A new type of rainfall sensor (the intervalometer), which counts the arrival
of raindrops at a piezo electric element, is implemented during the Tanzanian
monsoon season alongside tipping bucket rain gauges and an impact
disdrometer. The aim is to test the validity of the Poisson hypothesis
underlying the estimation of rainfall rates using an experimentally determined
raindrop size distribution parameterisation based on

Africa, and particularly Sub-Saharan Africa, is one of the most vulnerable
regions in the world to climate change

For example, a recent review of weather index insurance for smallholder
farmers (some of the world's poorest people) found that the sparsity of
ground-based weather stations is a large challenge for insurers in Sub-Saharan
Africa

Satellite retrievals face another issue for areas with a lack of ground-based
data for validation. Since both active (radars) and passive (radiometers or IR
sensors) onboard sensors do not measure rainfall directly, information about
the microstructure of precipitation is needed in order to develop robust
rainfall retrieval algorithms. Information about the drop size distribution
(DSD) in particular is needed to retrieve rainfall rates (

An assumption that is seldom explicitly mentioned in the presentation of these
parameterisations is the homogeneity assumption

To overcome these difficulties, two different approaches have been
proposed. Some researchers

The aim of this study is to formally assess the adequacy of the homogeneous
Poisson hypothesis and its importance in deriving rainfall estimates from
ground-based measurements in a tropical climate. The intervalometer, a new
kind of inexpensive rainfall sensor, is introduced and tested for its
suitability in providing ground-based rainfall estimates in Sub-Saharan
Africa. To this end, nine intervalometers were deployed over a 2-month period
during the Tanzanian tropical monsoon. The

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

In total, eight sites were selected along the southern coast of Mafia Island,
Tanzania. Figure

The eight intervalometer sites on Mafia Island, off the coast of Tanzania. Each site contains one intervalometer. Pole Pole also had a co-located tipping bucket and impact disdrometer. MIL1 also had a co-located tipping bucket.

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

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

Record of the time periods during which the intervalometers collected data for each intervalometer site and the total rainfall amount [mm] from the tipping bucket at Pole Pole.

For the truncated DSD, the

Using the

Sources of measurement error for the intervalometer are the calibration of the
parameter

There is also model error that arises from the assumption that the DSD is
adequately described by the

Three separate exponential parameterisation are tested. First is the
self-consistent

The natural logarithm of

The third model uses a power law to relate the intercept parameter

The observed DSD over the entire observational period of the disdrometer is compared
to the three parameterisations of the DSD. The self-consistent

The self-consistent

It should be noted that the value of

Ratios of 0.8, 0.9 and 0.95 (i.e. an underestimate within 5

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

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

The number of raindrops arriving at the surface over non-overlapping time intervals is statistically independent.

The number of raindrops arriving at a surface during a time interval [

The probability of more than one raindrop arriving at a fixed surface
over a time interval [

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

A patch of rainfall, with a coherence time of 20

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

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

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.

The total rainfall amount [mm] observed by the co-located tipping bucket,
intervalometer and disdrometer at the main site (Pole Pole) for the longest
“online” period of the three instruments. Panel

The coherence time or window length over which the Poisson tests were
performed ranged from 2 to 22

The percentage of all rainfall patches, measured by the intervalometer, that fail each of the hierarchical tests as well as the mean rainfall arrival rate for each group. The presented data are an average, weighted by the length of each patch, across all of the intervalometer sites.

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, 45.9

Of the remaining 47.1

The occurrence of Poisson rain in the rainfall records of three sites. The complete observational period is plotted in the left-hand column and a single large-scale storm which was common to all three sites is plotted in the right-hand column. The observational period in Meremeta and Pole Pole is much longer than that in Chole Mjini due to an instrument failure at that site.

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.

Trends in mean drop size for Poisson and non-Poisson rain are presented as
well as the percentage of drops that fail each of the Poisson tests. Panel

The disdrometer drop size measurements can be used to characterise Poisson and
non-Poisson rainfall patches further and are presented in
Fig.

The bottom panel of Fig.

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

The intervalometer and disdrometer had different sensors. The intervalometer
has a smaller minimum detectable drop size than the disdrometer (0.8 and
1

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 (

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

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

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.

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

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

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

This research leads to the following conclusions:

The majority of rainfall and almost all the convective-type rainfall,
which contributed most to total rainfall amount in this study, did not exhibit
characteristics that are consistent with the Poisson hypothesis. Patches that
complied with the Poisson hypothesis were characterised by low mean rainfall
arrival rates during periods of sustained stratiform-type rainfall. No
examples of Poisson-distributed rain patches, with

There appear to be genuine examples of Poisson rainfall that occur during consistent light stratiform-type rainfall conditions. However, small sample sizes were an issue in this study and may have resulted in the statistical tests failing to reject the null hypothesis of Poisson at low arrival rates for many rainfall patches, making it hard to differentiate between genuine and spurious Poisson rainfall. Increasing the patch length is not a suitable solution to increase the number of observations. Fine structures are observed in rainfall at very small scales, and sampling across such structures with different means may actually lead to increased uncertainty in the mean. New sampling techniques or observation methodologies are needed to increase the effective sample size.

Total cumulative rainfall estimates derived from the disdrometer drop
counts with the best-performing

Total cumulative rainfall estimates derived from the best-performing

It is possible to retrieve rainfall rates using an intervalometer. The intervalometer principle shows potential for providing ground-based rainfall observations in remote areas of Africa. The main advantage of this instrument is its low cost. However, further improvements are needed to make the sensor more robust, as several instruments were damaged by water during this study. The results also show that it is necessary to verify the DSD model with observed drop size data from within the local climate with an instrument that has the same sensitivity as the intervalometer.

Data and corresponding Python scripts for analysis are available from the corresponding author upon request.

RH and NvdG contributed to the designs of the disdrometer and intervalometer. NvdG and MCtV were responsible for acquiring funding. DdV and NvdG designed the experiment. Data were collected by DdV. Analysis was performed by DdV, with contributions from NvdG, MCtV and MS. DdV prepared the draft of the manuscript with contributions from all the co-authors.

The authors declare that they have no conflict of interest.

The opinions expressed in the document are of the authors only and no way reflect the European Commission's opinions. The European Union is not liable for any use that may be made of the information. Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The following are acknowledged (in no particular order) for their work in developing the disdrometer and intervalometer: Stijn de Jong, Jan Jaap Pape, Coen Degen, Ravi Bagree, Jeroen Netten, Els Veenhoven, Dirk van der Lubbe-Sanjuan, Wouter Berghuis, Rolf Hut and Nick van de Giesen. Special thanks are given to the hotels located on Mafia Island (Didimiza Guest House, Meremeta Lodge, Eco Shamba Kilole Lodge, Kinasi Lodge, Pole Pole Bungalows and the Mafia Island Lodge) and Chole Island (Chole Mjini Treehouse Lodge) for allowing access to their land and providing support in setting up the experiment.

This research has been supported by the European Community's Horizon 2020 Programme (grant no. 776691).

This paper was edited by Piet Stammes and reviewed by Remko Uijlenhoet and two anonymous referees.