Hydrological Uncertainty - Floods of Lake Eyre
Vincent Kotwicki and Zbigniew W. Kundzewicz
Kotwicki, V. & Kundzewicz, Z. W. (1995) Hydrological uncertainty - floods of Lake Eyre. In Kundzewicz, Z. W. (Ed.). New Uncertainty Concepts in Hydrology and Water Resources. Cambridge University Press, Cambridge, U.K., 32-38.
Kotwicki (1) and Zbigniew W. Kundzewicz (2)
(1) Water Resources Branch, Engineering and Water Supply Department, Adelaide, Australia
(2) Research Centre of Agricultural and Forest Environment Studies, Pol.Acad.Sci., Poznan and Institute of Geophysics, Pol.Acad.Sci., Warsaw, Poland
The uncertainty aspects of the process of floods of Lake Eyre are examined. The available records of floods cover the time span of fourty years only. As longer time series of precipitation records are available, one has extended the observed series of inflows to the Lake Eyre with the help of a rainfall-runoff model. Further reconstruction of the inflow series has been achieved with the help of proxy data of coral fluorescence intensity. However the limitations of these extensions and reconstructions of inflows are severe. The process of inflows to the Lake Eyre could be considered one of the most convincing manifestations of hydrological uncertainty.
Lake Eyre and its basin
The Lake Eyre, large depression in arid Australia, rarely filled with water, attracts the interest of limnologists, hydrologists, geomorphologists and ecologists all over the world. The process of inflows to the Lake Eyre has been studied recently by Kotwicki (1986). The following general information draws from the data assembled there.
The Lake Eyre basin spreads over 1.14 million km² of arid central Australia. Almost half of the basin area receives as little rainfall as 150 mm per year or less. The higher rainfalls of the order of 400 mm per year occur in the northern and eastern margins of the basin, influenced by the southern edges of the summer monsoon.
The annual potential evaporation as measured by Class A evaporometer ranges from 2400 to 3600 mm, with the value of pan coefficient for the Lake Eyre basin of the order of 0.6. The annual evaporation rate for the filled Lake Eyre ranges from 1800 to 2000 mm.
Since its discovery in 1840 until its first recorded filling in 1949 the lake was considered permanently dry and eventual reports on the existence of water in the lake were dismissed as observation errors. After 1949 a sequence of wet and dry spells have been observed. Amidst minor isolated floodings a major flood event was recorded, which began in 1973, reached its peak in 1974 and persisted until 1977. It is estimated that the peak water storage in the lake during this event read 32.5 km³. Until recently one used to consider the fillings of the Lake Eyre as rare and independent events. Now they are increasingly being looked at as a predictable manifestation of the global circulation patterns (e.g. El Nino - Southern Oscillations phenomena).
The Lake Eyre drainage basin is quite well developed and of persistent nature, due to favourable structural conditions. Much of this drainage pattern are disconnected relics from linked river systems which developed under wetter past climate and which became disorganized under the arid conditions.
The lake is mainly fed by its eastern tributaries, the Cooper Creek and the rivers - Diamantina and Georgina, featuring extreme variability in discharge und flow duration. Mean annual runoff of the Lake Eyre basin, of the order of 3.5 mm depth (i.e. 4 km³ volume) is the lowest of any major drainage basins in the world. This is some six per cent only of the value for the whole waterless Australian continent. Good demonstration of the aridity of the basin is its specific yield of 10 km³km²day-1in comparison to the value of 115 m³km²day-1 for the Nile. In the conditions of arid central Australia the rainfall of the volume of 50 mm is required to sustain a full channel flow and the frequency of such an event is less than once a year. Major events of filling the Lake Eyre are associated with rare cases of annual rainfall in exceedence of 500 mm, or, as happened in 1984 and 1989, by heavy localised storms with precipitation of some 200-300 mm in the vicinity of Lake Eyre.
Floods of Lake Eyre - records and reconstructions
Time series of recorded fillings of the Lake Eyre, shown in Fig 1 embrace a short span of four decades. This short record is also subject to significant uncertainties. The existing instrumentation and observation network is not adequate. Even now the inflows to the Lake Eyre are not measured directly in the lower course of either of its tributaries. The existing gauges measure runoff from as little as 40 per cent of the catchment area. The information on fillings before 1949 is practically non-existent. Therefore some means of extension of the observational records have been urgently required.
Kotwicki (1986) took a recourse to the use of a rainfall-runoff model for determination of past inflows to the lake. Using fourty years of available data for identification and validation of the model he managed to reconstruct a time series of inflows to the Lake Eyre for a century (1885-1984). However, the possibility of such a reconstruction is limited by the availability of rainfall data. The reliability of the results depends largely on the adequacy of the mathematical model available for transformation of rainfall into runoff. The idea of rainfall-runoff models has been developed for areas of humid or moderate climates. Therefore most of these models function satisfactorily under such climatic conditions and may not account the processes of water losses, essential in the Lake Eyre basin, with sufficiently good accuracy. Although it is believed that the particular rainfall-runoff model used (RORB3, cf. Laurenson and Mein, 1983), that has been developed and tested for the arid Australian conditions may be the best available method, it is still a source of some uncertainty. The results of Kotwicki (1986) are shown in Fig. 2. Fig. 3 shows the same data, that look rather erratic in the raw plot, in the moving average (11 terms) framework.
One of the possibilities of further extension of the available time series of inflows is to use the El Nino - Southern Oscillation link that manifests itself via some proxy data, thus allowing reconstruction of longer series of records.
Isdale and Kotwicki (1987) and Kotwicki and Isdale (1990) used coral proxy data to further reconstruct the inflows to the Lake Eyre. This is possible as the coral data reflect in some way the flows of the Burdekin River, draining a catchment of around 130 thousand km , directly adjoining the much larger Lake Eyre system. The process of the Burdekin River flow is strongly non-stationary, both in the yearly and over-yearly scale. The annual flows range from 3 to 300 per cent of the long term mean. During high flow periods the discharge of the Burdekin River moves northwards from the river outlet due to a longshore drift, and eventually reaches the shelf-edge reefs 250 km north of the mouth. The land-derived organic compounds, like humic and fulvic acids, are transported by the river and introduced to the marine system. These compounds taken up by corals and accomodated in their growing skeleton structures can be detected as they fluoresce under ultraviolet light. This fluorescence intensity provides a proxy measure of adjacent river discharge in the region of high flows. Dendrometric measurements along the depth of the core allowed dating of skeletal carbonate growth bands since 1724. The technique of dating the core resembles the method of dating yearly tree rings. Although the proxy data of the River Burdekin are shown to be statistically linked to inflows to the Lake Eyre, there are significant uncertainty elements involved. The basin of the River Burdekin, though adjacent to the Lake Eyre basin, may have been behaving quite differently for particular events, as pointed out by Isdale and Kotwicki (1987). The anomalies have been caused by the non-uniform storm coverage (heavy local rains). Moreover, the place where the coral reefs are analyzed is quite distant from the river outlet, what brings additional contribution to uncertainty. Kotwicki and Isdale (1990) observed a significant correlation between the time series of inflows to the Lake, the coral fluorescence intensities and the El Nino - Southern Oscillations (ENSO) index. The time series of coral fluorescence intensity for the period 1885-1980 are shown in the Figs. 4a and 4b in the raw form, and in the form of moving average (11 terms), respectively. It is clearly visible from Figs. 2 and 4a, and 3 and 4b, that there is some similarity of the two processes. The link between the two time series can be used to establish a relationship between the coral-proxy data and the inflows to Lake Eyre. This could help drawing from the entire coral proxy record available (Fig. 5) since 1735. There is, however, also a significant difference in behaviour of the 100-years series of inflows to the Lake Eyre and of coral fluorescence intensities. Coral intensity may be characterized by a continuous distribution, whereas the process of inflows is described by a mixed (discrete-continuous) distribution with a large part of the population (including the lower quartile) attaining zero value. This difference in behaviour of both series is shown in the form of box-plots (Figs. 6a and 6b) and in the form of frequency histograms (Figs. 7a and 7b). Both series contain a few (three or four) points laying significantly outside the upper hinge of the box-plots.
It seems that the standard two-parametric linear regression (Fig. 8a) is, in general, not the proper link between the coral-proxy and the flood data. This is so, because one is likely to obtain negative values of inflows to the lake for low values of intensity of coral fluorescence. This deficiency is eliminated via an one-parametric linear regression whose intercept has been set to zero (Fig. 8b). The fit can be improved if the linear regression excludes the zero yearly inflows to Lake Eyre. That is, one looks for the linear relation (Fig. 9):
inflow = function (coral intensity * inflow > zero).
However, it does not seem possible to identify if the condition of non-zero inflow is fulfilled, drawing from an exterior information. Fig. 10 shows the multiple box-plot of inflows to the lake Eyre for different classes of coral fluorescence. It can be seen that zero inflows occur for all magnitude classes of coral fluorescence intensity, whereas for the lowest class (coral fluorescence intensity under 100 units?) all three lower quantiles of inflows are zero. The probabilities of zero inflow in particular coral fluorescence classes read: 0.79 for the class from 0 to 100, 0.58 for the class from 100 to 200, 0.41 for the class from 200 to 300, and 0.08 for the class over 300.
The relation between the coral proxy data and the inflows to the Lake Eyre is the result of some dynamical process. Causal relationships call for an input-output dynamical model linking the flows of the River Burdekin (model input) and the coral fluorescence intensity (model output). If linear formulation is used, one gets the following convolution integral valid for an initially-relaxed case:
y(t) = i h(t-J) x(J) dJ = h(t) * x(t)
where x(t), h(t), y(t) denote - the input function (flows of the River Burdekin that are believed to be closely linked to the process of inflows to the Lake Eyre), impulse response (kernel function of the linear integral operator) and the output function (coral fluorescence intensity). The symbol * denotes the operation of convolution. However, what one needs is the inverse model producing the flows of the River Burdekin (and, further on - inflows to the Lake Eyre) from the coral data. Input reconstruction is a difficult, and - mathematically ill-posed problem. The theory warns that even small inadequacies in the data available for an inverse problem may render the result of identificatio unstable. Moreover, the available yearly data are not sufficient for identification of such dynamics. This results from the analysis of cross-correlation between the two series (Fig. 11), where a significant value is attained only for the lag zero.
The process of inflows to the Lake Eyre involves a very high degree of hydrological uncertainties. The most essential uncertainty aspects in the process of floods of the Lake Eyre are as follows:
i) The observations gathered until present (gauge records) pertain to runoff from a portion (some fourty per cent only) of the area of the Lake Eyre basin.
ii) The available records (biased as noted under (i)) cover the time span of fourty years only, that is the period of data for identification and validation of the rainfall-runoff model is very short. Therefore the recommended split-sample technique cannot be used.
iii) The accuracy of rainfall-runoff model (used for augmenting the available time series) for arid conditions may be lower than for humid or moderate conditions, where the idea of the unit hydrograph and alike concepts has been developed and widely applied.
iv) In order to validate the coral fluorescence-inflow relationship one disposes with fourty years of observations (cf. (i)) that can be augmented by results of rainfall-runoff modelling (see remarks (ii)-(iii)) to one hundred years. This may be still too little for the rigorous split-sample approach.
v) The coral proxy analysis is based on the assumption of the same climatological
forcing of both the River Burdekin and the Lake Eyre basins. However, the River Burdekin basin, although adjacent to the Lake Eyre basin, has not always been subject to a similar precipitation regime (anomalies identified by Isdale and Kotwicki, 1987). There were numerous periods of different behaviour of the process of flows of the river Burdekin and the inflows to the Lake Eyre - it was not uncommon that the spatial coverage of rainfalls did not embrace both basins. The Lake Eyre basin itself is huge, that is why different climatic conditions can occur simultaneously in various parts of the basin.
vi) It is only in cases of high flow that the land-derived organic compounds are transported to the Pandora Reef.
vii) There is a large distance between the gauge at the River Burdekin and the site of coral colonies studies. There may be additional uncertainty factors influencing the long range transport process.
viii) The process of inflows to Lake Eyre is extremely complex. It is driven by several mechanisms and therefore the sample is heterogenous. This is clearly seen in the examples of 1984 and 1989, when contrary to most of historical records, the bulk of inflows was provided by the ephemeral river west of Lake Eyre. These rivers were typically dry during other events. It is believed that after removing the heterogenity of available records, a significally better correlation with coral proxy data would be achieved.
Fig. 12 shows the cause-effect structures used for extension of available records. There are uncertainties contained in the rainfall-runoff analysis shown as the line 1 (e.g. difficulties in obtaining the average rainfall, questionable applicability of a lumped model with average precipitation as the lumped input). The cause-effect structure of the relationships between the coral proxy data and the inflows to the Lake Eyre is far more complicated (line 2 in Fig. 12). It is an inverse problem combined with inferring on adjacent system. That is coral proxy reconstruction of past inflows to the Lake Eyre is almost a hopeless task. However, despite all the uncertainties in the cause-effect links, Isdale and Kotwicki (1987) had the right to draw the following corollaries from their studies:
- The vast area of the Australian landmass comprising both the Lake Eyre and the Burdekin River basins endure a common broad climatic forcing, which is the Southern Oscillations or some function of it and.
- The coral proxy record may be used to hindcast annual scale paleohydrological sequences (and ENSO periodicities) for several centuries before the modern instrument period.
One could say that in consideration of inflows to Lake Eyre there is possibly some analogy to the Galilean statement on infinitely complex movement of a single droplet of water. This complexity does not hamper the specialists to forecast routinely the movement of water masses (flood waves) in open channels. The complexity at the microscale turns into simplicity at the macroscale. It is not to say that on this scale the problem suddenly turns simple: however, it is possibly closer to our scale of perception and its solution more adequate to the roughness of data which can be collected.
Isdale, P. and Kotwicki, V., 1987, Lake Eyre and the Great Barrier Reef: A paleohydrological ENSO connection, South Australia Geographical Journal, 87: 44-55.
Kotwicki, V., 1986, Floods of Lake Eyre, Engineering and Water Supply Department, Adelaide, 99 pp.
Kotwicki, V. and Isdale, P., 1990, Hydrology of Lake Eyre: El Nino link, accepted for publication in Palaeogeogr., Palaeoclimatol., Palaeoecol.
Laurenson, E.M. and Mein, R.G., 1983, RORB VERSION 3 - User Manual, Monash University, Department of Civil Engineering.
List of Figures (not shown on this website):
Fig. 1 - Observed inflows to the Lake Eyre (records since 1949)
Fig. 2 - Observed inflows to the Lake Eyre (since 1949) augmented with inflows reconstructed with the help of the rainfall-runoff model (1885-1980)
Fig. 3 - Moving average (11 terms) of inflows to the Lake Eyre
Fig. 4 - Observed coral
fluorescence intensities (1885-1980) in the form of
a - time series
b - moving average (11 terms)
Fig. 5 - Available time series of coral fluorescence data (since 1735)
Fig. 6 - Box-plots of
a - inflows to the Lake Eyre
b - coral fluorescence data
Fig. 7 - Frequency histograms of
a - inflows to the Lake Eyre
b - coral fluorescence data
Fig. 8 - Linear regression of inflows to the Lake Eyre vs coral fluorescence intensity
a - two-parametric linear regression
b - one-parametric linear regression with supressed intercept
Fig. 9 - Linear regression of inflows to the Lake Eyre vs coral fluorescence data for non-zero inflows
Fig. 10 - Multiple box-plot for different zones of coral fluorescence intensities
Fig. 11 - Estimated cross-correlations between coral fluorescence
and inflows to the Lake Eyre
Fig. 12 - Cause-effect structures for extension of available records