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

 

ABSTRACT  

 

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.

 

 

Conclusions

 

 

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.

 

References

 

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