Estimating flow duration curves in perennial and ephemeral catchments using a disaggregated approach

The authors proposed a methodology for estimating flow duration curves (FDC) for perennial and ephemeral catch‐ ments in islands using a disaggregated approach. The pro‐ posed method is approached statistically and uses no complex parameters in order to reduce uncertainty and retain simplicity. Firstly, the FDC was disaggregated into three parts (high, middle and low) and, for the purpose of this study, it focusses on the low flow section. Initially, the mean monthly flow was used for estimating runoff in both types of catchments. The results show the mean monthly flow provided proper estimates in the perennial catchments, but for the ephemeral catchments the estimates were sub‐ standard. Therefore, a different approach using climate indices such as aridity and a precipitation index was used in a generalized regression equation. The results show the majority of the ephemeral catchments responded properly to the climate indices indicating climate as a major control‐ ling factor at the lower end of the FDC.


INTRODUCTION
Water is one of the most critical resources in island catchments, hence efficient water use and water resource planning are perhaps even more indispensable than in continental catchments.Yet such use and planning is rather difficult in island catchments because of their smaller sizes and more limited hydrological data (Falkland, 1991(Falkland, , 2002)).Toward resolving these problems, one possible way forward is the development of a methodology to estimate flow duration curves (FDCs) in ungauged catchments.
Methods for estimating FDCs in gauged catchments have been suggested in the literature, including exploring the statistical relationships between the shapes of FDCs with geologic and climatic effects (Musiake et al., 1975), dominant soil types (Ward and Robinson, 1990), vegetation type (Burt and Swank, 1992), and morphometric, soil, land use, climate types (Sefton and Howarth, 1998), and morphology and climate characteristics (Castellarin et al., 2004a).
Following the stochastic representation of FDCs by Castellarin et al. (2004b), Botter et al. (2007) derived FDCs with a stochastic-dynamic model and explained how hydrological process can contribute to the shape of FDCs.This approach was extended by Muneepeerakul et al. (2010) to estimate fast and slow flow components of FDCs.Using a process-based lumped model, Yokoo and Sivapalan (2011) further extended the FDC separation approach and showed slow flow component of FDCs can be estimated from the mean monthly flow in humid catchments.Leong and Yokoo (2017) demonstrated fast flow components of FDCs can be estimated from precipitation data and curve number models in humid catchments.Residual problems include (i) how to estimate the low flow section of FDCs and (ii) how to estimate FDCs in arid catchments.How can we estimate the low flow parts of FDCs?Low flows are the result of complex interactions between climate and catchment processes, including topography, vegetation variety, evapotranspiration rates and soil characteristics (Smakhtin, 2001;Laaha et al., 2013).Hence low flow should be estimated from the physiographic characteristics of a catchment.The statistical literature to estimate low flows by Laaha et al. (2013) include regression models (e.g.Vogel and Kroll, 1992;Draper and Smith, 1998;Laaha and Bloschl, 2006), index low flow models (e.g.Clausen and Pearson, 1995;Madsen and Rosbjerg, 1998), and geostatistical models.In addition, recent literature suggests that the base flow index is a dominant indicator (Smakhtin, 2001;Mohamoud, 2008).The aforementioned studies indicate that the statistical investigation between low flows and physiographic characteristics of catchments would help to estimate not only low flows but also identify dominant rainfall-runoff processes.
Hence this study aims to develop a methodology for estimating the whole shape of a disaggregated FDC with emphasis on low flows.The study focusses on using simple and less parameterized models in perennial and ephemeral island catchments.In addition, by learning from perennial and ephemeral FDCs in this study, we also pursued broader applicability not only to islands but also to continental catchments under humid and arid climates.

Study area and data
This study was based in the Hawaiian Islands and later expanded to Australian catchments.Figure 1 shows the location of two catchments in Hawaii in an underlying mean annual rainfall distribution map.The rainfall mapping datasets were downloaded from University of Hawaii: Coastal Geology Group (2007aGroup ( , 2007b)).The runoff and rainfall datasets used in the study were taken from Farmer et al. (2003) and US Geological Survey (USGS, 2012a), the catchment boundaries were taken from USGS (2012b) and the Rainfall Atlas of Hawaii taken from Giambelluca et al. (2013).The catchments were chosen as examples of perennial and ephemeral catchments.The hydrographs for the catchments in Figure S1 show the perennial and ephemeral differences between the catchments.

Estimating the shape of the FDC
This study uses a conceptual framework proposed by Yokoo and Sivapalan (2011) to disaggregate and reconstruct a flow duration curve while identifying process controls at each disaggregated section.Detailed information on the framework can be viewed in the original study but here we discuss it briefly.The framework partitions (or disaggregates) the FDC into two parts (high and low) and shows that a strong relationship exists between high flows and rainfall and low flows and mean monthly flow (MMF).The MMF is the mean monthly runoff (12 months) which mimics daily runoff in a FDC approach specifically at the low flow ends as identified in Yokoo and Sivapalan (2011).It is a simple and important tool given that the FDC does not take into consideration the sequence of runoff occurrence.A subsequent study by Leong and Yokoo (2017) disaggregated the FDC into three parts namely high, middle and low and attempted to estimate each section independently using simple hydrologic methods.The Leong and Yokoo (2017) study focused on the high and middle sections and found that the Curve Number (CN) method is suitable for estimations in the high flow section and the MMF for the middle sections.The demarcation of the FDC sections is also discussed in Leong and Yokoo (2017).The CN method uses a rainfall runoff equation to estimate runoff using a single CN parameter.The CN parameter is dependent on the hydrologic soil group with different infiltration rates.A more detailed explanation for the CN method can be found in Ponce and Hawkins (1996).
This study puts emphasis on the low flow section and, based on the Yokoo and Sivapalan (2011) framework, the MMF was first used to estimate low flows in both the perennial Hanalei catchment and the ephemeral Makaha catchment in Hawaii.Subsequently, further investigation was performed in ephemeral catchments.Here, only the ephemeral Makaha catchment in Hawaii was combined with the four Australian ephemeral catchments (Stones, Bass, Bull and Maxon) as in Table SI.The analysis into the characteristics of the combined ephemeral catchments led to two noticeable points (1) the number of rainfall days were quite low and (2) the annual evapotranspiration exceeded the annual rainfall.Therefore, for simplicity, it was decided that the climate indices of aridity (AI, Equation 1) and precipitation (PI, Equation 2) in a generalized regression approach at intervals of 5% exceedance probability (EP) would be suitable to make compound estimations in ephemeral catchments.A regression analysis was performed for the catchments' observed runoff data against both AI and PI (that is q obs versus AI; q obs versus PI) at 5% EP up to 70% maximum EP because the EP threshold for the combined ephemeral catchments' ability for potential runoff was cutoff at this point.At each 5% EP intervals (Q 5 , Q 10 , Q 15 , ..., Q 70 ) a regression equation was obtained and the results with the best R 2 values were used to make runoff estimates.
The aridity index (AI) in this study is the ratio of potential evapotranspiration (PET) estimated by the Thornthwaite (1948) method to precipitation (P) at an annual scale as in Equation 1.
The precipitation index (PI) is a simple function of number of precipitated days annually, where p n is the number of days with precipitation and T t is the total time (d) at annual scale.

Mean monthly flow estimations
The results indicated that the MMF was suitable for making low flow estimations in perennial catchments but not in ephemeral catchments.This is shown in Figure 2, where the MMF provides a proper estimation in the perennial Hanalei catchment at the low ends but has sub-standard performance in the ephemeral Makaha catchment.The results added proof to the Yokoo and Sivapalan (2011) framework for estimating low flows using the MMF.However, this framework is only suitable to perennial catchments and not ephemeral catchments.

Reviewing ephemeral catchments
Two simple indices relating to normal ephemeral catchment characteristics, the aridity index (AI) and precipitation index (PI) previously explained, were used in an attempt to make proper runoff estimations in the ephemeral catchments.The AI regression analysis was seen to have better results when a power function was assigned to it, whereas the PI regression analysis had better results with a linear Figure 2. Comparison of runoff components.(a) Hanalei (b) Makaha.q obs is the observed runoff and q mmf is the mean monthly flow function.The generalized regression equations are shown in Table SII.The PI was seen to be consistent with previous literature showing a linear relationship between rainfall and runoff indicating that runoff is dominantly controlled by rainfall as explained in the comparative study of Hanasaki et al. (2008).A comparison of regression analysis for the two indices is shown in Figure 3, where it can be seen that: at 5% (top section) both indices responded poorly with low R 2 values, from 10% to 50% the two indices are comparable, and from 55% onwards PI has better R 2 at the low flow ends.Figure 4a for the Makaha catchment shows AI and PI both providing proper estimations when approaching the low flow section but for majority of the other ephemeral catchments, as in Figure S2 (panels a, c, e), PI provides more appropriate estimations than AI especially in the low ends.However, runoff was unsuccessfully estimated in the Bull catchment in Figure 5a, which is explained in the discussion section.Although in some catchments runoff estimations due to AI (q AI ) or PI (q PI ) seem departed from the q obs on the semi-logarithmic graph, on a normal axis the values are comparable.Also added to Figure 4 (panel b), Figure S4 (panels b, d, f) and Figure 5 (panel b), are Curve Number (CN) runoff estimations (q cn ) which is also discussed in the next section.

CONCLUDING DISCUSSION
The study attempted to make runoff estimations for the whole shape of a disaggregated FDC using simple methodologies towards the context of prediction in ungauged basins (PUB; Sivapalan, 2003).It is an extension of Leong and Yokoo (2017), which disaggregated the FDC into three parts and estimated runoff in the high and middle sections.The emphasis here is on the low flow section.The demarcation of each section at different EP and its reasoning is also mentioned in Leong and Yokoo (2017).Figure 6 outlines the method used for the present study.The chosen catchments were examples of perennial and ephemeral catchments.
Figure 3.Comparison of aridity index (AI) and precipitation index (PI) for the studied ephemeral catchments from 5-70% exceedance probability The study succeeded in identifying possible methods for estimating low flow runoff in a disaggregated FDC and completing the low flow part of a three-part disaggregated FDC study.The advantage of this study is the simplistic statistical approach, as the models used restricted heavy parametric dependency.Banasik (2011) stated that highly parametrized models are not feasible in small catchments that are sensitive to environmental changes because instability is caused by the variability of its components.The present study intends to provide first order runoff estimations to give insights into simple model applicability and less emphasis on high accuracy.
The results in the present study of low flows, together with Leong and Yokoo (2017), show the MMF can be used to make proper runoff estimations in the middle and low end sections of the FDC of perennial catchments but is unfavorable in ephemeral catchments.This study uses a statistical MMF approach.However, studies that focus on global hydrology have made considerable advancements in modeling MMF without the MMF statistical data as mod-Figure 4. Comparison of several runoff estimations for Makaha catchments.q obs , q PI , q AI and q cn are observed runoff, estimated precipitation index runoff, estimated aridity index runoff and estimated curve number method runoff, respectively eled and explained by Hanasaki et al. (2008).This new MMF modeled approach can potentially be one of the simplest of hydrological tools that can be used towards the challenge of estimating runoff in ungauged catchments.
The substandard runoff estimations in ephemeral catchments can be explained using Yokoo and Sivapalan (2011), which found that the slope of the FDC is highly influenced by the nature of seasonality indicators such as rainfall and evapotranspiration, especially when out of phase with each other as in arid climates where evapotranspiration becomes dominant.Furthermore, when low rainfall and high evaporation occurs (out of phase seasonality) in arid or semi-arid climates, the consequences are ephemeral runoff and sparse river networks, showing climate has a major influence on low flows (Laaha et al., 2013).Based on the findings from past studies, it is evident that ephemeral catchments have annual evapotranspiration exceeding annual rainfall; a phenomenon common to arid and semi-arid areas where there is also an abundance of non-rainfall days.
Figure 5.Comparison of several runoff estimations for Bull catchment.q obs , q PI , q AI and q cn are observed runoff, estimated precipitation index runoff, estimated aridity index runoff and estimated curve number method runoff, respectively For simplicity and considering natural phenomena, two climate indices, AI and PI, were used in an attempt to estimate runoff at 5% intervals using a best fit generalized regression function.Figure 3 shows that the PI linear regression function provided a better result than the AI power regression function, especially from 55-70% EP.Figures 4a and S4 (panels a, c, e) confirm this, with PI providing proper runoff estimations especially in the low ends of the FDC.This indicates that climate is the dominant control for low flows in ephemeral FDCs.However, the Bull catchment in Figure 5a had substandard runoff estimations.The Bull catchment data only reaches 15% EP which can be regarded to be in the top section of the FDC.In this catchment rainfall is the dominant control (Yokoo and Sivapalan, 2011) and therefore it is not possible to make proper runoff estimations with AI or PI in Bull catchment.As an alternative approach for Bull catchment, the authors used the CN method (adopted from Leong and Yokoo, 2017).In Figure 5b it can be seen that the CN method provides proper runoff estimations in the FDC top section especially within the 10% EP of Bull catchment.Figures 4b  and S4 (panels b, d, f) provide supporting evidence from other catchments for the use of the CN method to estimate runoff within 10% EP of the FDC (hashed lines on all CN related graphs).Given 10% EP range is related to possible flooding or first runoff reaction to rainfall, the result is supported by the literature which suggests that the CN method is suitable for storm loss and related runoff (Boughton, 1989;Ponce and Hawkins, 1996).The effectiveness of the CN method for modeling high flows is accepted in previous studies.However, this study used a disaggregated FDC while previous studies did not.Therefore, previous studies only mention the effect of the CN method on high flows but not the exceedance probability range of applicability on Figure 6.An overview of the study.The perennial and ephemeral catchments were identified by the shape of their FDCs.Within 10% exceedance probability (EP) of the FDC, the Curve Number method can be used to make runoff estimations regardless of the type of catchment.For the perennial catchments, the mean monthly flow can make estimations in the middle and low flow sections but for ephemeral catchments, the precipitation index (PI) can be used for estimating low flows.The PI here represents the number of rainy days annually the FDC.Therefore, for this study, we claim that for any catchment, regardless of whether perennial or ephemeral, the CN method can be used effectively for estimating runoff in the top section of the FDC, especially within 10% EP where flooding and high flows are prevalent.
In this study, we have demonstrated from a statistical approach the use of MMF, AI and PI to make proper runoff estimations in perennial and ephemeral catchments, respectively.However, in the context of PUB there is potential for all three to be used to make first order runoff approximations in ungauged catchments.As explained earlier, recent developments in global hydrology studies have made MMF available which can be used in ungauged catchments.On the other hand, the climate data needed to formulate the indices AI and PI are accessible through the initiative of the World Climate Research Programme (WCRP) which developed a superlative global inter-comparison climate project to share and compare climate models and data: such as the fifth phase of the Coupled Model Intercomparison Project (CMIP5, Taylor et al., 2012).For example, CMIP5 has the capacity to provide precipitation data which can be used in PI and temperature data which can be used in the Thornthwaite (1948) model to estimate AI.This approach is not part of the present study, and therefore has not been verified, but serves as an insight into possible runoff estimation in ungauged catchments and is encouraging for potential future study.

Figure 1 .
Figure 1.Maps showing two of the studied catchments in the Hawaiian Islands.(a) Hanalei on Kauai Island (b) Makaha on Oahu Island with underlying mean annual rainfall distribution maps Figure S1.Hydrographs of the perennial and ephemeral catchments Figure S2.Estimated runoff comparisons between catchments Table SI.Characteristics of ephemeral catchments used in the study Table SII.The generalized regression equations