the one hand, and the high and low regimes on the other. Such a criterion could obviously relate to the presence or absence ofa signifi-
394
DURY
1000¢n
high-flow
regime
E 50 0
o OOO;::oo ......
.
[
,,~O
O
~
I.
~
random
I
. . .
Iregime
digits
3" 2-
-o /
o 1-
1.01
1 1
1.58
2
recurrence
5
interval
10
20
50
Fig. 16: Magnitude-frequency graphs for high and low regimes, V'~inern-G6ta(146 items in all) and for a sequence of random digits (129 items in all).
cant difference between a block mean and a series mean. For immediate purposes it will suffice to consider the high-flow and low-flow regimes obviously identifiable in the record of the V'~inern-G6ta, 1807-1969, with the block 1951-64 alone excluded as averaging very close indeed to the series mean. This exclusion apart, there remain a total of 72 years included in low-flow blocks, and 74 years in high-flow blocks. Magnitude-frequency analysis for the high-flow and the low-flow regimes produces the results plotted in the upper part of Fig. 16. The interpretation of the uppermost end of the graph for the high-flow regime is a matter of some doubt, but between the recurrence intervals of 1.58 and 10 years the values for the high-flow regime run some 30 per cent greater than those for the low-flow regime. The lower part of the Figure shows outline magnitudefrequency graphs for the high and low regimes in a sequence of 129 normally distributed random digits, of which 69 occur in blocks of the high regime and 50 in blocks of the low. The points are plotted for the first appearance of a given digit in a reading up the scale of recurrence interval. Parallel results would be achieved by the plotting of last appearances, while a complete plot, needless to say, would rise in steps. The graphs obtained are similar to those drawn for the high-flow and low-flow regimes of the V'~inern-G6ta, the mean difference between high and low values, in the range of recurrence interval from 1.58 to 10, being only slightly less, in percentage terms, than that for the river. The outcome of this part of the analysis is, in view of the results of the analysis of departures in relation to block length, not at all surprising. It does however serve to show that the contrast between high-flow and low-flow regimes, as these are observed on rivers, can be replicated by random operation.
STREAMFLOW: STEP-FUNCTIONAL ANALYSIS 9.
395
CONCLUSIONS
Twenty years ago, CURRY (1962) advanced theoretical reasons for regarding climatic change as a random series. His comments, even though he was mainly considering major climatic variations, are certainly transferable to streamflow series. The present paper makes an empirical comparison between step-functioning streamflow series and the step-functional sequences obtained from normally distributed random digits. Peak streamflow on rivers with long records can, by reference to cumulative deviations from series means, and by the application of difference-of-means testing, be broken down into sequences of square waves. These sequences consist of blocks of years during which observed values averaged higher than, lower than, or occasionally close to a series mean. Successive blocks are separated by step-functional changes in regime. The real square-wave sequences, the real distribution of block length, the real distribution of the ratio between block length and percentage departure from series mean, the real secular trends, and the real distinction between high-flow and lowflow regimes, can all be simulated by random methods. The one uncertainty concerns the spacing of regime change on the Danube significantly greater than the average obtained from the random digits - and on the St Mary's, V'~inern-G6ta, and Vuoksi Rivers - significantly less. The observed spacing on these rivers can be matched from parts of the randomly generated sequences; but if the actual records are accurate representations of long-term behaviour, then the frequency of regime change on these four rivers should probably not be regarded as truly random through time. Contributions to distributional analysis include the identification of the frequency of block length as according with a Poisson distribution, and the provisional identification of the relationship of percentage departure from series mean to block length as forming a normal-log distribution.
10.
END-NOTE
The research project, the results of which are presented here, forms one of a series of analyses of the step-functional change of climatic and hydrologic variables. A parallel analysis of precipitation at Sydney, NSW, has already been cited (DURY, op.cit. 1980). Anlysis of the early precipitation records for Sydney and for Adelaide, S.A., is in hand. I have shown elsewhere that step-functional analysis can also be applied to tree-ring series and to the documentary records of harvest quality and of farming weather (DURY, 1981). The research was in part supported by a grant from the Graduate School of The University of Wisconsin-Madison (Project No. 181441). Professor Barbara Bell (Harvard) originally drew my attention to the usefulness of cumulative deviations, with which her own published work in part deals. Professors Waltraud Brinkmann and Grace Wahba (Madison) discussed, respectively, the use of transforms and the desirability of using normally, as opposed to regularly, distributed random numbers.
BIBLIOGRAPHY
CURRY, L. (1962): Climaticchange as a random series. Annals, AssociationofAmerican Geographers, 52, 21-31. DURY, G.H. (1980): Step-functionalchange in precipitation at Sydney. Australian Geographical Studies, 18, 62-78.
396
DURY
DURY, G.H. (1981): Climate and settlement in late-medieval central England. In: SMITH & PARRY (eds.): Consequences of Climatic Change, 40-53, University of Nottingham, Department of Geography. JACCHIA, L.C. (1975): Some thoughts about randomness. Sky and Telescope, 50, 371-375.
Anschrift des Autors: George H. Dury 46 Woodland Close, Risby, Bury St Edmunds IP 28 6QN, England, United Kingdom