In total, the experiment made use of nine intervalometers, one acoustic disdrometer and two tipping bucket rain gauges at eight different sites. The tipping bucket rain gauge was made by Onset (more info at https://www.onsetcomp.com/products/data-loggers/rg3, last access: 11 August 2021) in the US and was equipped with a HOBO data logger; the acoustic disdrometer was manufactured by Disdrometrics in Delft, the Netherlands; and the intervalometer was also made by Disdrometrics. The intervalometer is a device that registers the arrival of raindrops at the surface of a piezoelectric drum and can be constructed for less than USD 150. It has a minimum detectable drop diameter (Dmin) of 0.8 mm, determined in a lab experiment by Jan Pape. Typical values of Dmin for impact disdrometers are between 0.3 and 0.6 mm . The Dmin value of 0.8 mm for the intervalometer means that the instrument is likely to miss many small drops and underestimate rainfall rates. The advantage of the intervalometer over a standard rain gauge is that it provides drop counts as well as rainfall estimates. More information about the intervalometer can be found at https://github.com/nvandegiesen/Intervalometer/wiki/Intervalometer (last access: 11 August 2021). A similar instrument in terms of acoustic sensor is also described by . The acoustic disdrometer registers the kinetic energy of drop impacts at a drum and converts this to an estimate of the drop size. It is similar to an intervalometer but also provides individual drop size estimates. The minimum detectable drop diameter for the disdrometer was thought to be 0.6 mm but in practice was 1 mm. This is larger than what is typical for an impact disdrometer and means that it likely misses many small drops and underestimates the actual rainfall rate. The effect of truncation on rainfall estimation is discussed in Sect. 3.2. A good discussion of the pros and cons of impact disdrometers in general can be found in, e.g. and for tipping buckets in, e.g. . The tipping bucket rain gauge collects all drops over a known surface area and funnels them to a small bucket which tips whenever a fixed volume of water has been collected (typically 0.2 mm). The volume of each tip is verified in situ via a field calibration experiment.

In total, eight sites were selected along the southern coast of Mafia Island, Tanzania. Figure  presents the experimental layout. Sensors were placed in an approximate line, such that a rectangle 3.1 km in length and 500 m in width would cover all the sites. The dimension of the long axis of the experiment was chosen to approximate that of the spatial resolution (approximately 5 km) of the GPM dual polarisation radar (DPR) instrument.

Rainfall measurement sites were chosen to comply as much as possible with World Meteorological Organisation guidelines within the constraints of accessibility and landscape. Ideally, this means that all of the sensors should be placed in vegetation clearings, sheltered as much as possible from the wind at a height of 1.5 m off the ground and 1.5 m to the nearest instrument (if co-located) and between 2×H and 4×H from the nearest object, where H is the difference in height between the nearest obstacle and the rainfall measurement instrument. All guidelines were followed except for the H requirement due to dense vegetation within the entire observational area. In practice, the distance to the nearest object ranged between H and 4×H. No instruments where placed at sites where the nearest obstacle was ≤H away. Tipping buckets were calibrated in the field by dripping 100 mL of water (from a tripod stand) at a rate slower that 20 mmh-1 onto the instrument and recording the number of tips. The calibration experiment was repeated five times for each tipping bucket to determine the mean volume and the standard deviation (hereafter called SD error) of each tip in the field. At higher rainfall rates than 20 mmh-1, the rainfall accumulation amounts may be underestimated .

There were some issues over the course of the experiment with the various instruments that affected the availability of data. The disdrometer picked up on a oscillating signal from 20 May 2018 onward that resulted in total corruption of the data. Some intervalometers experienced water damage, particularly in storms with high rainfall intensities, which caused the instruments to go offline for certain periods of time. Two were damaged beyond repair. The tipping bucket gauges experienced no known issues. Figure  presents an overview of the data available.

present an excellent review of the exponential DSD parameterisation as well as the derivations of relevant rainfall quantities. A small summary mostly derived from their work is presented below. The raindrop size distribution in a volume of air NV(D) [mm-1m-3] is defined such that the quantity NV(D)dD represents the average number of drops with diameters between D and D+dD per unit volume of air. proposed to model NV(D) using an exponential model of the form: 1NV(D)=N0exp⁡(-ΛD)2Λ=4.1R-0.21[mm-1]3N0=8×103[mm-1m-3]. If raindrops are assumed to fall at terminal velocity, then NV(D) can be related to the DSD of drops arriving at a unit surface area, NA(D) [mm-1m-2s-1], by v(D) [ms-1], which describes the relationship between drop diameter and terminal fall velocity. NA(D) is the form of the DSD that is observed by disdrometers and intervalometers . 4NA(D)=v(D)NV(D) showed that v(D) can be approximated by a power law, v(D)=αDβ, with α=3.778 [ms-1mm-β] and β=0.67 [–] providing a close fit to the data collected on the terminal fall velocity of drops in stagnant air by for 0.5mm≤D≤5.0 mm. The mean raindrop arrival rate, ρA [m-2s-1], is defined as the integral over all drop sizes of NA(D). For the intervalometer, this is the integral between Dmin=0.8mm and ∞ since the instrument has a minimum detectable drop diameter of 0.8 mm. 5ρA=∫Dmin∞NA(D)dD=αN0∫Dmin∞Dβexp⁡(-ΛD)dD=αN0Γ(1+β,ΛDmin)Λ1+β, where Γ is the upper incomplete gamma function . presented an equation relating the rainfall rate (R) to Λ,β,α and N0, and noted that for self-consistency purposes the left- and right-hand sides of Eq. () should be equal: 6R=6π×10-4αN0Γ(4+β)Λ4+β. This equation can be inverted to give the self-consistent Λ–R relation. If the DSD is truncated (in this case at Dmin), then Eq. () is modified as follows: 7R=6π×10-4αN0Γ(4+β,ΛDmin)Λ4+β.

For the truncated DSD, the Λ–R relation must be solved for numerically. Equation () is used in conjunction with the α and β values presented by and the N0 value and a Dmin of 1 mm. The rainfall rate is varied from 0.1 to 100 mmh-1 in increments of 0.1 mmh-1 and for each value of R, Λ is solved for numerically. The approximate Λ–R power law relation is derived from a best fit of the Λ–R data points and is presented in Eq. (): 8Λ=4.06R-0.203[mm-1].

Using the α,β values and the modified R–Λ relationship in Eq. (), the rainfall rate (R) can then be calculated from the drop arrival rate (ρA). The values of α,β,N0 and Dmin are fixed and for a given value of ρA, Eq. () is used to numerically solve for Λ(ρA|α,β,N0,Dmin). The rainfall rate (R) can be estimated by re-arranging the Λ–R relation in Eq. () so that R=(Λ4.06)-4.926.

Sources of measurement error for the intervalometer are the calibration of the parameter Dmin and the measurement of ρA. Errors in the determination of Dmin affect the ρA–R relationship. Errors in the ρA measurement can result from splashing of drops from outside the sensor onto the sensor surface during high-intensity rainfall (resulting in overestimated rain rates), spurious signals from something other than rain falling on the sensor (resulting in overestimated rain rates) or from edge effects (resulting in underestimated rain rates). Edge effects occur when drops with D>Dmin land near the edges of the sensor, where the signal is damped and may not be recorded properly, especially if D is close to Dmin.

There is also model error that arises from the assumption that the DSD is adequately described by the exponential parameterisation rather than some other parameterisation. The parameterisation of the DSD with a fixed intercept parameter (N0=8000 [mm-1m-3]) and a slope parameter Λ depending on rain rate according to a power law (Λ=4.1R-0.21 [mm-1]), such as proposed by derived from stratiform rainfall in Montreal, Canada, may not be applicable in Tanzanian rainfall, which is of a largely convective nature. Model error will be accounted for by comparing three type exponential parameterisations of the DSD. Many parameterisations for the DSD have been proposed and tested in the literature, of which the most widely used are the exponential (of which the model is a special case), gamma and lognormal distributions . These other parameterisations will not be investigated as the focus of this study is to test the homogeneity assumption that underlies these models rather than compare different DSD parameterisations.

Three separate exponential parameterisation are tested. First is the self-consistent parameterisation with the Λ–R relation presented in Eq. (). The second model uses an experimentally determined value for the intercept parameter N0 over the entire observational period. This can be determined from a linear fit of the drop diameter (D) vs. the natural logarithm of NVD for the entire observational period. The experimentally determined value of N0 is 4342 [mm-1m-3] and the corresponding self-consistent Λ–R relation is Λ=3.56R-0.204.

The third model uses a power law to relate the intercept parameter N0 to the rainfall rate. found that the value of N0 can vary greatly depending on the rainfall event and even within rainfall events. These “jumps” in N0 mean that an average N0 value for the entire observational period may not be sufficient to accurately describe the DSD between or within rainfall events. In that case, the value of N0 for each rainfall event is determined from a linear fit of the drop diameter (D) vs. the natural logarithm of NVD for the rainfall event as shown in Fig. . The observed values of N0 vary from less than 2000 [mm-1m-3] to more than 15 000 [mm-1m-3] within the different rainfall events. A power law is fit to the R vs. N0 values for the different rainfall events and results in the following relation: 9N0=5310R-0.366[mm-1m-3].

The self-consistent Λ–R relation for Eq. () is Λ=4.13R-0.32. These two relations form the basis of the experimentally determined power law parameterisation. The three parameterisations as well as the observed DSD are plotted in Fig. .

It should be noted that the value of Dmin for both the intervalometer (0.8 mm) and disdrometer (1 mm) is larger than is typical for impact disdrometers. This means that many small drops will not be counted towards the rainfall arrival rate or rainfall rate, resulting in underestimates. developed a method for estimating the effects of the truncation of the DSD on two rainfall integral variables, the liquid water content and the reflectivity factor. These variables were chosen because they are a function of integer powers of the drop diameter, D3 and D6, respectively. presents a contour diagram showing the ratio of the rainfall integral variables, with a truncated DSD, to the same rainfall integral variable, without any truncation of the DSD, as a function of the integration limits ΛDmin and ΛDmax. The rainfall rate (R) is not a function of an integer power of the drop diameter, and therefore the integration limits ΛDmin and ΛDmax are approximations for this rainfall integral variable. shows that the simple approximation still gives excellent results for the rainfall rate. The approximation is used to investigate the effect of DSD truncation on the rainfall rate derived from the disdrometer data. The ratio (Rratio) of the truncated DSD rainfall rate to the rainfall rate without truncation effects is given by the ratio of Eqs. () to (). This simplifies to Eq. (). 10Rratio=Γ(4+β,ΛDmin)Γ(4+β)

Ratios of 0.8, 0.9 and 0.95 (i.e. an underestimate within 5 %) correspond to ΛDmin values of 2.8, 2.2 and 1.8, respectively. For the disdrometer (Dmin=1 mm), this means that any values of Λ>2.8, 2.2 or 1.8 will result in rainfall estimates that are within 20 %, 10 % or 5 %, respectively, of the “true” rainfall rate. These threshold values for Λ can be used to calculate threshold rainfall rates using each of the three Λ–R relations presented for the three exponential parameterisations. At the 95 % level, the threshold rainfall rates for the self-consistent , experimentally determined N0 and the N0 power law are 55.0, 28.3 and 13.4 mmh-1, respectively. At the 90 % level these values decrease to 20.5, 10.6 and 7.2 mmh-1, respectively, and for the 80 % level they decrease still further to 6.2, 3.2 and 3.4 mmh-1, respectively. All observed rainfall rates greater than the threshold rate will be affected by the effects of truncation by less than 20 %, 10 % and 5 %, respectively. During the course of the observational period of the disdrometer, 61.1 % of the total rainfall amount fell at a rainfall rate greater than 13.4 mmh-1 and 82.6 % of the total rainfall amount fell at a rainfall rate greater than 7.2 mmh-1. This means that the majority of the observed rainfall fell at rainfall rates greater than the 90 % ratio level for both the experimentally determined N0 and the N0 power law parameterisations and greater than the 95 % level for the N0 power law parameterisation. Therefore, the contribution of DSD truncation to the error in rainfall estimates of these models is expected to be minimal for the N0 power law and small (<10 %) for the experimentally determined N0 parameterisation. This is not the case for the self-consistent model which is expected to significantly underestimate the rainfall amount as a result of truncation of the DSD at 1 mm. Since the Dmin of the intervalometer is less than that for the disdrometer, the effect of truncation is even lower.

The concept of a drop size distribution depends on the assumption that at some minimum spatial or temporal scale (the primary element) the rainfall process is homogeneous. Homogeneity in a statistical sense implies that the data within the primary element follow Poisson statistics . In particular, some key assumptions must hold:

The rainfall process is stationary; i.e. it has a constant mean raindrop arrival rate.

In this study, a rainfall event is defined as a period of rainfall in which the interarrival time between consecutive raindrops does not exceed 1 h. Each rainfall event is subdivided into N patches of length τ and the fundamental Poisson assumptions can be tested on each individual patch consisting of 10 s drop count observations. A hierarchical test is used, where a patch of rainfall of length τ must pass each test before moving onto the next test and all tests must be passed in order for a patch to be classified as Poisson. The system of hierarchical tests is as follows.

Tests for stationarity. The augmented Dickey–Fuller (ADF) and Kwiatkowski–Phillips–Schmidt–Shin (KPSS) tests for stationarity are used with a p value of 0.05. The KPSS test is used to test the null hypothesis that the process is trend stationary . The number of lags considered is equal to 12×nobs10014 . The ADF test is used to test the null hypothesis that the process has a unit root . The lag is determined using the Akaike information criterion . The approach to unit root testing implicitly assumes that the time series to be tested can be decomposed into the sum of a linear deterministic trend, a random walk and a stationary error. The presence of a unit root will result in a trend in the stochastic component and the series will drift away from the deterministic trend value after a perturbation, whereas a process without a unit root will not drift after a perturbation. A more complete discussion is presented by , and . If the null hypothesis for the KPSS test is accepted and the null hypothesis for the ADF test is rejected, then the process is assumed to be strictly stationary .

Tests 1 and 2 assess the stationarity and independence assumptions of a Poisson process. Test 3 checks that the distribution matches a Poisson distribution, and Tests 4 and 5 are quality checks. The quality checks are used because the sample size over which each test is conducted is often quite small. Figure  shows a good example of a patch of rainfall that passes all of the tests and can therefore reasonably be assumed to comply with the Poisson homogeneity hypothesis.

The rainfall rate is plotted in the top panel and can be characterised by uncorrelated fluctuations around a constant mean rate of arrival, in this case 220 m-2s-1. The corresponding PDF of this patch of rainfall along with the expected PDF of a Poisson process with the same mean arrival rate is plotted in the bottom panel. The auto-correlation function of the patch is plotted in the middle panel.

The total rainfall amounts [mm] measured by the co-located tipping bucket, intervalometer and disdrometer at the main site (Pole Pole) for the longest “online” period of the three instruments are presented in Fig. . Estimates of total rainfall are derived for both the disdrometer and intervalometer from the arrival rates using the three type exponential parameterisations that were presented. For the disdrometer, the self-consistent parameterisation underestimates the co-located tipping bucket rainfall amount by more than 50%. The parameterisation with a fixed experimentally determined N0 underestimates the co-located tipping bucket rainfall amount by approximately 48%. The power law parameterisation shows good agreement with the tipping bucket record and only underestimates the co-located tipping bucket rainfall amount by approximately 4%. The results of the intervalometer are similar. The self-consistent parameterisation underestimates the co-located tipping bucket rainfall amount by more than 70%. The parameterisation with a fixed experimentally determined N0 underestimates the co-located tipping bucket rainfall amount by approximately 64%. The power law parameterisation overestimates the co-located tipping bucket rainfall amount by approximately 12%.

The coherence time or window length over which the Poisson tests were performed ranged from 2 to 22 min across all eight sites, with a typical length being in the order of 6 min. Using the tests defined in Sect. 3.4, we determined the rainfall patches that can reasonably be assumed to be representative of a Poisson process.

The proportion of rainfall patches, averaged across all the intervalometers, that do not conform with the Poisson hypothesis as well as the mean arrival rate for each group is presented in Fig. . Overall, 28.3 % of all patches can reasonably be assumed to be Poisson distributed. These are patches of stationary rainfall that exhibit no correlation between drop counts within a 95 % confidence interval, match a Poisson distribution very well and have a mean dispersion of approximately 1. The KL divergence of the Poisson patches was between 0.01 and 0.07 for all sites and only between 0 % and 7 % of all those patches had a KL divergence greater than 0.2.

Overall, 45.9 % of all patches failed the stationary tests and 7.0 % did not pass the independence test, indicating the presence of correlations between drop counts on scales as small as 2 min. It should be noted that these patches of rainfall are characterised by higher arrival rates (e.g. the rainfall that fails the independence test has a mean ρA that is approximately 4 times higher than the ρA of Poisson rain).

Of the remaining 47.1 % of rainfall patches, 17.7 % did not follow a Poisson distribution. Only a very small subset (1.2 %) did not pass the dispersion criteria and mostly because the observed variance was larger than expected for Poisson statistics. Again, these patches were characterised by higher raindrop arrival rates than the ones that passed.

Based on the previous results, it appears that most rainfall patches with higher raindrop arrival rates are inconsistent with the Poisson hypothesis. This can be clearly seen in the two middle panels (c, d) of Fig.  as well as panel (b). The time series in Fig.  clearly show that the mean rainfall arrival rate is a reasonable predictor of whether a given patch is likely to be Poisson or not. Figure  shows the total rainfall record for Pole Pole, Chole Mjini and Meremeta in the left-hand column and a single large-scale storm that was observed at all three sites in the right-hand column. This storm is characterised by sustained stratiform-type rainfall with low arrival rates and little fluctuation over time. This type of rainfall pattern is quite atypical for the rainfall record as a whole. Chole Mjini was only online for a relatively short period of time between 30 April and 8 May 2018, and this period happened to contain this atypical storm. The much longer time series for Pole Pole (panel a) and Meremeta (panel e) show that the observational record is dominated by intermittent rain events with sharp peaks and lots of convective rainfall followed by longer dry spells. Figure  also shows that most rainfall patches and in particular patches of rain with high rainfall arrival rates are typically not classified as Poisson, whereas many patches of rainfall with sustained low arrival rates (below 500 m-2s-1) are classified as Poisson. This is especially evident in panels (b) and (d) where the two rainfall peaks do not pass the Poisson tests but the lower intensity patches in between them do.

The disdrometer drop size measurements can be used to characterise Poisson and non-Poisson rainfall patches further and are presented in Fig. . The mean drop size of each of the 10 s drop counts is plotted. The larger variance in mean drop size at lower arrival rates is due to the fact that these 10 s drop counts contain fewer drops, and therefore the mean is more susceptible to random sampling effects. The trend in mean drop size with rainfall arrival rate for Poisson and non-Poisson rain is presented in the top panel. It shows again that Poisson rain is characterised by low arrival rates. No examples of Poisson rain are found at ρA>1500 m-2s-1. The data also show a positive correlation between the mean drop sizes and the arrival rate.

The bottom panel of Fig.  presents, for each data point, the test that it fails. It shows that Poisson rain is found mostly at the lower end of the arrival rate range, ρA≤500 m-2s-1. This range of rainfall arrival rates contributes little to the total rainfall; 69 % of all drops fall in this range but only contribute 16 % to total rainfall. At arrival rates between 500 and 1300 m-2s-1, the rainfall is a mixture of Poisson rain and mostly patches of rainfall that fail the χ2 test. Data that fail the χ2 test are patches of stationary rainfall with uncorrelated fluctuations about the mean. However, the data are over- or underdispersed compared to the expected Poisson value of 1 and do not match the Poisson distribution. Mostly, these data are overdispersed; i.e. the variance is greater than expected by Poisson statistics. As arrival rate increases to between 1300 and 2000 m-2s-1, a higher proportion of rainfall (in this subrange) fails the stationarity and independence tests indicating that rainfall is becoming more and more dynamic (rapid changes in the mean and correlations between drop counts). At arrival rates greater than 2000 m-2s-1, the patches of rainfall predominantly fail the stationarity test. Arrival rates greater than 1000 m-2s-1 systematically fail the independence tests and arrival rates greater than 2000 m-2s-1 systematically fail the stationarity tests. This rainfall is characterised by correlations between drop counts and fluctuations in the mean arrival rate on scales of 2 to 22 min.

Three parameterisations of the DSD were presented. Of the three, the experimentally determined power law parameterisation resulted in the best estimates of the co-located tipping bucket rainfall amount for both the intervalometer (overestimate of 12 %) and the disdrometer (underestimate of 4 %). The other two parameterisations result in large underestimates of the total rainfall amount. The poor estimate of the total rainfall amount by these parameterisations is due to the poor agreement with the observed drop size distribution, in particular at larger drop sizes. These larger drop sizes contribute most to the rainfall amount. In the case of the self-consistent parameterisation, the effect of truncation of the DSD also significantly contributes to the large underestimate of the rainfall amount. In Fig. , the three parameterisations are plotted against the observed drop sizes for the entire observational period. The parameterisation with a fixed value of N0 determined from the entire observational period fits the observed data best. This is because it was derived from the data. However, whilst the agreement with the observed data is best overall, it underestimates the larger drop sizes of D>2.5mm. The self-consistent parameterisation shows the poorest fit with the observed data and also results in the worst rainfall estimates. The self-consistent parameterisation overestimates small drop sizes and largely underestimates larger drop sizes. The experimentally determined power law parameterisation underestimates smaller drop sizes but correctly estimates larger drop sizes and overestimates very large drop sizes. Since the larger drops contribute most to the rainfall amount, the parameterisation which models this part of the DSD best, which is the N0 power law parameterisation, results in the best rainfall rate estimates. These results clearly show the importance of accurately modelling the DSD, particularly at larger drop sizes, for rainfall estimation.

The intervalometer and disdrometer had different sensors. The intervalometer has a smaller minimum detectable drop size than the disdrometer (0.8 and 1 mm, respectively). This can be clearly seen in Fig. , where the intervalometer registers higher arrival rates than the disdrometer for every observed rainfall event. The different minimum detectable drop sizes for each instrument indicate that they observe different DSDs. Therefore, parameterisations derived from the disdrometer are not optimal for use with the intervalometer. Despite this challenge, the estimate of the rainfall amount by the power law is quite reasonable and shows promise for the intervalometer concept. Furthermore, the estimate of rainfall amount using the disdrometer measurements by the N0 power law parameterisation shows excellent agreement with the co-located tipping bucket. Note that this estimate was derived by using the disdrometer in intervalometer mode; i.e. only the drop counts were used to estimate rainfall amount. These results show the potential for using intervalometers to measure rainfall; however, they also highlight the need for proper calibration of the DSD model using data from a similarly sensitive instrument from the local climate that the intervalometer will be placed in.

The results of the hierarchical tests show that the majority of rainfall tested does not comply with the Poisson homogeneity hypothesis. This is because the rainfall record is dominated by dynamic convective storms that are characterised by high arrival rates that are fluctuating on very short timescales (<2 min in some cases). This rainfall is also characterised by correlations between drop counts on these timescales. This convective-type rainfall, which contributes most to the total rainfall amount (>80 %) in this study, is almost never classified as Poisson and does not exhibit characteristics that are consistent with Poisson statistics.

Another type of rainfall is also observed in the rainfall record. This stratiform-type rain is characterised by sustained periods of consistent low-intensity rainfall that has few fluctuations in the mean arrival rate. Rainfall of this type is often classified as Poisson and appears to exhibit characteristics that are consistent with Poisson statistics, yet it contributes less than a fifth to the total rainfall amount.

How do we explain the fact that rainfall estimates based on a parameterisation which has been defined independently of a notion of scale, and therefore implies homogeneity, are quite good for both the disdrometer and intervalometer arrival rates? At the same time, the majority of rainfall does not comply with the Poisson hypothesis. Is something fishy going on?

The regime of tests implemented in this study aims to assess the validity of the Poisson hypothesis in rainfall estimation. That is, the tests are binary (yes vs. no) in nature. We find that for most of the rainfall the Poisson hypothesis is not strictly true. However, the usefulness of the Poisson hypothesis is not tested. This approach may be too short-sighted and other, more practically oriented diagnostic tools could be designed to determine the conditions under which the Poisson hypothesis is likely to result in good estimates of rainfall rates (or drop diameters). So, whilst the Poisson model may not be strictly true for the rainfall observed in this study, it does appear to be a good approximation and highly useful for estimating rainfall rates.

There is also the issue that the regime of tests used in this study is likely biased such that rainfall with lower arrival rates is much more likely to be classified as Poisson than rainfall with higher arrival rates. This is due to inherent differences between low and high rainfall arrival rates and also the failure of the χ2 goodness-of-fit test to reject the null hypothesis at small sample sizes. The majority of low-arrival-rate rainfall generally occurs in patches of rainfall characterised by reasonably stationary mean arrival rate and uncorrelated fluctuations around this mean. High arrival rates occur in highly dynamic patches of rainfall that have changes in the mean at smaller timescales than most of the patches tested in this study. Consequently, almost no rainfall with high arrival rates passes the stationarity and independence tests, whereas a very large proportion of rainfall with low arrival rates does. The χ2 goodness-of-fit test is then conducted almost exclusively on patches of rainfall with low arrival rates. These patches have small sample sizes, and the power of the χ2 test to reject the null hypothesis is limited at these sample sizes.

This is well understood in statistics and has led to various sampling criteria, such as a minimum of five observations per rainfall arrival rate class for the χ2 goodness-of-fit test . This criterion is not used in this study. However, as pointed out by , , and , rainfall conditions are changing rapidly, sometimes on temporal scales smaller than 2 min. The presence of these fine structures within rainfall would be obscured by larger sampling windows. Furthermore, sampling across such structures with different means may actually lead to increased uncertainty in the mean. noted that on the temporal resolution, some experiments will pick up the super-Poissonian variance and some will not. Similarly, at longer timescales, the auto-correlation can no longer be calculated, making it hard to define patches on which the Poisson assumptions can be tested. This increased uncertainty in the mean over an entire rainfall event would make it almost impossible to test the homogeneous Poisson hypothesis because rainfall is very rarely stationary over longer time periods.

The high acceptance rate of the Poisson hypothesis at low arrival rates observed in this study may be driven by the failure of the statistical tests to reject the null hypothesis at low sample sizes. However, despite the presence of spurious patches of Poisson rainfall, there are also many examples of patches that are likely to be genuine representations of the Poisson distribution, such as in Fig. . It is difficult to differentiate between these patches with the statistical tests given the small sample sizes. It is also not clear whether these genuine Poisson patches occur because the homogeneous Poisson hypothesis is applicable under certain rainfall conditions, e.g. consistent light stratiform-type rainfall, or whether these patches arise through randomness due to the sheer number of rainfall patches tested. This should be investigated further.

These findings highlight some limitations in how rainfall is observed with ground-based instruments. The intervalometer and disdrometer used in this study had a surface area of 9.6 and 14.5 cm2, respectively. Consequently, the number of drops that is observed is quite low, and the number of 10 s drop counts for a coherence time of 2 min is only 12. Practically, this means that statistical tests do not have enough power to reject the null hypothesis. Furthermore, increasing the length of the coherence time is not a suitable solution. The presence of these fine structures within rainfall would be obscured by larger sampling windows.

New sampling techniques or observation methodologies are needed to increase the effective sample size. One way of increasing the number of available observations is by increasing the effective surface area of the measuring instruments. This can be done by using many co-located instruments. In this way, the number of observations per window of time could be increased and the aggregation bin could be decreased to 5 or 1 s, thus increasing the number of drop counts available for testing at very short patch lengths. The number of observations could also be increased by increasing the sensitivity of the sensors to lower drop diameters. Another possibility would be to use adaptive sampling techniques, i.e. make sure each time interval has the same number of raindrops or rainfall amount, similarly to the idea proposed by . This would allow for a better interrogation of the Poisson hypothesis on the very fine rainfall structures present in convective storms.

Despite the issue with sample size and the fact that the Poisson hypothesis is likely not strictly true, the presence of significant amounts of homogeneous Poisson rain combined with the accuracy of derived rainfall estimates found in this study is compelling evidence for retaining the Poisson model. Furthermore, as was pointed out by , the observed presence of any non-clustering Poissonian structures in the rainfall conflicts with a fractal description of rain and is a good argument against abandoning the Poisson framework completely for a fractal description or some other model.