An MRC is commonly shown as a plot of daily streamflow (logarithmic scale) as a function of time as in Fig. 1. From A to B, the MRC is upwardly concave until the flow reaches the maximum recession constant (B) (location of minimum recession slope) after which the logarithm of flow decreases linearly (B to C). If there is transmission loss (stream evaporation or evapotranspiration from the adjacent riparian zone or leakage to the stream bed) the MRC follows B to D. The difference in the curves (B–D) and (B–C) is transmission loss, as defined by Boughton (2015).

An MRC represents the average rate of decline in streamflow following rainfall after surface runoff contributions have ceased. The average rate of decline is determined from the analysis of many individual recessions. The recession shape at any point in time depends on the initial water content of the unconfined aquifer and other sub-surface processes that together feed the stream. Also, as noted above, transmission losses affect recession shape. The initial water content of the aquifer/s and sub-storage systems is a result of the preceding climate and weather conditions and aquifer antecedent storage, as observed by Bart and Hope (2014). In our analysis we recognise that while the MRC is normally considered to represent the average rate of decline, it is sensible to characterise the natural variability (or aleatory uncertainty) in the MRC that arises from variable antecedent hydroclimatic conditions and heterogeneous storage conditions in the aquifer. Hence, in contrast to others (Tallaksen, 1995; Duncan, 2019), we hypothesise that there is a continuum of recession curves that can be represented either by a single MRC representative of average conditions, or else by a continuum of curves that result from variability in the initial water content of the aquifers feeding the stream and from the spatial heterogeneity in their physical configuration. These storages range from being full following a very wet period to a near empty condition following a long dry period. In our analysis, these two extreme conditions are notionally represented by MRCs for 90th percentile and 10th percentile exceedance frequency curves respectively. Three intermediate conditions are also included: 75th percentile represents recessions following a moderate wet period, whereas 25th percentile is for conditions following a moderate dry period. The “typical” initial condition is represented by a 50th percentile MRC. Here, we express variability in terms of an exceedance probability rather than in terms of a physical variable, for example, antecedent rainfall. That is, in a given catchment in a moderately dry condition, it might be reasonable to expect that an individual recession might follow an MRC that is exceeded 25 % of the time, and another individual recession following wetter conditions might follow an MRC that is exceeded 75 % of the time. We adopted this approach in McMahon and Nathan (2025). It is not inconsistent with Fiorotto and Caroni (2013), who analysed the structure of the recession curve within a stochastic rather than in a deterministic framework.

An important parameter associated with an MRC is the maximum recession constant (Kmax). As observed in Fig. 1, it is the point where the recession curve reaches minimum slope and without transmission losses would continue recessing at that slope. It has been used by many researchers under different guises. For example, it is related to the characteristic time (Brutsaert and Lopez, 1998), and used to estimate residence time (Chapman, 2003). Boughton (2015) utilised the maximum recession constant in estimating transmission losses in streams and McMahon and Nathan (2025) adopted Kmax as a key to their procedure to estimate hydraulic properties of unconfined aquifers from daily streamflow data. In Sect. 4, we adopt Kmax to represent the slopes of the continuum of MRCs as they move from wet to dry antecedent conditions.

It is rare for observed daily streamflows to be available for long periods without being impacted by rainfall or observational tolerance issues, or by measurement malfunction, and so plots of actual recessions (streamflow volume Q vs. time t) rarely show a complete shape of the recession. Virtually all investigators are confronted with these problems.

The exceedance percentile values for the MRC are estimated from binned values of daily Kj (recession constant) and Qj pairs following the correlation method (Hall, 1968; Beran and Gustard, 1977; Boughton, 2015) which is adopted herein to compute an MRC. The key steps in the correlation approach are:

Daily recession constants (Kj) (where Kj=Qj/Qj-1, for 0<Kj<1), and Qj (daily discharge at time j) are ranked as pairs (by Qj) after values potentially affected by rainfall are discarded. The number of rain-impacted days to be discarded is a function of catchment area guided by Linsley et al. (1982, Eq. 7-4). We have not included adjustment for data imprecision (Rupp et al., 2009) because we compare the computed MRC with observed recession data.

In accepting the daily observed recessions for plotting (as in Fig. 3), we adopted the following guidelines:

Choose at least ten consecutive decreasing daily discharges that are for days unimpacted by rainfall. The first x d are not used in plotting the observed recession to reduce the potential for surface runoff to affect the recession. Here, “x” d is defined by Linsley et al. (1982, Eq. 7-4).

The comparison between the computed MRCs and observed recessions was made for four streams as set out in Table 1. They span a wide range of hydrology across Australia as defined by mean annual catchment runoff from about 3 to 2000 mm yr⁻¹.

As noted above, the number of raindays to be eliminated prior to plotting the recessions is a function of catchment area; for Gibbo, Northern Arthur and South Johnstone Rivers 3 d were eliminated, whereas for Myall River the initial 4 d of recessions were eliminated.

The observed recessions for all four streams (shown in Figs. 3 and 4, and in the Supplement, a total of 169 across the four streams) plot neatly on the computed MRCs. This is expected, of course, as both sets of results are drawn from the same data. Furthermore, the range of behaviour between the 10th percentile to 90th percentile MRCs provides much scope to ensure the observed recessions fit reasonably well to one of the five curves. The plots for Myall River (Fig. S2) are an exception where eight recessions were not allocated to one of the five MRCs. Most would have plotted on intermediate MRCs.

We have not found plots similar to Fig. 3 in the literature comparing observed recessions with probabilistic curves, although Kienzle (2006) identified two very different MRCs for a 7675 km² watershed in Alberta, Canada. Fiorotto and Caroni (2013, Fig. 6) incorporated a range of recessions as a function of probability by interpreting the master recession in terms of a stochastic process. On the other hand, Yang et al. (2019) developed separate MRCs for four flow regime conditions, high, moist, low, and dry and, separately, for the four climate seasons. The approach by Gao et al. (2023), inter alia, is similar to our method but they do not map the observed data to the developed recessions.

Our plots confirm that the adopted correlation method to estimating MRCs is a valid alternative to other approaches and is easily adapted to characterising natural variability and uncertainty. An important feature of Fig. 3 is that the observed recessions extended beyond the day of their respective maximum recession constants, even for the extremely wet South Johnstone catchment with a mean annual runoff of more than 2000 mm yr⁻¹ (Fig. S1) where there is evidence of transmission loss. Some studies (for example, Millares et al., 2009) involving recession analysis (Q∼t) that produce Q∼t diagrams do not extend the time scale much beyond the maximum recession constant, presumably in the belief that the curve continues in a concave manner. We highlight this point as few journal papers dealing with Q∼t recession analysis address this important feature of an MRC.

Another point to note relates to the confidence with which we can estimate the different percentile MRC curves. As pointed out earlier, we are suggesting the different percentile MRC curves represent the natural variability in the initial level of stored water in the different aquifer units feeding the stream; in other words, the source of variability is the aleatory uncertainty due to antecedent conditions and its influence on a catchment with spatially heterogeneous physical properties.

However, our ability to identify the different percentile MRCs is limited by the period of available observations, where the uncertainty around the derived MRCs increases as the sample size decreases. This source of uncertainty represents the epistemic uncertainty due to sampling variability and can be characterised by non-parametric bootstrapping. To this end, the uncertainty around a given percentile MRC is characterised by resampling (with replacement) estimates of the recession constant within each bin. An example of the epistemic uncertainty around the median MRC for Gibbo River is shown by the (dashed) confidence lines in Fig. 3. For the available data, it is seen that the epistemic uncertainty is much smaller than the aleatory uncertainty.

The characterisation of aleatory uncertainty as discussed herein can be applied to any application of an MRC analysis. One example of this is the estimation of the frequency of occurrence of transmission losses in streams, where the losses are based on MRCs (Boughton, 2015). Another example is the estimation of storage residence times in unconfined aquifers derived using the maximum recession constant, which is a key component of an MRC (McMahon and Nathan, 2025).

Analysis of recessions separately for the wet and dry 6-month periods for Gibbo River (Fig. 4) shows that the observed dry period recessions extend across the five computed MRCs whereas the observable recessions for the wet period cover mainly the 25th, 50th, 75th and 90th percentile MRCs. This distribution across the two periods is observed even though the dry 6-month period yields about one quarter the runoff generated during the wet period. The figure is important in that it demonstrates the difficulty in classifying catchment conditions in terms of general wet and dry periods, probably because we are examining the behaviour of individual events.

Two plots are provided for Northern Arthur. Figure 5a shows the recession data for the winter to mid-spring period whereas Fig. 5b is for the summer to early autumn period. The recession variability between summer and winter, which is characteristic of catchments in south-west Western Australia with perched water tables, is discussed in detail by George and Conacher (1993). Noting that at the Lake Toolibin stream gauging station for Northern Arthur River, there are, on average, only 43 d of flow per year and so the number of plausible recessions is limited. It is also noted that the minimum flow plotted in these figures is about five times larger than the minimum that can be measured under field conditions (McMahon and Peel, 2019). It is comforting to note that for this stream the observed recessions in Fig. 5 are, overall, consistent with the computed MRCs. The episodic nature of the Northern Arthur River hydrology is evident in Fig. 5 where the two of the three unallocated recessions are for climate conditions more extreme than the 10 % to 90th percentile plotted range.

Another way to explore seasonality is to relate the continuum of MRCs to a measure of antecedent climate; a simple measure is antecedent rainfall. The results of such an analysis are shown in Figs. 6 and S4a, b, c and d where the maximum recession constant (Kmax) is plotted against antecedent (monthly) rainfalls for specific exceedance percentiles. Here antecedent rainfalls for stations within or near the catchment are defined as the 10 % percentile exceedance of one month, 3-month sums, 5-month sums, 7-month sums and 11-month sums for very dry antecedent conditions, and the same range of antecedent rainfalls for 25 % (dry), 50 % (“average”), 75 % (wet) and 90 % (very wet). The rainfalls are for independent sums rather than overlapping sums. It is observed in Fig. 3 that Kmax (the minimum slope of a given recession) is an appropriate metric to represent the continuum of MRCs because as Kmax increases in slope the MRCs move from wet to dry antecedent conditions as illustrated in Figs. 3 and S1, S2 and S3. In Fig. 6, Kmax is strongly related to 3-month sums of rainfall. It is equally correlated with 1 month, 5 months, 7 months and 11 months sums (see Fig. S4a, b, c and d). Because of the high correlations among the 1 month, 3 months, 5 months, 7 months and 11 months sums (Table S1 in the Supplement), we cannot say without further analysis which duration is the most appropriate for a specific catchment. Such an analysis is beyond the scope of this study. What is evident, however, is that the continuum of MRCs is strongly correlated to antecedent rainfalls thus confirming our hypothesis that MRCs are the result of varying antecedent climate and weather conditions and are represented by a continuum of MRCs rather than a single average MRC.

But what is the explanation for the variability exhibited in Figs. 3 to 5? Our proposed hypothesis is that, in the catchments we are considering, there are many sub-catchments (they have Strahler stream orders greater than three; Strahler, 1957) with unconfined aquifers and delayed sub-surface flows. These sub-surface storages vary greatly in terms of dimensions, hydraulic properties, location and elevation relative to the nearest stream channel. This heterogeneity within a catchment combined with a varying weather and climate is hypothesized to produce a continuum of MRCs as illustrated in the Figs. 3 to 5.

Because Northern Arthur is, hydrologically speaking, a well-instrumented and researched catchment (Callow et al., 2007, 2020), it is used to assess a qualitative model explaining the observed variability (aleatory uncertainty) in the MRCs (Fig. 5; see also Plates S1 and S2 in the Supplement which show, respectively, the mid-western catchment and the stream gauging station location). Consider Fig. 5a and b which show the recessions for Northern Arthur River. Two periods of activity are identified; the number of recessions over the winter to mid-spring period are twice that over summer to early autumn (Fig. S5). We deduce that the very steep 10th percentile MRC that occurs over about 15 d following rain on a very dry catchment occurs from an unconfined aquifer (denoted as “A”) with very high hydraulic conductivity. On the other hand, the considerably flatter 90th percentile MRC occurs after a very wet period and is likely fed from an aquifer (denoted as “B”) that exhibits a much more attenuated response to rainfall due to having either larger storage or lower hydraulic conductivity. For simplicity, we assume that these two aquifers, A and B, are in parallel and are the only groundwater sources discharging to the stream. After a very wet period that fills both aquifers, aquifer A will empty very quickly while aquifer B will dominate the recession. However, if aquifer A, the less attenuated aquifer, is located in the catchment such that after a long dry period there is sufficient rain to replenish it yet insufficient for aquifer B to discharge, a steep recession will result. This explanation must be tempered by noting that stream evaporation transmission losses probably will be high during the summer to early autumn periods (average area potential evapotranspiration (AAPE) ∼5 mm d⁻¹) but much less during the winter to mid-spring periods (AAPE ∼2.5 mm d⁻¹) (Wang et al., 2001). Adopting two parallel storages for aquifer modelling is not unusual (see for example, Moore, 1997, and Gao et al., 2017, although they assumed linear systems).

The above simple qualitative two-stage model of the surface water-groundwater connectivity proposed for the Northern Arthur catchment is not inconsistent with the three-stage surface water-groundwater connectivity and vertical recharge processes described by Callow et al. (2020). Their description is based on intensive instrumentation of the catchment and detailed analyses. As described by Callow et al. (2020), during the winter period soils saturate and macropores close, matrix flow dominates, and surficial aquifers become connected; although a vertical hydraulic gradient continues, the system acts as a semi-confined aquifer, and as the aquifers connect, a transition to the bottom-up groundwater discharge occurs. This description applies to the winter to mid-spring period resulting in the observed recessions depicted in Fig. 5a. From late spring, groundwater levels fall and, as the system dries, de-coupling between the surface and groundwater occurs, and macropores re-develop. Valley floor areas dry with large surface cracks. Top-down recharge occurs as a result of high infiltration facilitated by the macropores and surface flows. Observed recessions in Fig. 5b are the result of these processes. In their discussion of the surface-subsurface processes involved, Callow et al. (2020) describe the transitions between the two main stages of low flow generation in Northern Arthur catchment. Referring to Fig. 5, we surmise that some of the intermediate recessions (75 %, 50 % and 25th percentiles) represent these transitions as the system passes from a wet period to a dry period and back again to the next wet phase.