The methodology comprises three parts: creating the annual cycle climatology of rainfall and determining intermittent dry periods, identifying seasonal windows, and calculating temporal and intermittent dry period rainfall characteristics for each year. For temporal characteristics, one context is examined in this section and a summary of the methodology in this context can be found in Fig. 1. In addition, Fig. 1 includes a summary of the adaptations of the methodology in other contexts (e.g. Absolute Meteorological Characteristics, and Agronomical Characteristics), which can be found in Sects. 4 and 5, respectively.
Create annual cycle climatology of rainfall and determine intermittent dry periods
The annual cycle climatology of rainfall is calculated from the raw daily (unsmoothed) data, excluding leap days. A harmonic filter then smooths the annual climatology. Harmonic filters have been used to investigate the number of wet seasons over a given region (e.g. Wang 1994; Wang and LinHo 2002; Liebmann et al. 2012). The number of harmonics used depends on the region under study. Previous studies suggest the use of the first 4 to 12 Fourier harmonics (Wang 1994; Wang and LinHo 2002; Liebmann et al. 2012). Here, a five harmonic (5-term Fourier) filter is used to smooth the climatology.
For the purposes of the analysis, the calendar year should be shifted also, to better capture the full cycle of the annual climatology and all transitions between wet and dry seasons (e.g., Bombardi et al. 2017; B20), particularly for regions where a rainfall season bridges two calendar years. The shift is determined by the peaks and troughs of the rainfall cycle that are identified in the smoothed annual climatology. In the smoothed annual climatology, the minima that separates the dry and wet seasons, and the final maximum of the rainfall season are identified. The former denotes the beginning of the climatological annual rainfall cycle. Next, the start and end dates of the analysis are determined. Selecting the start and end dates are subjective choices. Recent studies used 20–50 days before the beginning of the climatological annual rainfall cycle as their start date (D16; B20). However, some determine the start date from tradition (e.g. Allen and Mapes 2017; CIMH deems the start of the Caribbean rainfall season as April 1st). The same steps are applied to determine the end date for the analysis; the end date must be some days after the final climatological peak of the annual rainfall cycle. Recent studies use 20–50 days after the final climatological peak of the annual rainfall cycle to determine the end date (D16; B20), or use a traditional date that is commonly used to describe the end of the rainfall cycle (e.g. M19, M20). These dates will also be used in the year-to-year analysis. For the Caribbean, the data is shifted such that Day 1 is March 1st and Day 365 is February 28th of the following year. For the Guianas, the data is shifted such that Day 1 is February 1st and Day 365 is January 30th of the following year. Daily data for each sub-region was calculated by averaging their stations’ daily data. The start date for the Northwestern and Western Caribbean is April 1st. Since the transition from less-wet to wet rainfall season begins in late March/early April for the Central and Eastern Caribbean (M19), the start date is March 1st. The start and end date in the Guianas is the day of the climatological minima of the annual cycle prior to the first rainfall season, or March 1st, through March 5th of the following year. The end date is November 30th in the Northwestern Caribbean. The Central Caribbean end date is December 30th and in the Eastern and Western Caribbean the end date is January 30th of the following year, as the demise of their rainfall cycles is later in the year than in the Northwestern Caribbean (M19). A summary of the start and end dates for each sub-region can be found in Table 1.
Intermittent dry periods in the smoothed annual climatology are identified in order to determine the modality of the annual climatology. Between the start and end dates, an intermittent dry period is identified if the difference between either rainfall peak and the minima between peaks is greater than 1 mm, following similar steps from B20. Next, any identified intermittent dry period is classified as subtle or distinct. If the intermittent dry period minimum is 1 mm less than the climatological annual mean, the intermittent dry period is distinct. Otherwise, the intermittent dry period is subtle. A step-by-step example of the process for identifying and classifying intermittent dry periods using the smoothed annual climatology is given for the Northwestern Caribbean (Fig. 2a) and Guianas (Fig. 2b). In the Northwestern Caribbean there are two rainfall peaks with a minimum between peaks. The difference between the minimum and either rainfall peak is greater than 1 mm; therefore, the Northwestern Caribbean experiences one intermittent dry period. The intermittent dry period is not 1 mm less than the climatological annual mean (~ 3.8 mm/day); therefore, the intermittent dry period is subtle. Similarly, the Guianas has two rainfall peaks with an intermittent dry period. However, the intermittent dry period minimum in the Guianas is 1 mm less than the climatological annual mean (~ 6 mm); therefore, the intermittent dry period is distinct. The number of intermittent dry periods and their classification determines the modality of the climatological mean annual rainfall cycle. The Northwestern Caribbean experiences one subtle intermittent dry period; therefore, the modality of the rainfall cycle is unimodal dual maxima, or a subtle bimodal rainfall cycle. The Guianas experiences one distinct intermittent dry period; therefore, the modality of the rainfall cycle is bimodal. Whether or not an intermittent dry period is subtle or distinct determines whether one is able to calculate multiple onsets and demises between the start and end dates of the interannual analysis (Sect. 3.3). For example, in the Northwestern Caribbean, only one onset and demise can be determined in the interannual analysis because the intermittent dry period is subtle. In the Guianas, two onsets and two demises can be determined in the interannual analysis because the intermittent dry period is distinct. A summary of the classified modalities for each sub-region can be found in Table 1.
Identify seasonal windows
Using the smoothed annual climatology, seasonal windows are introduced for the calculation of interannual characteristics associated with each rainy season of the annual rainfall cycle. For unimodal regimes, the seasonal window is the analysis start and end dates. For unimodal dual maxima and bimodal regimes, the first window is set between the start date to the date of the smoothed annual climatology rainfall minimum within the intermittent dry period. The second window is set between the smoothed annual climatology minimum within the intermittent dry period and the end date. An intermittent dry period window is set by determining the midpoints between the smoothed annual climatology rainfall peaks and the minima of the smoothed annual climatology intermittent dry period. For example, two seasonal windows are set for the Northwestern Caribbean (Fig. 2a): the ‘Early Rainfall Season’ window, which is between the analysis start date and the date of the climatological intermittent dry period rainfall minimum, or Day 141 (July 18th), and the ‘Late Rainfall Season’ window, which is between the date of the climatological intermittent dry period rainfall minimum and the analysis end date (Fig. 2a). The intermittent dry period window, or Mid-Summer Drought window, is Day 117 (June 26th) to Day 164 (August 12th).
Calculate rainfall characteristics for each year
First, the temporal location, amplitude, and width of every peak and trough in the gaussian-filtered (smoothed) daily data for each year is calculated. Second, inflection points are determined in the year-to-year smoothed daily data, and are classified as candidate meteorological onset or demise dates. For candidate onsets, the method considers every trough-to-peak within the smoothed daily data, and finds the minimum day, if any, that has both of the following user-input conditions: (1) the day is above an inflection point mm threshold, and (2) the rate of change between the inflection point and X days exceeds a given mm/day threshold. The latter is done to avoid inflection points in which no clear changes in slope above the mm threshold exist in subsequent days. For candidate demises, the method inspects every peak-to-trough and finds the minimum day, if any, that has its value below the inflection point mm threshold. To define a wet vs. non-wet day, most studies use a threshold value from 0.85 mm (Stern et al. 1981) to 1 mm (Moron and Roberstson 2014), but in some regions with unclear dry seasons, values as high as 2.5 mm are used (Nandargi and Mulye 2012). Therefore, it is recommended to set the inflection point mm threshold between 0.85 mm and 2.5 mm. For the onset calculation, the rate of change requirement is similar to methods that use a criterion to check for persistence of rainfall, and typically check approximately 10 days out. However, those methods do not check for slope and only ask how many days after the date are also at or above the mm threshold. Onsets are associated with a change in intensity of rainfall; hence they should also depend on the change in slope. This method is the first to put a mm/day tendency threshold. In the application of this method, the tendency is calculated as the average change in daily rainfall amount between the smoothed annual climatology trough and peak that is associated with the onset.
To demonstrate the calculation of the accumulated precipitation anomalies, onsets, and demises, a step-by-step illustration of the method using the 2011 rainfall cycle in the Northwestern Caribbean (Fig. 3) and the 1985 rainfall cycle in the Guianas (Fig. 4) are shown. In the Northwestern Caribbean, a candidate onset is determined between any trough to peak if the inflection point is above 1 mm and if there is a rate of change of 0.102 mm/day between the date of the inflection point and 10 days out (Fig. 3a). The latter was determined by taking the difference between the smoothed climatological peak (7 mm) and trough (1.9 mm) that is associated with the onset divided by the number of days between the smoothed climatological trough to peak associated with the onset (50 days). A demise inflection point is determined between any peak to trough if the inflection point is below 1 mm (Fig. 3a). A similar procedure is done for the Guianas (Fig. 4a), except (1) the inflection point threshold is 1.5 mm because it is common for the region to not have consistent rainfall values below 1 mm, and (2) determining candidate onsets are different between the first and second rainfall seasons, as the second rainfall season has a separate mm tendency threshold.
The context examined in this section is an adaptation of the approach used in Liebmann and Marengo (2001) and Bombardi et al. (2017). Unlike their methods, the present method does not calculate interannual onsets and demises using a mean threshold that is based on the climatology of the annual rainfall cycle. Onsets and demises using the present method will be referred to as the Relative Meteorological Onset (RMO) and Demise (RMD). Using the unsmoothed data for each year, the same equation from Bombardi et al. (2017) is applied with some modifications to calculate year-to-year RMOs and RMDs:
$${S}_{\mathrm{window}}\left(n\right)=\sum_{i={t}_{\mathrm{window}}}^{n}P\left(i\right)-{P}_{\mathrm{window}}$$
(1)
where Swindow(n) is the anomalous accumulated precipitation at precipitation day “i” to day “n”. P(i) is the daily precipitation at day “i”. Pwindow is the mean daily value of the rainfall in that year calculated over the particular seasonal window; this differs from other methods that use Pwindow as the annual climatological daily precipitation rate (Liebmann and Marengo 2001; Bombardi et al. 2017; D16; B20). Finally, twindow is the date related to the start, or before the start, of the seasonal window being used. Similarly to Bombardi et al. 2017, for onsets it is recommended to place twindow in the dry season that precedes the rainfall season in order to accurately depict the transition from dry to wet seasons. Depending on the modality of the smoothed annual climatology, one or multiple Swindow (n) are used when calculating interannual onsets and demises. For unimodal patterns, only one Swindow(n) is used and is used for calculating both onset and demise. For unimodal dual maxima patterns, two Swindow(n) are used: one is used for the onset, and the other is used for the demise. For bimodal patterns, four Swindow(n) are used: each for each onset and each demise calculation. For onset, the minimum of Swindow(n) is identified. Next, the candidate onsets from the given year’s smoothed daily data are utilized by finding the candidate onset that relates to the date of the minimum of Swindow(n). This is done by finding the latest date of a candidate onset that is (1) earlier than the date of the minimum of Swindow(n), and (2) the magnitude of the rainfall on the candidate onset day is within the Rth percentile of the rainfall amount, relative to that year, over the window. The first criterion does not include the location of the onset to be later than the start date of the seasonal window, as this would restrict the location of the onset to be solely after the climatological intermittent dry period minima (for multi-modal cases) or start date. The latter is done in case the smoothing filter has several “kinks” between the date of the minimum of Swindow(n) to the actual onset. A recommended value for the onset Rth percentile is the 33rd to 50th percentile. The candidate onset that satisfies both criteria is deemed the RMO.
The approach to find the RMDs follows that for the RMOs, with a few key differences. The maximum of Swindow(n) is found. Using the candidate demises, the timing of the RMD is determined by finding the minimum candidate demise that: (1) is later than the date of the maximum of Swindow(n), and (2) the candidate demise value is within the Rth percentile of the rainfall amount, relative to that year, over the window. The first criterion does not include the location of the demise to be earlier than the end date of the seasonal window, as this would restrict the location of the demise to be solely before the climatological intermittent dry period minima (for multi-modal cases) or end date. The second criterion is done in case the smoothing filter has several “kinks” from the date of the maximum of Swindow(n) to the actual demise. A recommended value for the demise Rth percentile is the 10th to 33rd percentile.
In the Northwestern Caribbean, two Swindow(n) are determined:
$${S}_{\mathrm{ERS}}\left(n\right)=\sum_{i={t}_{\mathrm{ERS}}}^{n}P\left(i\right)-{P}_{\mathrm{ERS}}$$
(2)
$${S}_{\mathrm{LRS}}\left(n\right)=\sum_{i={t}_{\mathrm{LRS}}}^{n}P\left(i\right)-{P}_{\mathrm{LRS}}$$
(3)
tERS is Day 1 (March 1st) and tLRS is the date of the climatological intermittent dry period minima, or Day 141 (July 18th). Since the Northwestern Caribbean has a unimodal dual maxima pattern in its smoothed annual climatology, only the first window is used to determine yearly onsets, and the second window is used to determine yearly demises. The date of the minimum of SERS(n) is calculated, and is on day 90 (May 30th) (Fig. 3b). The date of the minimum of SERS(n) is used as a reference date to determine the timing of the RMO (Fig. 3c). There is one candidate onset that is before the date of the minimum of SERS(n) and its value is within the 33rd percentile of the rainfall amount, relative to the year, over the ERS window. Therefore, the RMO is on day 80 (May 20th). The date of the maximum of SLRS(n) is calculated, and is on day 248 (November 4th) (Fig. 3b). There are two candidate demises that are after the date of the maximum of SLRS(n) and its value is within the 10th percentile of the rainfall amount, relative to the year, over the LRS window (Fig. 3c). Therefore, the RMD is the earlier of the two candidate demises, or day 255 (November 11th).
The Guianas experience a bimodal pattern in its smoothed annual climatology; therefore, four Swindow(n) are determined in the interannual analysis (Fig. 4b). In the interannual analysis, when finding the onset of the first rainfall season and demise of the second rainfall season, Eq. 1 is used. For finding the demise of the first rainfall season and onset of the second rainfall season, Eq. 1 is altered, such that the mean daily value of the rainfall in that year calculated over the intermittent dry period window, or PIDP(n), is used. For bimodal patterns the demise of the first rainfall season (RFS1) and onset of the second rainfall season (RFS2) fall under the intermittent dry period window, where using PIDP(n) better characterizes the transition between each rainfall season. The minimums of SRFS1(n) and SRFS2(n) are calculated and RMOs for each season are determined (Fig. 4c). The maximums of SRFS1(n) and SRFS2(n) are calculated and RMDs for each season are determined.
Characteristics related to the intermittent dry period
The intermittent dry period window is used to calculate the magnitude and duration of intermittent dry periods in the interannual analysis. For each year, the magnitude of the intermittent dry period is estimated from the unsmoothed data as the mm/day average within the intermittent dry period window. For each year, the duration of the intermittent dry period is the total number of unsmoothed data days within the intermittent dry period window that has a mm value less than the 66th percentile of the unsmoothed intermittent dry period magnitude from the annual climatology. For the Northwestern Caribbean, the unsmoothed annual climatology magnitude of the intermittent dry period is 4.70 mm, and the dates categorized as less than the 66th percentile of the unsmoothed intermittent dry period magnitude from the annual climatology would be those with amounts less than 5.60 mm.