Daily streamflow observations utilized in this study are from the Global Runoff Data Centre (GRDC, 2007), specifically those stations located along the global hydrology model's drainage network. Since station records that are missing even short periods may affect how a high-flow season is defined, we have excluded years with any daily missing values. In this study, a minimum of 20 hydrological years is required for a station to be retained, leaving 691 stations from all continents except Antarctica, with upstream basin areas ranging from 9539 to 4 680 000 km2 and periods of record between 20 and 43 years across 1958–2000 (Fig. 1). Although this criterion is admittedly quite strict (no missing 20-year daily data), including stations with missing records does not add a significant number. These stations are mostly located on large rivers; the annual streamflow of 75 % of stations is larger than 100 m3 s-1, 35 % of stations are larger than 500 m3 s-1, 20 % of stations are larger than 1000 m3 s-1, and 5 % of stations are larger than 5000 m3 s-1.

In this study, we evaluate simulations of daily streamflow over the period 1958–2000 taken from Ward et al. (2013), carried out using PCR-GLOBWB (PCRaster GLOBal Water Balance), a global hydrological model with a 0.5∘ × 0.5∘ resolution (Van Beek and Bierkens, 2009; Van Beek et al., 2011). Although the PCR-GLOBWB model is not calibrated, and simulations may contain biases and uncertainty at course spatial resolution, the long time series of streamflow provided globally has been deemed sufficient to estimate long-term flow characteristics with spatial consistency (Winsemius et al., 2013). Additionally, this model has been validated in previous studies in terms of streamflow (Van Beek et al., 2011) and terrestrial water storage (Wada et al., 2011) at stations along major rivers in the world. The model's extreme discharges are also evaluated by Ward et al. (2013) with fair to good performance at stations with large drainage area (≥ 125 000 km2), corresponding to 24 % of GRDC stations used in this study, excepting overestimation in several arid regions. Note that for the simulations used in this study, the maximum storage within the river channel is based on geomorphological laws that do not account for existing flood protection measures such as dikes and levees.

For the simulations used in this study, the PCR-GLOBWB model was forced with daily meteorological data from the WATCH (Water and Global Change) project (Weedon et al., 2011), namely precipitation, temperature, and global radiation data. These data are available at the same resolution as the hydrological model (0.5∘ × 0.5∘). The WATCH forcing data were originally derived from the ERA-40 reanalysis product (Uppala et al., 2005), and were subjected to a number of corrections including elevation, precipitation gauges, timescale adjustments of daily values to reflect monthly observations, and varying atmospheric aerosol loading. It is possible that this may have some minor effect on streamflow simulation, likely providing more realistic outcomes. Full details of corrections are described in Weedon et al. (2011).

In the last few decades, a number of studies have investigated the timing of peak flows in the context of analyzing flood seasonality, frequency and trends. Generally, two main properties are emphasized regarding flood timing: peak volume and peak timing. Considering peak volume, the occurrence dates are commonly recorded for a fixed time period or specific amount of peak volume, often in the context of trend analysis. For example, Hodgkins and Dudley (2006) use winter–spring center of volume (WSCV) dates to analyze trends in snowmelt-induced floods, and Burn (2008) uses percentiles of annual streamflow volume dates as indicators of flood timing, also for trend analysis. For peak timing, two sampling methods are frequently applied in hydrology. The first and most common is the annual-maximum (AM) method, which samples the largest streamflow in each year. The second method is the peaks-over-threshold (POT) method (Smith, 1984, 1987; Todorovic and Zelenhasic, 1970), in which all distinct, independent dominant peak flows greater than a fixed threshold are counted. In contrast to the AM method, POT can capture multiple large independent floods within a single year, including the annual maximum flow, but may not capture the annual maximum flow in years in which streamflow is less than the pre-defined threshold; this threshold can either be defined based on a specific average number of floods or a specific mean exceedance level over the entire period (Cunderlik et al., 2004a; Institute of Hydrology, 1999; Lang et al., 1999). The PM selected, therefore, is dependent on the peak properties (volume, timing) considered. For a local study, selecting the PM can be based on well-defined climatic or hydrologic characteristics (e.g., rainy season, snowmelt, etc.); however, no single global method can be uniformly applied to define the PM everywhere. Thus, to define the HS, and specifically the PM, globally, both peak volume and peak timing aspects need to be considered (Javelle et al., 2003). To do this, we adopt a volume-based threshold (VBT) technique. This technique is similar to a streamflow volume-based technique in terms of capturing the days (Julian dates) when streamflow exceeds the pre-defined threshold (percentile of flows) and associated volume (Burn, 2008). The major difference, however, is that the VBT applies the threshold over the entire time series (available record) concurrently instead of on a year-by-year basis. In other words, for the 95th percentile, instead of annually calculating the 95th percentile, it is calculated using the entire period of record. The common volume-based technique thus records events every year surpassing the threshold; however, for the VBT approach, every year need not have a peak above the threshold. This approach emphasizes capturing the key peaks across the entire available time series (as in a peaks-over-threshold approach). VBT thus contains both volume and timing characteristics for defining the peak month (PM). Here, the month containing the greatest number of occurrences over the specified percentage of flows across all years (1958–2000) is defined as the PM, and subsequently the HS is designated as the period containing the PM plus the month before and after the PM. Figure 2 provides an example based on 7 years of synthetic streamflow with the volumetric threshold set at the top 5 % of flows; the number of days surpassing the 5 % threshold is listed for each month. In this example, August has the largest number of days over the threshold (105 days); thus, August is defined as PM and July–September is defined as HS.

To evaluate the defined HS objectively, by evaluating the number of annual maximum flows captured, we develop a simple evaluating statistic called the percentage of annual maximum flow (PAMF). PAMF is computed as shown in Eq. (1): PAMF(i)=∑j=i-1i+1nAMF(j)∑k=112nAMF(k),1≤i≤12, where nAMF(i) denotes the number of annual maximum flows that occur in month i across the full record. In Eq. (1), when i is 1 (January), i - 1 in the summation is 12 (December), and when i is 12 (December), i+1 is 1 (January). Here the PAMF provides the percentage of annual maximum flows occurring in the defined HS across the evaluation period. The PAMF is relatively simple, yet provides a clear indication of how well the PM selected represents the occurrence of annual peaks across the time series. For example, a high PAMF indicates that the HS is highly likely to contain the annual maximum flood each year. In contrast, a low PAMF indicates that the timing of the annual maximum flow is more likely to vary temporally, and may be a result of bimodal seasonality, consistently high or low streamflow throughout the year, streamflow regulated by infrastructure or natural variation. In this study, we subjectively classify HS PAMF values as high (80–100 %), moderate (60–80 %), low (40–60 %) and poor (0–40 %). The PAMF is calculated for both the observed streamflow at the 691 selected GRDC stations and the simulated streamflow at the associated 691 grid locations.

The VBT technique is compared with the common volume-based technique and POT technique to gauge performance. Four volume-based durations, namely V01 %, V03 %, V05 % and V10 %, and three POT techniques averaging 1, 2, and 3 peaks per year (POT1, POT2 and POT3, respectively), are selected. For the V01 % technique, the HS is simply centered on the PM containing the largest number of occurrences of the top 1 % of annual streamflow volume across the total years available. The V03 %, V05 % and V10 % techniques are similar to the V01 % approach, respectively using 3, 5 and 10 % of annual streamflow volume. Comparatively, techniques with a shorter time component (1–3 % of annual volume) favor identifying the PM by peak timing, since the top 1–4 days of streamflow tend be located near the peak, while techniques with longer time components (5–10 % of annual volume) favor identifying the PM based on duration and peak volume, since the top 19–33 days of streamflow tend to be located near the volumetric centroid of the hydrograph, rather than the peak, if they differ. The VBT technique is an attempt to bridge these two criteria. For the POT techniques, independence criteria are applied to avoid counting multiple peaks from the same event (Institute of Hydrology, 1999). For example, two peaks must be separated by at least 3 times the average rising time to peak, and minimum flow between two peaks must be less than two-thirds of the higher one of the two peaks. More details of independence criteria are described in Lang et al. (1999).

An analysis examining sensitivity of selected threshold levels to the VBT technique is also undertaken. Performances of thresholds representing 1, 3, 5 and 10 % exceedance across the entire period of record, named VBT1 %, VBT3 %, VBT5 % and VBT10 %, respectively, are compared.

To compare techniques and thresholds, the PMs are defined at the 691 selected stations and associated model grids. The locations where the PMs differ (by at least one technique) are of most interest. This occurs at 61 % of stations and 54 % of associated grids. Cross-correlations of PM between the four common volume-based techniques clearly indicate the tendency of the defined PM to shift from peak timing dominated to peak volume dominated as the time component increases (Table 1). Correlation between VBT techniques and volume-based techniques are quite similar and consistent (0.82–0.86 and 0.84–0.86 for observed and simulated streamflow, using VBT5 %; Table 1), preliminarily indicating some success in capturing both timing and volume properties, while correlations between the VBT techniques and POT are less strong (0.78–0.81 and 0.79–0.83 for observed and simulated streamflow, respectively, using VBT5 %; Table 1). The PAMF is also useful for comparing techniques, such that the technique having the highest average PAMF typically contains more annual maximum flow events in their defined HSs. The VBT5 % is superior to other VBT and POT techniques for both observed and modeled streamflow, having the highest PAMF values; however, the volume-based techniques indicate similar or even slightly better performance than VBT5 % (Table 2). This is not unexpected as the volume-based techniques are designed to capture annual peak flows on a year-by-year basis, whereas the POT and VBT record significant peaks across the full time series, and may not capture annual peaks in some years in which that peak is small relative to all peaks throughout the available record. Thus VBT tends to select PMs that contain the most significant peaks overall, and subsequently have the highest potential for capturing probable flood seasons for flood-prone basins, a desirable outcome for this study. To illustrate this in the context of the PAMF, if all years are ranked for each location based on the annual peak flow, and the top 50 % (half) are retained, the PAMF actually favors the VBT approach, surpassing the volume-based approach by 5–6 % for PMs and 2–3 % for HSs.

Finally, techniques may be evaluated by comparing the temporal difference (number of months) between model-based and observed PMs; closer is clearly superior. The VBT3 % and VBT5 % techniques produce the greatest degree of similarity between model-based and observed PMs (81 % of stations having ±1 month difference; Table 3). Overall, the VBT technique demonstrates superior performance as compared with the POT techniques by all comparisons. The VBT technique is also on par with or slightly superior to the common volume-based technique, especially considering the 5 % threshold; thus, the remainder of the analysis is carried out utilizing the VBT5 % technique only.

In addition to evaluating the HS at the 691 grid cells based on model outputs, the PM and HS can also be defined at the sub-basin scale globally where observations are present. Previous studies have investigated flood seasonality as it relates to basin characteristics; for example, basins are delineated/regionalized and grouped according to similarity/dissimilarity of streamflow seasonality (Burn, 1997; Cunderlik et al., 2004a), or conversely, flood seasonality is occasionally used to assess the hydrological homogeneity of a group of regions (Cunderlik and Burn, 2002; Cunderlik et al., 2004b); thus, evaluating at the sub-basin scale is warranted.

While defining a single PM for a large-scale basin may be convenient, it may be difficult to justify given the potentially long travel times and varying climate, topography, vegetation, etc. Additionally, infrastructure may be present to regulate flow for flood control, water supply, irrigation, recreation, navigation, and hydropower (WCD, 2000), causing managed and natural flow regimes to differ drastically. This becomes important, as globally more than 33 000 records of large dams and reservoirs are listed (ICOLD, 2009), with geo-referencing available for 6862 of them (Lehner et al., 2011). Nearly 50 % of large rivers with average streamflow in excess of 1000 m3 s-1 are significantly modulated by dams (Lehner et al., 2011), often significantly attenuating flow hydrographs and flood volumes (20 % of GRDC stations fall into this category). The PAMF, as previously defined, can aid in identifying stations affected by upstream reservoirs through low PAMF values. This is applied with the assumption that reservoir flood control disperses the annual maximum flows across months rather concentrated within a few months (e.g., akin to natural flow). In this study, we used the global sub-basins from the 30′ global drainage direction map (DDM30) data set (Döll and Lehner, 2002) with separation of large basins (Ward et al., 2014).

To define a sub-basin's PM, the maximum PAMF and associated PM for each station within the sub-basin are considered according to the following:

if multiple stations exist within the sub-basin, the PM is defined as the PM occurring for the largest number of stations;

In contrast, the model-based simulated streamflow produces a high PAMF at STA06 (97 %), as the Itezhi-Tezhi dam is not represented in the simulations used for this study, and subsequently does not account for modulated streamflow. Across other stations, the PAMF is also high; however, an equal number of stations select February and March. In this case, February is selected as the final basin PM given its higher average PAMF value (96 % vs. 91 %).

By this approach, all 691 GRDC stations are grouped into 223 sub-basins to define the PM (Fig. 6); 58 % of sub-basins are defined by a single station, only 7.6 % (observations) and 8.1 % (model) of sub-basins have ties when defining PMs, and only one sub-basin has a tie between PMs and average PAMF values.

Ideally the model-based and observed GRDC stations have fully or partially overlapping HS periods. If so, this builds confidence in interpreting HSs at locations where no observed data are available. For comparing modeled PMs to observations, the defined PMs and calculated PAMF are represented globally at the station scale (Figs. 4–5) and sub-basin scale (Fig. 6) with temporal differences of PMs (modeled PM – observed PM). In the southeastern United States, GRDC stations express relatively lower PAMF values for observations (40–60 %) than model outputs (60–80 %), due to the high level of managed infrastructure. In the central–southern US and Europe, low PAMF values are computed for both observations and modeled output (Fig. 5) with notable temporal differences (Fig. 4c). For observations, this is attributable, at least in part, to reservoirs and dams along the Mississippi, Missouri and Danube rivers. Additionally, relatively constant streamflow patterns are identified in both observations and modeled output, consistent with previous studies reporting these flow regimes as uniform or perpetually wet (Burn and Arnell, 1993; Dettinger and Diaz, 2000; Haines et al., 1988). Minor high-flow seasons may also play a role. Model biases also affect PM selection; for northwestern North America, PMs for many points are defined on average 1 month earlier than with observations, producing moderate PAMF values (60 % and higher). In northern Europe, especially southern Finland, this becomes much more pronounced, with large differences between PMs from observations and the model, on the order of 4 months (Figs. 4c, 6c, and 8a). In western and northern Australia, PMs are modeled 1 month later on average than observations, except for two occurrences in the west (5-month difference) due to both observed and modeled low-flow conditions. Such low-flow regimes are also apparent in southeastern Australia, causing large differences between PMs (4–5 months). The differences in PMs between observations and modeled outputs are also compared at the continental scale (Fig. 7). In North America, 38 % of stations and 51 % of sub-basins produce identical PMs, growing to 82 % of stations and 93 % of sub-basins when considering a ±1 month temporal difference (e.g., HS; Fig. 7). In Asia 65 % of stations and 70 % of sub-basins have identical PMs, growing to 90 % of stations and 92 % of sub-basins with ±1 month temporal difference (Fig. 7). In central Russia, a large difference between PMs (±3 months) is attributable to reservoirs on the Yenisei and Angara rivers and model bias (Fig. 4c). In Africa, 48 % of stations and 60 % of sub-basins produce identical PMs (Fig. 7), 30 % of stations and 27 % of sub-basins are modeled 1 month earlier, and 7.4 % of stations and 6.7 % of sub-basins are modeled 1 month later than observation (Fig. 7). In South America, with only five stations, 40 % have the same month, 40 % are modeled 1 month earlier, and 20 % of stations are modeled 2 months earlier than observations.

Comparing observations and modeled output globally, 40 % of the locations share the same PM. The model's bias is one of the main reasons for this moderate performance; other important contributors include minor high-flow seasons, perpetually wet or dry regions, and anthropogenic effects such as reservoir regulation. Considering a difference of ±1 month, this jumps to 81 %, and 91 % for ±2 months (Fig. 7). From a sub-basin perspective, the similarities are even stronger (50 % identical PM, 88 % ±1 month and 92 % ±2 month), indicating a relatively high level of agreement. For locations having dissimilar PMs (≥ ±3 months, 9 % of locations and 8 % of sub-basins), a substantial number are located downstream of reservoirs directly, such as STA06 in the Zambezi example (Table 4), or are low-flow (dry) or constant-flow locations, both producing exceedingly low PAMF values. Differences in PMs are not unexpected for low-flow and constant-flow locations, given the propensity of the annual streamflow maximum to potentially occur in a wide number of months. Overall, however, as more than 80 % of both stations and sub-basins have similar PMs (±1 month), it appears that the global water balance model performs appropriately well in defining high-flow seasons globally at locations where observations are available.

This may be subsequently extended to defining PMs and PAMF at all grid cells (Fig. 8). Generally, low and poor PAMF values (0–60 %) indicate a naturally unstable annual maximum flow (no clear high-flow season), which occurs in cases of constant flow, low flow, bi-modal flow and regulated flow. All cases, except regulated flow, are simulated within the PCR-GLOBWB simulations used; thus, the cell-based PAMF values (Fig. 8b) can provide a sense of confidence for the defined PM (Fig. 8a). Examples of low-flow regions include the central United States and Australia, having low PAMF regional values (Fig. 8b). Bi-modal regions, such as much of eastern Africa and southern South America with their two rainy seasons, and constant-flow regions, such as Europe, also indicate low PAMF values (Fig. 8b). These flow regimes are further investigated as minor HS in Sect. 5.

Model-based PMs may also be verified (subjectively) by surveying historic flood records. One such source is the Dartmouth Flood Observatory (DFO), a large, publicly accessible repository of major flood events globally over 1985–2008, based on media and governmental reports and instrumental and remote-sensing sources. Delineations of affected areas are the best estimates (Brakenridge, 2011). The DFO records provide the start time, end time and duration of each flooding event, as defined by the report or source, and represented as the occurrence (start) month (Fig. 9). DFO flood events and grid-cell-based PMs (Fig. 8a) may be compared outright; however, their characteristics differ slightly. The DFO covers 1985–2008, while the model represents 1958–2000. Also, the model-based PM represents the month most likely for a flood to occur; the DFO is simply a reporting of when the event did occur, regardless of whether it fell in the expected high-flow season or not. Nevertheless, model-based PMs and historic flood records illustrate similarity (compare Figs. 8a and 9), particularly when both the major and minor high-flow seasons are considered, further indicating merit in the ability of the proposed approach to identify the PM. Consistently, regions with high model-based PAMF (80–100 %), such as eastern South America, central Africa and central Asia, tend to agree well with DFO records, while poor or less than poor PAMF (0–60 %) regions, such as central North America, Europe, and eastern Africa, tend not to be in agreement with DFO records. In these low PAMF regions, however, DFO records also illustrate floods occurring sporadically throughout the year, further supporting accordance between cell-based PAMF and DFO records (Figs. 8b and 9).