1. Introduction
The tropical oceanic region is of vital importance for the study of the changing climate. Perhaps the most important variable that needs accurate measurement is rainfall because of its direct association with the amount of latent heat released in the tropical atmosphere. The significance of measuring tropical rainfall is evident from the development of many algorithms used to convert satellite-received radiance from clouds or raindrops to rainfall amount (Ebert and Manton 1998). In addition, the National Aeronautics and Space Administration’s (NASA’s) Tropical Rainfall Measuring Mission (TRMM) satellite was launched in 1997, and its sole objective the observation of tropical rainfall and the study of its characteristics (Simpson et al. 1988). The primary instrument on the TRMM satellite is a downward-looking radar uniquely designed for observing tropical rainfall (Kummerow et al. 1998). It has long been understood that remotely sensed estimates of precipitation require verification using surface observations, which typically involve the use of in situ rain gauges because of the inherent uncertainties in the algorithms used in the estimation of rainfall from satellite radiance data (Schumacher and Houze 2000; Wolff et al. 2005; Wang and Wolff 2010). However, rain gauges have their own sources of random and systematic error arising from various effects, such as evaporation and rainfall undercatch. Verification of satellite-estimated rainfall over the tropics is especially difficult because of the scarcity of surface observations. Rain gauges located on islands are subject to varying degrees of topographical effects potentially biasing the measurements, so that the data may not be representative of open-ocean conditions. In addition, in situ gauges on these islands are also subject to calibration issues associated with the availability of resources to maintain them.
In an effort to provide reliable open-ocean estimates–of surface rainfall, the National Oceanic and Atmospheric Administration’s (NOAA’s) Pacific Marine Environmental Laboratory (PMEL) and the Japan Agency for Marine-Earth Science and Technology (JAMSTEC) have deployed rain gauges on many of the moored buoys that are part of the Tropical Atmosphere Ocean (TAO) and Triangle Trans-Ocean Buoy Network (TRITON), or TAO–TRITON, buoy array located between 10°N and 10°S on a number of north–south transects centered at different points along the equatorial Pacific (McPhaden et al. 1998). In addition, rain gauges are also deployed in the Atlantic on the Prediction and Research Moored Array in the Tropical Atlantic (PIRATA) buoy network (Bourlès et al. 2008) and in the Indian Ocean as part of the Research Moored Array for African–Asian–Australian Monsoon Analysis and Prediction (RAMA; McPhaden et al. 2009). These buoy-mounted rain gauges (Serra et al. 2001, hereafter S01) are R. M. Young self-siphoning, capacitance-type gauges and are mounted 3.5 m above the ocean surface on the buoys.
Since the initial deployment of the Young siphoning gauges on the buoys, several studies have been conducted using the capacitance gauge measurements to conduct validation studies of rainfall algorithms applied to measurements from different instruments onboard the TRMM satellite (Serra and McPhaden 2003; Bowman 2005; Bowman et al. 2005; Bowman et al. 2009). The spatial resolution of satellite rain estimates differs substantially from the point buoy measurements, and, therefore, the estimation of bias in the satellite rain estimates has been the primary focus of most of these studies. Thus, it is important that confidence can be placed on the accuracy of long-term averages of the buoy-deployed capacitance gauges.
While the magnitude of systematic errors associated with different land-based rain gauges can be studied using a test bed of closely spaced, varied instrumentation (e.g., Duchon and Essenberg 2001), it is extremely difficult to determine the magnitude of the error from rain gauges situated on remotely located buoys. Only one comprehensive study has been conducted to date on the nature and sources of error in the TAO–TRITON buoy capacitance gauges, and that was done by S01. The S01 study investigated the gauge error using both field and laboratory measurements. The gauge works on the principle that the capacitance between a steel probe in the gauge and the volume of collected rainwater are correlated. S01 demonstrated that capacitance gauges are not complicated devices, and noted that error in measurements arise from 1) noise created from water disturbances within the gauge, 2) temperature variations, and 3) algorithm error in converting capacitance to rain amount and the effects of siphoning out the rainwater after the gauge has filled. S01 also noted that small negative rain rates are observable in the 1-min measurements resulting from instrument noise and evaporation.
It was also noted that a substantial amount of uncertainty in the measurements result from ocean wave effects and wind-induced undercatch. Unlike a land-based station, which can use an Alter-type shield to minimize such an undercatch (Alter 1937), this is simply not feasible on a buoy-borne gauge. In addition, the effects of wave motion, which produce a temporally varying zenith angle between the gauge and the perpendicular to the earth’s surface, raises a number of questions as to the magnitude of undercatch during varying weather conditions. The effects of this motion on rain catchment by these gauges have not, to the best of our knowledge, been well studied. Thus, while the capacitance gauges investigated in S01 had similar errors in magnitude, indicating gauge-to-gauge consistency, the complexity of the estimation procedure along with the above-mentioned potential error sources strongly suggests that data from deployed capacitance gauges be compared with a different type gauge of a simple design to firmly establish confidence in the measurements from these gauges. Ideally, a test bed of simple, accurate rain gauges located on a stable platform in the close proximity of one or more deployed capacitance gauge is needed to assess the true bias in the capacitance gauge (excepting wind bias). Unfortunately, such a direct side-by-side field comparison would be not only extremely costly, but would be very difficult to design, operate, and maintain.
Martin (1964), using 20 rain gauge stations in Central America, was perhaps the first to notice a consistent relationship over a given period between the cumulative amount of rainfall and the cumulative percent frequency of days with rain. Jackson (1986), in a related study, found a similar consistent relationship in tropical regions between the number of rain days in a month and total monthly rainfall. Morrissey et al. (1994, hereafter MKM94), using island and buoy-mounted optical rain gauge data, found that the relationship between the fractional time in rain (FTR) during a month at a point in space and monthly total rainfall at that point for the tropical equatorial western Pacific becomes much stronger when higher-resolution measurements are used. MKM94 also found that this relationship proved independent of island topography and gauge separation distance, both of which varied significantly from island to island. Thus, a reasonable hypothesis was made that because this relationship appeared independent of these two variables it should be valid over the open ocean as well. A method by which this relationship can be used to estimate monthly rainfall using ocean acoustic gauges was proposed by McCollum and Krajewski (1997), which they referred to as the Fractional-Time-in-Rain method.
Given the results of MKM94, a study of the capability of the buoy capacitance gauge measurements to reproduce the FTR–monthly rainfall relationship that was found using island-based rain gauges may provide additional insight into the capacitance gauge biases (relative to the island gauges). In fact, MKM94 used this approach to determine that six experimental optical rain gauges (ORGs), mounted on six Tropical Ocean Global Atmosphere (TOGA) moorings in the western Pacific from 1992 through 1993 overestimated monthly rainfall by approximately a factor of 2 (Table 1 in MKM94). These results were supported by Bradley et al. (2001) in a reanalysis of the TOGA Couple Atmosphere Response Experiment (COARE) rainfall data, which included the data from the ORGs. These biases are one of the reasons why the use of optical gauges was discontinued and replaced with capacitance gauges in the tropical moored buoy arrays.
The approach presented in this study is similar to that of the one conducted in MKM94; however, in the current study the objective is focused on assessing biases in the capacitance gauges instead of the ORGs. Fifteen years of additional hourly data from the U.S. National Weather Service (NWS) tipping-bucket rain gauges located on six western Pacific island Weather Service Offices (WSO) are now available, together with approximately 12 yr of data from buoy-mounted capacitance gauges in the tropical Pacific (hereafter the island-based tipping-bucket gauges will simply be referred to as TB gauges). Included in this study, but not the original MKM94 work, are hourly TB data from the NWS WSO in Pago Pago, American Samoa.1 Unfortunately, there are no NWS TB gauges in the tropical Atlantic or Indian Oceans. However, to obtain an initial assessment of the regional consistency of the FTR–monthly rainfall relationship among the capacitance gauges in another ocean basin two PIRATA capacitance gauges in the Atlantic with relatively long records are also included in this study.
It has been noted by both S01 and Bowman et al. (2009) that the issue of wind bias in rain gauge undercatch has been given little attention in the literature. Undercatch is a very complicated issue involving wind dynamics, pressure forces, gauge shape and orientation, and other factors. In this study it will be assumed that wind biases among the TB and capacitance gauges are the same, though admittedly this assumption is very speculative. Given the importance of this issue, the reader is encouraged to review the work of Duchon and Essenberg (2001) and Nespor and Sevruk (1999).
2. Data
a. TAO–TRITON buoy capacitance gauges
The Autonomous Temperature Line Acquisition System (ATLAS) buoys that comprise the bulk of the TAO–TRITON array in the equatorial Pacific were developed, tested, and deployed by PMEL. Of the 67 TAO–TRITON sites, 37 are routinely deployed with capacitance gauges (25 on ATLAS moorings and 12 on TRITON moorings). In addition, capacitance gauges are mounted on all PIRATA and RAMA buoys in the Atlantic and Indian Oceans. A detailed technical explanation of how the capacitance gauges operate is given by S01. A short synopsis of the rainfall measurement physics is given in the appendix. However, because some of the operating details are particularly pertinent to the present study a short summary of their operating design will be given below.
In this study we will focus mainly on the Pacific where gauges were initially deployed from 1995 through 1998. Since then virtually all of the gauges have gaps of missing values. Most of the gauge data are of “default” quality, which indicates that gauge calibration was applied only during predeployment or postdeployment. For this study only data from gauges with complete months, that is, with no missing data in a month and no “accumulated values,” were used. Accumulated values are defined as data representing rainfall that occurred over a period of several successive hours, but not necessarily continuously over that period. Thus, information pertaining to whether it rained or not in a particular hour in this accumulation period is lost. Gauges with the largest number of complete months in their record, located along the 165°E transect, were selected for this study. One additional buoy gauge located at 5°N, 140°W was selected for this study because of its proximity to the equator and its reasonably long record length. Figure 1 shows the TAO–TRITON gauges in the Pacific selected for this study. Most other capacitance gauges have much shorter records. As mentioned previously, two buoy capacitance gauges in the Atlantic PIRATA array were tested to observe any regional differences in gauge performance and climatological differences on the FTR–monthly rainfall relationship between the two oceans. Unfortunately, the capacitance gauges from the RAMA array in the Indian Ocean generally had record lengths less than 2 yr and, thus, were considered unsuitable for inclusion in this study. A capacitance gauge is shown deployed on a buoy in Fig. 2.
TAO–TRITON buoy locations that have capacitance gauges (solid squares). The eight Pacific buoys with capacitance gauges used in this study (solid squares surrounded by a rectangle). Island TB gauges used in this study are located by an ×. Note buoys located at 8°N, 125°W; 8°N, 110°W; and 8°N, 180°W were only equipped with gauges for a few years and, thus, have limited records.
Citation: Journal of Atmospheric and Oceanic Technology 29, 6; 10.1175/JTECH-D-11-00171.1
A capacitance gauge shown mounted near the top of this TAO–TRITON buoy.
Citation: Journal of Atmospheric and Oceanic Technology 29, 6; 10.1175/JTECH-D-11-00171.1
b. Island TB rain gauges
During the mid-1980s standard 8″ NWS TB rain gauges were deployed at most of the Pacific island WSOs. These islands include those located in the Federated States of Micronesia (FSM), the Republic of the Marshall Islands (RMI), Palau, the Commonwealth of the Northern Marianas Islands, Guam, Wake Island, Hawaii, and the South Pacific American territory of American Samoa. These gauges use a two-sided tipping-collector apparatus designed to tip after 0.254 mm of rain falls. The number of tips during a given hour is recorded, resulting in a resolution of 0.254 mm h−1. The data from these gauges are stored at NOAA’s National Climatic Data Center (NCDC) in dataset TD-3240 and were provided via the World Data Center for Meteorology at NCDC (see http://www.ncdc.noaa.gov/oa/wdc/; Hammer and Steurer 2000).
The TD-3240 dataset contains several Pacific island TB gauge sites, with nearly complete records starting around 1984. Given the need for long record lengths and for the sites to be located in the equatorial Pacific region, an inspection of TD-3240 determined that six island WSO sites had sufficient TB data suitable for this study. The sites are shown in Fig. 1 and detailed in Table 1. As with the capacitance gauges only those months with no missing, accumulated, or bad records were used in this study. An example of a NWS TB gauge located at the Majuro WSO is shown in Fig. 3.
Island NWS WSO stations used in this study. Marshall Islands: MI, Federated States of Micronesia: FSM.
The tipping-bucket gauge on Majuro. (Courtesy of R. White, director of the Majuro WSO.)
Citation: Journal of Atmospheric and Oceanic Technology 29, 6; 10.1175/JTECH-D-11-00171.1
3. The FTR–monthly rainfall relationship
However, the accuracy of the FTR estimator depends largely upon the time resolution of the data used in its estimation. Using discrete data, the FTR within T is generally overestimated because any given interval with rain will be flagged as “raining,” even if it rained only during a portion of that interval. This produces a positive bias in 
One effect of this bias in the FTR estimator can be observed using hourly rainfall accumulations to estimate monthly rainfall in Majuro (Fig. 4). Note that the relationship appears to be nonlinear. While there is little physical justification for the fitted power-law relationship over another nonlinear relation, MKM94 showed that a power-law relation consistently explained a larger amount of variance than did a linear relationship using TB gauges at five other WSOs in Micronesia. One question that comes to mind is “why a nonlinear relationship”? It would first appear from (3) that if 
The power-law least squares fit to the TB gauge located on Majuro atoll.
Citation: Journal of Atmospheric and Oceanic Technology 29, 6; 10.1175/JTECH-D-11-00171.1
By assuming that sequential hours with rainfall represent a single event, the average rainfall event duration can be plotted against the average single-hour rainfall amount associated with a given event duration. Using hourly TB data from Majuro2 WSO it can be observed from Fig. 5 that rainfall intensity per hour does indeed increase with event duration. The upper portion of the plot shows the deviations of the values about the best least squares fit curve giving a rough indication of the quality of fit, with symmetric deviations suggesting a reasonable fit. The increase in the variance of these values about the curve with duration is most likely due to sampling error. In other words, there were fewer long-lasting rain events resulting in increasing uncertainty in the average hourly rainfall estimates with duration. Because it is quite likely that rainfall is continuously falling during long-duration, heavy rainfall events, this implies that the bias in 
The average rainfall accumulation per hour vs rainfall events defined by consecutive hours with rain at the Majuro atoll WSO. The upper portion of the graph shows the deviations of the values from the best fit function.
Citation: Journal of Atmospheric and Oceanic Technology 29, 6; 10.1175/JTECH-D-11-00171.1
4. Constructing a useful fractional time raining estimator
To investigate any biases between the island TB and buoy capacitance gauges it is important that the FTR estimator be relatively unbiased between the two gauge types. The results in section 3 suggest that this could be done if the gauges could be made to have the same measurement resolution. Of course, this cannot be done exactly because the TB and capacitance gauges have resolutions of 0.254 mm h−1 and approximately 0.2–0.3 mm h−1 (S01), respectively. Note that while these two resolutions are similar, any small differences may classify an hourly measurement from one gauge to be “raining” while the other may be “nonraining” under the same conditions. In addition, of particular concern in constructing an unbiased FTR estimator is the noise amplitude in the capacitance gauges. As noted by Bowman et al. (2009) even after using a Hamming filter applied to the 1-min data, noise is not completely removed from the 10-min values. Thus, any positive 10-min value within a single hour makes that hour raining, even if it is noise. This will contribute to a high bias in the FTR estimator. It is generally assumed that any 1-min values less than or equal to the absolute value of 1 mm h−1 is noise (as in S01). This leaves considerable uncertainty as to the actual resolution of the capacitance gauges because noise is considered random with an unknown or assumed distribution about zero. It may initially be thought that noise in the capacitance gauges could be removed via statistical means. However, Bowman et al. (2009) showed that the amplitude of the noise varies with rainfall amount and cannot be removed in any “simple fashion.” Thus, a direct comparison of the FTR–monthly relationship between the capacitance and island TB gauges using Eq. (3) is likely to be contaminated, given the uncertainty resulting from resolution differences and the noise magnitude in the capacitance gauges.
One way to account for these problems is to raise the threshold hourly rain amount to a common value for which a particular hour can be considered raining for both gauge types. This would tend to filter out light rain events in the FTR estimator while also reducing the effects of noise. Studies by Garstang (1966) and Riehl (1967) and a summary of these results by Ramage (1995) showed that a disproportionate amount of tropical monthly rain falls in approximately 10% of the time with rain in a month. This, together with the above results, suggests that light rainfall events lasting less than 1 h contribute little to the total monthly rainfall. Thus, while raising the threshold value will underestimate the total monthly rainfall, the bias in 
5. Analysis and results
Beginning with the maximum island TB measurement threshold resolution of 0.254 mm h−1 for a value of x in (4), this value was increased until the scatter diagrams of FTR versus monthly rainfall totals for the TB gauges became clearly linear. As discussed above, the degree of nonlinearity is an indicator of the bias in the FTR estimator. After trying various values a threshold of x = 3.81 mm h−1 (or 0.15 in. h−1) was selected. While this threshold produces minimal bias in the FTR estimate in a month, as evidence by the linearity of the resulting relationships, it also most certainly exceeds the noise values for the capacitance gauges and, thus, can be applied equally to both gauges so that comparisons can be confidently made with the knowledge that P[r(t) > 0 | ra > 3.81 mm h−1] ≈ 1.
Using (4) as an estimator of FTR the analysis was carried out on both the island TB and the buoy capacitance gauge measurements. The results are shown in Figs. 6–8, and the statistics associated with the least squares linear fit are given in Tables 2 and 3. Table 4 shows the results for two capacitance gauges with relatively long record lengths selected from the PIRATA array in the Atlantic for comparison purposes.
Graphical results from the six Pacific island WSO stations with TB gauges. The upper portion of the graph shows the deviations of the values from the best-fit linear function.
Citation: Journal of Atmospheric and Oceanic Technology 29, 6; 10.1175/JTECH-D-11-00171.1
As in Fig. 6, but using the data from the buoy transect along 165°E.
Citation: Journal of Atmospheric and Oceanic Technology 29, 6; 10.1175/JTECH-D-11-00171.1
As in Fig. 6, but using data from (top) two widely space buoys in the Pacific and (bottom) two gauges from the PIRATA array in the Atlantic.
Citation: Journal of Atmospheric and Oceanic Technology 29, 6; 10.1175/JTECH-D-11-00171.1
Statistics for a linear fit to the fraction of raining hours to the monthly totals for the island stations. Error limits are taken at the 95% confidence level.
Statistics for linear fit to the fraction of hours raining and the total monthly rainfall for the buoy capacitance gauges. Error limits are taken at the 95% confidence level. The error bound about the mean is computed from twice the standard deviation.
As in Table 3, but for the two PIRATA capacitance gauges located in the equatorial Atlantic.
The first observation one makes is that the relationship is quite linear in all cases, as was our objective. The residual plots in each graph show a quasi-symmetric distribution of points about the regression line, indicating a reasonably good fit. In addition, the scatter diagrams do not appear to be heteroskedastic because the variance about the regression appears quite constant with increasing monthly rainfall. Note that the points are likely to be somewhat dependent, lessening the number of the actual degrees of freedom. Thus, the errors bars about the statistics, representing the 95% confidence level, are likely to be underestimated.
Comparisons of the statistics shown in Tables 2 and 3 and the graphs provide some very interesting results. The slope values among the island TB gauges are very similar and only vary by 4.5% about the overall mean, that is, (412/9219) × 100 ≈ 4.5% at the 95% level of confidence. The intercept values are close to zero, which implies that the contribution to monthly rainfall from hours with accumulations less than 3.81 mm h−1 is insignificant for the gauges tested, supporting the studies cited earlier. The physical meaning for the slope is the time average of the conditional rain rate at an instance in time when ra > 3.81 mm h−1. While the slope values are given in millimeters per month, it is probably easier to think of the slope as a measure of gauge catchment of rainfall per hour given that ra > 3.81 mm h−1. Thus, the slope values are also given in Tables 2–4 in units of millimeters per hour, assuming 720 h month−1. Given the selected value for threshold x in (4), the linearity of the relationship implies that 
Scatter about the regression lines is due to recording errors, random gauge error, and some variation of the actual 
The slope values for the buoy capacitance gauges are approximately 9123.0 ±~854 mm month−1 (Table 3) and, as with the TB gauges, are quite similar to each other with an approximately 9% variation about the mean, suggesting gauge-to-gauge consistency. The mean TB and capacitance gauge slope values are virtually identical at 12.9 and 12.7 mm h−1, respectively. It is noted that the number of useable months for each capacitance gauge is approximately one-half of the island TB gauges, resulting in fewer degrees of freedom. The slope values for the capacitance gauges are not statistically significantly different from each other (at the 99% confidence level), and moreover are also very similar to those from the TB gauges. The very small standard deviation values about both these mean values strongly suggest that all the tested gauges are relatively unbiased among the same gauge type and between the two different gauge types. The correlation values between the 
These results may be regionally dependent. Table 4 and Fig. 8 both show the results of two capacitance gauges from the Atlantic PIRATA array. A comparison of the slope values between these two gauges indicate that they appear to be somewhat higher than those in the Pacific although the differences are not statistically significantly different. Considering that these gauge’s slope values are close to each other, one can summarize that these two gauges are consistent with the Pacific gauges. Obviously more work is required to establish whether this is the case.
6. Conclusions
An examination of the FTR–monthly rainfall relationship determined from western Pacific WSO TB gauges together with a selective number of buoy-mounted capacitance gauges strongly suggests that the two types of gauges are relatively unbiased with respect to each other. Of course, both gauge types are undoubtedly biased low to some degree from wind effects.
Raising the threshold value above which an hour was considered raining not only linearized the relationship, but also supported past conjectures that light rainfall contributes very little to tropical monthly rainfall in regions where rainfall is relatively high. These results are especially significant when one considers the MKM94 study, where, using a similar methodology, the early buoy-mounted ORGs showed a considerable bias compared to the same island TB gauges. The results of the TB gauges compare closely with what MKM94 found. Given an approximately 150% increase in the number of data since 1994 the FTR–monthly rainfall relationship also appears to be temporally homogeneous.
It is noted again that studies of wind-induced biases in in situ rain gauge measurements are needed to assess the magnitude of these biases. The importance of accurately estimating the true wind bias in open-ocean measurements cannot be overstated for the development of accurate satellite rainfall algorithms. Nespor and Sevruk (1999), using a numerical simulation of the airflow around different size gauge catchment areas, found that rain undercatch is a function of the wind speed, rainfall rate, and raindrop size distribution. While the first two variables can be measured at high temporal resolution, the latter variable will prove to be most difficult to accurately estimate because it varies both temporally and spatially.
Acknowledgments
The authors would like thank Mr. Reginald White, Director of the Majuro, U.S. NWS WSO for providing important information on the tipping-bucket gauge located in Majuro. The tropical moored buoy arrays, as well as the work of MJM and HPF, are supported by NOAA’s Climate Program Office. This work was funded by NOAA’s Climate Observations & Monitoring Program Grant 105-140500.
APPENDIX
R. M. Young Capacitance Gauge
The R. M. Young capacitance gauges have an 11.3-cm-diameter collecting area and a 100-cm2 catchment volume. A stainless steel Mylar-covered tube is located in the center of the catchment cylinder. The Mylar acts as a dielectric between the steel probe and the water. As water raises in the catchment tube the surface area of rainwater in touch with the Mylar increases, thus increasing the capacitance between the water and the inner tube. Capacitance is then converted to frequency and then to digital 1-min counts using onboard circuitry. An algorithm is used to convert these counts into 1-min rain volume (S01). One-minute rainfall totals are assessed by differencing 1-min accumulations. As S01 note, the amount of water in contact with the Mylar constantly varies not only due to rainfall, but also water motion induced by buoy motion. This creates a certain amount of noise, which is minimized, but not eliminated; using a 16-min Hamming filter the 1-min data stream was applied, producing 10-min averages. Once the data are recovered they are flagged for errors and missing data. The gauge self-siphons when the water volume reaches the 500-ml level in the collecting container. The siphoning takes approximately 30 s. S01 estimated the noise to be about zero for 1-min measurements to have an amplitude of roughly 1 mm h−1 . They also estimated the magnitude of error resulting from temperature, water volume sensitivity, and evaporation, and the data are available from the PMEL website (http://www.pmel.noaa.gov/).
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Pago Pago is the only NWS WSO located in the Southern Hemisphere, and its data were not incorporated in the original MKM94 study.
Majuro was selected because it is a very low-elevation atoll with a large lagoon and extremely small landmass as compared to the size of the lagoon. Its topography, in all likelihood, allows the gauge measurements to be somewhat representative of open-ocean conditions.
