1. Introduction

Recent satellite missions to measure ocean near-surface salinity (NSS) have been mainly justified with the idea that they could be used to measure changes in rainfall more effectively than previously possible. This is the idea of using NSS as an “oceanic rain gauge” articulated by Lagerloef et al. (2008). In the original concept, this rain gauge operated on climate time scales and would be used to test the hypothesis that the global water cycle is accelerating due to a warming atmosphere (Yu et al. 2020). While the oceanic rain gauge is one of the original driving concepts for remote sensing of NSS, the information collected by these missions has found application in improving our understanding of ocean circulation and variability, the global hydrologic cycle, mesoscale circulation, ocean prediction, and ocean biogeochemistry (Vinogradova et al. 2019). The present work provides a step in this direction by developing an oceanic rainfall detector using salinity measurements.

The ability to measure NSS from space has greatly improved in the years since the 2009 launch of the Soil Moisture and Ocean Salinity (SMOS) satellite, the 2011 launch of the Aquarius satellite instrument, and the 2015 launch of the Soil Moisture Active Passive (SMAP) satellite. In that time period there have been many studies linking rainfall and NSS on much shorter time (hourly) and smaller space (a single point) scales (Henocq et al. 2010; Boutin et al. 2013, 2016; Clayson et al. 2019; Supply et al. 2020; Li and Adamec 2009). Indeed, two large field campaigns have been dedicated to studying this issue: Salinity Processes in the Upper-Ocean Regional Studies–1 (SPURS-1) in the subtropical North Atlantic (Lindstrom and Schmitt 2015) and SPURS-2 in the eastern tropical North Pacific (Lindstrom et al. 2019). One of the main results of these efforts is the understanding that NSS responds almost instantaneously to rainfall (Boutin et al. 2013), and that rain-induced freshening events are mostly confined to the upper couple of meters of the ocean in the first hours or day following the rain, depending on wind speed and preexisting upper-ocean stratification (Drushka et al. 2019; Rainville et al. 2019; Volkov et al. 2019; Reverdin et al. 2020; Thompson et al. 2019; Iyer and Drushka 2021a). The freshening is only confined to the upper few meters when salinity stratification (a stable near-surface layer) is formed, which only happens when the rain is strong or long enough and/or the wind is weak enough. Preexisting upper-ocean stratification can also impact how quickly a stable layer forms or mixes away (Iyer and Drushka 2021a). Some of these results have been incorporated into models (Jacob et al. 2019; Song et al. 2015) and used to improve satellite retrievals under rainy conditions (Meissner et al. 2018).

The clear connection between rainfall and upper-ocean freshening leads to the obvious next step, which is to attempt to use NSS to detect rainfall occurrence at an hourly time scale. The notion of using NSS to detect and quantify rainfall occurrence has been explored recently by Supply et al. (2018) using satellite precipitation and NSS measurements. They used satellite-derived salinity from SMOS and SMAP to estimate rain rate. The values were compared with satellite-derived rain rate from passive microwave data. A complicated algorithm was developed to determine the NSS anomaly from a larger-scale background. These anomalies were then collocated with SMOS and SMAP satellite passes to produce a training dataset. The determination of rain rate was done with a simple linear relationship between NSS anomaly and rain rate. Correlations between predicted and measured rain rate were 0.6–0.7. These comparisons were done using instantaneous values of NSS from SMOS and 15-min rainfall estimates from satellites both at 0.2° and 1° spatial scale. The current effort is an attempt to formulate a rainfall detection algorithm over smaller space and time scales using NSS and precipitation data from an individual mooring, and to examine the trade-offs involved in setting the criteria for rainfall detection. Specifically, the patterns of NSS in response to rainfall events that motivate our approach are illustrated in Fig. 1 via Integrated Multi-satellitE Retrievals for GPM (IMERG) and a Regional Ocean Modeling System (ROMS) simulation (Li et al. 2019) of NSS at the SPURS-2 mooring site in the Pacific Ocean. A patch of very low NSS appears to the southeast of the site on 30 June as a result of heavy rainfall. These dynamics suggest that an NSS response at short time scales may contain information about the presence or absence of a rainfall event at a given time.

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(left) Daily rainfall (mm) from IMERG for 29 Jun 2011. Color scale is at right, with no color meaning no rainfall. Also shown are ROMS simulations of NSS from (center) 29 and (right) 30 Jun 2011. The red x (left panel) and black dots (middle and right panels) denote the location of the SPURS-2 mooring. Heavy rainfall moved over the mooring, resulting in a decrease in NSS at the mooring location on 30 Jun relative to the day before.

Citation: Artificial Intelligence for the Earth Systems 2, 3; 10.1175/AIES-D-22-0009.1

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Satellite rainfall measurements have been available for many years (Huffman et al. 2007). However, there have been concerns that satellite-derived rainfall might not be as accurate in the tropics as it could be, especially at high rain rate (Pfeifroth et al. 2013) and a different method of determining rainfall may be valuable. The simple rain occurrence classification based on mooring data could be useful for validating satellite rainfall measurements.

The paper is organized as follows. Section 2 describes the SPURS-2 dataset with NSS used as an input to the rainfall detection algorithm and rainfall measurements used for validation. The proposed unsupervised classification approach is introduced in section 3. A model validation strategy is described based on verified rainfall events available at the SPURS-2 mooring. A receiver operating characteristic (ROC) curve is used to visualize the trade-off of the true positive and false positive rates of the classification approach. Section 4 summarizes the results of the proposed two-stage rainfall classification method and makes recommendations for future work. Conclusions are outlined in section 5.

2. Data

All values of salinity in this paper are given in the unitless 1978 practical salinity scale, and thus we omit the commonly used term “psu” (Millero 1993).

We use precipitation and NSS data from the SPURS-2 central mooring (Farrar 2020). SPURS-2 was a field experiment carried out in the eastern tropical North Pacific in 2016–17 (Lindstrom et al. 2019) whose purpose was to understand the connection between rainfall and NSS in a precipitation-dominated region where the intertropical convergence zone resides for part of the year (Melnichenko et al. 2019). The central mooring was deployed at 10°N, 125°W in August 2016 and recovered in November 2017 (Farrar and Plueddemann 2019). NSS was measured at ∼1-m depth by a Sea-Bird SBE-37 MicroCat instrument. Rainfall was measured by an R.M. Young Co. 50202 Self-Siphoning Rain Gauge (Serra et al. 2001; Bigorre et al. 2013). Both variables were recorded at 1-min intervals, with 1-h running mean values used for input and validation of the proposed rainfall detection method. The entire precipitation and NSS records are displayed by Farrar and Plueddemann (2019, see their Fig. 3). We focus on rain events where the hourly averaged rate exceeds 5 mm h⁻¹. Rain events over this threshold account for 62% of the accumulated precipitation at the SPURS-2 mooring over the deployment time of August 2016 to November 2017. We examined wind and evaporation data from the mooring as well but found that the explanatory power of the proposed approach did not improve when including these variables in either of the two stages of classification. Therefore, in the present work we focus solely on NSS as a predictor of rainfall.

As stated above, the NSS data are measured at a depth of ∼1 m. Technically, this is not NSS, which is commonly used as the salinity at the skin surface (Boutin et al. 2016), but more of a bulk near-surface value. As many have pointed out (e.g., Iyer and Drushka 2021a,b; Drushka et al. 2016; Reverdin et al. 2020; Drushka et al. 2019) there can be significant salinity stratification even over the top 1 m of the ocean under rainy conditions. One important result from modeling the surface layer is that the depth of penetration of salinity anomalies into the surface layer depends on the rain rate (Drushka et al. 2016), but salinity anomalies associated with rain events with high rates can penetrate at least 1 m into the ocean except under very light wind conditions. In this paper, we only study events with relatively large rain rates, >5 mm h⁻¹. We do not know what the impact might be on the proposed method of using measurements made at 1-m depth as opposed to the skin surface. Some previous studies (see references earlier in this paragraph) have shown that heavy rain events can penetrate that far. Prior studies have shown that freshwater can extend farther by several tens of meters, but this is observed in the extended time period of several hours after rainfall and in high wind speed conditions (e.g., >10–15 m s⁻¹) when vertical mixing is very strong. The purpose of this story is to capture rain detections on 1 h time scales, so our focus is on salinity at 1-m depth. All observations and model results show that rain freshening should be detectable at 1-m depth relatively soon after the rain event, depending on the wind speed. If anything, using salinity at 1 m means there will be a delay in the detection of rainfall, since it takes a finite amount of wind, diffusion, and or time for freshwater to mix down to 1 m (Asher et al. 2014; Drushka et al. 2016, 2019; Reverdin et al. 2020; Iyer and Drushka 2021a,b). Most operational and research moorings measure at 1-m depth, and our focus here is only on rain events of >5 mm h⁻¹. Thus, in this context, we continue forward using the best and most commonly available measure of in situ salinity, at 1 m, which we dub NSS.

We also make brief use of the IMERG (Huffman et al. 2019) 0.1°, daily mean satellite rainfall measurements and the level 3, 9-day V4 SMOS satellite NSS product produced by the Laboratoire d’Océanographie et du Climat: Expérimentations et Analyse Numérique (LOCEAN; Boutin et al. 2018). Rainfall from a single day is displayed in Fig. 1a. The 2011–16 mean rain rate is shown in Fig. 2a and the 2011–16 mean NSS is displayed in Fig. 2b. We considered the possibility of using IMERG satellite rain measurements, which are available in a 10 km × 10 km region around the SPURS-2 location, as an alternative to the mooring gauge. However, the distance from the location of the satellite measurement to the mooring is such that any NSS anomalies produced in response to nearby rainfall across the 10-km grid box will be unlikely to reach the mooring within 1 h of the beginning of the nearby rainfall event at ambient current speed of 10–20 cm s⁻¹ (Farrar and Plueddemann 2019). We therefore restrict our attention to the mooring rain gauge for validating our model.

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The (a) 2011–16 mean rain rate (mm h⁻¹) from IMERG daily data, and the (b) 2011–16 mean NSS from SMOS data. Color scales are included at the right of each panel. Large dots in the middle of each panel are the location of the SPURS-2 central mooring.

Citation: Artificial Intelligence for the Earth Systems 2, 3; 10.1175/AIES-D-22-0009.1

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3. Methods

The connection between NSS and precipitation has long been known and is described by the salinity balance equation and its many variants (e.g., Delcroix et al. 1996; Yu 2011). However, such models are complex with components that vary both in space and over time. Empirical models of NSS do not have this drawback, although they do not formally incorporate theoretical relationships between NSS and precipitation.

To detect rainfall occurrence at a fixed location in the ocean from hourly averaged NSS data, we propose a two-stage unsupervised learning approach using the SPURS-2 mooring as a case study. The first stage of our approach broadly identifies candidate time periods of rainfall using an empirical analysis of NSS extremes at the location of interest. For each time point under consideration, we compute the empirical distribution of NSS from a sliding time window with fixed width. If the measured NSS falls below a fixed empirical quantile, we consider the time point as potentially containing a rainfall event. The method yields consecutive rainfall candidate time sequences ranging in length from 1 to 21 h. This first-stage classifier is able to detect the majority of hourly averaged rainfall events where the rate exceeds 5 mm h⁻¹ but tends to misclassify periods of NSS recovery as containing rainfall, thus overestimating rainfall duration. Even after rainfall has stopped, NSS takes time to recover beyond the fixed quantile threshold we have set, likely due to the formation of stable near-surface fresh layers (Asher et al. 2014; Rainville et al. 2019; Volkov et al. 2019; Reverdin et al. 2020). Furthermore, this approach alone does not distinguish between salinity dips due to rainfall versus those that result from transient eddies. It may be possible to devise ways to make this distinction, perhaps by making use of simultaneous temperature data, or considering the timing of the low salinity event. We leave this work for future studies. The second stage of our approach corrects our initial estimates by incorporating this physical knowledge into the classifier by locally fitting a mechanistic model of the dynamics of NSS. Instead of considering complex dynamical models such as the Generalized Ocean Turbulence Model (Drushka et al. 2016; Iyer and Drushka 2021a), we target a statistical approach by locally fitting a simplified salinity balance equation to the candidate segments of NSS data to estimate the presence or absence of rainfall. This second model-based classifier reduces false positive detections by refining rainfall prediction within the candidate rainfall intervals identified in the first stage of estimation. The result is a time series estimating the presence or absence of rainfall over each 1-h interval.

a. Model validation

Before introducing the proposed rainfall detection method, we briefly discuss model validation, which serves two important purposes. First, it allows us to evaluate the performance of the classifier by comparing it with verified precipitation when a physical rain gauge is available. Second, it provides information for choosing auxiliary parameters for rainfall detection as new data become available.

We evaluate the proposed method by comparing the predicted rainfall patterns with those measured at the SPURS-2 site. A verified rainfall event is established when hourly averaged precipitation rate exceeding 5 mm h⁻¹ is measured within 1 h before or after the detection. Table 1 summarizes different rainfall scenarios at the SPURS-2 site and provides the corresponding classification as either true positive (TP), false positive (FP), true negative (TN), and false negative (FN). Note that we make an allowance for rainfall detection to be shifted by 1 h if an observed rainfall event is present (boldface font in Table 1). From this we can construct so-called ROC curves, which are plots of the proportion of true positive against false positive classifications under different classifier settings. The coordinates of each point on an ROC curve consist of the true positive rate, TPR = TP/(TP + FN), and false positive rate, FPR = FP/(FP + TN), for a given choice of classifier settings, which can give a sense of the overall performance of the classifier and can help to select an optimal range of settings for future classification tasks.

Table 1.

Table summarizing different rainfall scenarios and their associated classification as true positive (TP), false positive (FP), true negative (TN), and false negative (FN) both at the mooring location and nearby. The meaning of the boldface type is explained in the text.

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b. Empirical NSS quantile model of precipitation

We now describe the first stage of the proposed rainfall detection approach. Quantiles are convenient summaries of the distribution of NSS, because they can be used to study its shape (e.g., by producing boxplots) and detect outliers. An estimator, or empirical quantile, is obtained as follows. Consider hourly NSS measurements S₁, …, S_h within a fixed window of length h hours. The empirical α quantile, where 0 < α < 1, is given by

${\widehat{q}}_{\alpha }=\inf \left(q:\frac{1}{h}{\displaystyle \sum _{i=1}^{h}I}\{S_i\le q\}\ge \alpha \right),$(1)

where I{} is the indicator function. The empirical α quantile is interpreted as the smallest NSS below which a proportion α of the observations are located.

The first stage in the proposed rainfall detection approach is based on the observation that NSS values in the lower tail of the empirical density within a sliding window of fixed duration are associated with precipitation events. Figure 3a illustrates that a negatively skewed distribution of NSS, that is, one with many low outliers, is strongly associated with heavy rainfall (Supply et al. 2020; Bingham et al. 2002). We therefore propose to classify any time period as possibly containing precipitation when NSS falls below the empirical α quantile, calculated from a 2n-day window centered at the time period of interest. Centering the sliding window on the candidate time point decreases the sensitivity of the empirical distribution of NSS to short (on the order of hours) fluctuations that are likely not due to rainfall. However, one drawback is that this only works for retrospective classification, and a lagging sliding window may be more useful (although less accurate) when short-term (hourly scale) prediction is required. The optimal quantile values α and window half-width n for prediction are unknown quantities, which can be calibrated by using existing rainfall data at the location of interest or some nearby representative location. Further discussion of this issue is provided in the subsection on model validation.

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(a) SPURS-2 maximum observed hourly precipitation over a 2-day period (mm h⁻¹) vs NSS sample skewness over the same time period. Each symbol corresponds to a single 2-day period in the SPURS-2 record. Median values of skewness for maximum precipitation above and below 5 mm h⁻¹ are respectively located above and below the horizontal black line. (b) A short piece of record from 17 to 22 Sep 2016. Orange curve and right axis: measured NSS. Blue curve and left axis: measured precipitation. Gray areas represent time periods where rainfall was detected using the empirical quantile classifier with a sliding window of n = 1 day and quantile threshold α = 0.3. Band at the bottom identifies TP, TN, FP, and FN classification via the quantile method.

Citation: Artificial Intelligence for the Earth Systems 2, 3; 10.1175/AIES-D-22-0009.1

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While the quantile-based classifier is effective in identifying intervals containing rainfall, it tends to overestimate the duration of precipitation events due to the relatively slow recovery of NSS following rainfall. Figure 3b illustrates this feature of the quantile-based classifier on the SPURS-2 dataset over a period of 5 days in 2016. The gray bands represent time periods that were classified as rainfall events by the quantile-based method. Its performance is summarized in terms of TP (blue), TN (green), FP (red), and FN (yellow) classification on the horizontal stripe below the plot. Thus, a detection (gray band) can be either a TP (blue) or a FP (red), while temporal regions that are not shaded in gray can be either TN (green) or FN (yellow). The rainfall events associated with two large dips in NSS on 18 and 19 September are correctly identified by the quantile-based classifier (blue horizontal stripe segments) as possibly containing rainfall, resulting in true positive detections. However, the low NSS quantile values that led to the detections persist even after the rainfall associated with the dips in NSS has ended, leading to false positive detections (red horizontal stripe segments) immediately following each rainfall event. It is important to note that the relative length of the false positive detections is on average greater than the length of the true positive detections. This illustrates our empirical observation that although the quantile method is well suited to identifying time intervals containing precipitation, it is not able to correctly resolve the temporal location of the rainfall within each interval. We address this problem by proposing a second-stage classifier that fits a parametric model of the dip and recovery of NSS to check each interval identified by the quantile-based classifier and refine the rainfall prediction.

c. Local precipitation-forced salinity balance equation model of NSS

Next, we restrict our attention to the continuous time periods (segments) identified via the empirical quantile classifier as possibly containing rainfall, illustrated by the vertical gray bands in Fig. 3b. Because we are interested in rainfall identification at relatively short time scales, we split longer intervals into consecutive segments of length Δt or less, which is set to 20 h in our analysis. The local salinity balance equation model will be fit separately to NSS data within each time segment under the assumption that the model parameters do not vary substantially over small intervals of length Δt or less. The magnitude of the resulting estimated precipitation occurrence time series $\widehat{P}(t)$ will be used to classify a given time period as either containing rainfall greater than 5 mm h⁻¹ or not.

A simplified version of the salinity balance equation in the upper-ocean models the NSS tendency (d/dt)[S(t)] as a function of the mixed layer depth H(t), evaporation E(t), and precipitation P(t) as

$\{\begin{array}{ll}\frac{d}{dt}S(t)\hfill & =c+\frac{S(t)}{H(t)}[E(t)-P(t)],\hspace{0.17em}t\in \left(0,T\right]\hfill \\ S(0)\hfill & =S_0\hfill \end{array},$(2)

where S₀ is the initial value of NSS. The evaporation rate observed at the mooring was negligible in comparison with the precipitation rate during rain events, and we assume that the mixed layer depth is constant over 1-h time scales considered here, allowing the following simplification of the salinity balance equation over short time intervals of length Δt:

$\{\begin{array}{ll}\frac{d}{dt}S(t)\hfill & =c_1+c_{2}S(t)[c_3-P(t)],\hspace{0.17em}t\in \left(t_0,t_0+\mathrm{\Delta }t\right]\hfill \\ S(t_0)\hfill & =S_{t_0}\hfill \end{array},$(3)

where c₁, c₂, c₃, and S_t0 are unknown constants. To fit the model in Eq. (3) to data, a parametric representation is needed for the precipitation function P(t) within the small interval [t₀, t₀ + Δt]. In principle, it is possible to model P(t) as a smooth function and register NSS as functional data. However, since our current goal is binary classification, this level of resolution is not required. Furthermore, we can reduce the computational cost of our two-stage classifier by utilizing a simple, low-dimensional model of P(t), as illustrated in Fig. 4. We consider a fixed number R of component rainfall events occurring at central times τ_r, r = 1, …, R with intensities m_r over a small time interval [t₀, t₀ + Δt]. We model precipitation as

$P(t)={\displaystyle \sum _{r\mathrm{=}1}^R{m}_r{K}_h(t,\tau ),\hspace{1em}t\in [t_0,t_0\mathrm{+\Delta }t]},$(4)

where rainfall is modeled by a uniform kernel,

$K_h(t,{\tau }_r)={\rm I}\left\{\left|\frac{t-{\tau }_r}{h}\right|\le 1\right\},\hspace{1em}t\in [t_0,t_0\mathrm{+\Delta }t],$(5)

where i represents the indicator function and the half-width h corresponds to the duration of the rainfall event. To simplify estimation, we set m_r = 5 mm h⁻¹ and h = 1 h and let the number of events R vary between 0 and a fixed upper limit. This model can create longer or more intense rainfall events by allowing τ_rs to occur concurrently or sequentially. Figure 4 gives examples of this function in different precipitation scenarios, illustrating the flexibility of the model to capture precipitation events occurring at various times, of varying length, and intensity. A longer rainfall consists of multiple rainfall events whose centers are spaced 2-h time steps apart, while a more intense precipitation event may be obtained by allowing for multiple component events to be centered at the same time point.

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Illustration of the parametric model for rainfall in three different scenarios. (top left) Two separate rain events (R = 2) centered at times t₃ and t₇, each with an intensity of 5 mm h⁻¹, and (top right) R = 3 rainfall components centered at times t₃, t₅, and t₈. Since the first two components are adjacent, they create a single, longer rain event with intensity of 5 mm h⁻¹, which is separate from the rain event at time t₈. (lower left) The R = 3 rainfall components, but with two overlapping components centered at t₃, which result in an intense rainfall event of 10 mm h⁻¹, followed by another smaller rainfall centered at time t₈ with intensity 5 mm h⁻¹.

Citation: Artificial Intelligence for the Earth Systems 2, 3; 10.1175/AIES-D-22-0009.1

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Under the above assumptions, fitting the parametric model in Eq. (3) to NSS data requires estimating the vector of central times τ = (τ₁, …, τ_R)^T defining the rainfall pattern, duration, and intensity, as well as the vector of nuisance parameters c = (c₁, c₂, c₃)^T defining the local salinity balance equation dynamics (we set the initial salinity $S_{t_0}$ equal to the first NSS measurement within the time segment). Let t_obs = (t₁, …, t_N)^T represent the vector of observation times within the interval [t₀, t₀ + Δt] and let S_obs = [S(t₁), …, S(t_N)]^T be a vector of NSS measurements. Model fitting is done by minimizing a cost function consisting of two components: a weighted model discrepancy component and a penalty term that increases with the number of rainfall events to guard against overfitting. Therefore, within each time interval [t₀, t₀ + Δt] we estimate the unknown parameters τ and c by minimizing the negative penalized log-likelihood,

$\ell (\mathbf{c},\mathit{\tau })=0.5{\widehat{\sigma }}^{-2}{[S({\mathbf{t}}_{\mathrm{obs}};\mathbf{c},\mathit{\tau })-{\mathbf{S}}_{\mathrm{obs}}]}^{\mathrm{T}}[S({\mathbf{t}}_{\mathrm{obs}};\mathbf{c},\mathit{\tau })-{\mathbf{S}}_{\mathrm{obs}}]+\log (R!)-R\hspace{0.17em}\log (\lambda \mathrm{\Delta }t),$(6)

where the sample variance is

${\widehat{\sigma }}^2=\frac{1}{N-1}{({\mathbf{S}}_{\mathrm{obs}}-{\mathbf{1}}_N{\overline{S}}_{\mathrm{obs}})}^{\mathrm{T}}({\mathbf{S}}_{\mathrm{obs}}-{\mathbf{1}}_N{\overline{S}}_{\mathrm{obs}}),$(7)

ensuring that any NSS dips we observe are measured relative to the overall NSS variability within that local interval; S(t_obs; c, τ) is the solution to the local salinity balance in Eq. (3) with parameter vectors c and τ evaluated at the observation locations t_obs. The Poisson penalty on the number of rainfall events R is the probability mass function of the Poisson distribution with mean parameter λ > 0, a distribution on counts of rare events within an interval,

${\mathrm{\mathbb{P}}}_R(r)=\frac{{\lambda }^r{e}^{-\lambda }}{r!},\hspace{1em}r=0,1,\dots .$(8)

The larger we set λ to be, the larger the number of rainfall events is that are allowed within each interval without imposing substantial penalty. Selecting a small value of λ both guards us from overfitting the data and reduces the computational cost of the classification technique.

The model parameters estimated above are then used to produce a plug-in estimate $\widehat{P}(t)$ of the time series P(t) in Eq. (4). The proposed classification method uses the following rule to determine the presence or absence of a rainfall event: when $\widehat{P}(t)$ exceeds 5 mm h⁻¹, a rainfall event is said to have occurred at time t, and is said to not have occurred otherwise.

We emphasize that the role of fitting the simplified salinity balance equation to NSS data is only to extract information about the pattern of presence or absence of precipitation within a particular candidate interval. Although this model does not capture the complex physical dynamics of the ocean surface, it is flexible enough to reproduce the observed dip and recovery pattern observed in NSS data after rainfall. For this reason, the estimated parameters defining the salinity balance equation are not themselves of interest, nor can they be interpreted to extract physical meaning.

The proposed approach is summarized in algorithm 1, given in the appendix. First a number of candidate segments of NSS are identified by the quantile-based classifier under a fixed threshold α. Any continuous time interval in which NSS falls below the α quantile threshold is considered a candidate. Each candidate segment is subdivided into small local intervals of length Δt or smaller. Then the local parametric model in Eq. (3) is fit to NSS data within each segment by minimizing the log penalized likelihood in Eq. (6). This reduces the number of false positive detections as compared with the quantile classifier alone, while retaining a comparable true positive detection rate.

Figure 5 illustrates the output of the proposed two-stage algorithm at the SPURS-2 mooring for a sample time period. We also take this opportunity to remind the reader that the goal of our approach is only to determine the presence or absence of rainfall over time scales between one and 20 h. The estimation of the precipitation function P(t) within the algorithm serves only this goal, and thus the performance of the algorithm is not evaluated based on the goodness of fit of the estimated function $\widehat{P}(t)$ to the precipitation data, but rather the correct identification of the presence or absence of rainfall events. Results of the two-stage approach shown in Fig. 5 cover the time period considered in Fig. 3b. Results are shown under four different parameter settings (quantile threshold α = 0.1, 0.3 and penalty term λ = 5, 10). Solid blue and orange lines represent hourly averaged precipitation and NSS measured at the mooring. The dashed orange line represents predicted NSS, while predicted rainfall $\widehat{P}(t)$ is not shown to avoid confusion. The gray rectangles indicate the time periods where rainfall is detected by the two-stage algorithm. This occurs when 1) a candidate segment measures 1 or 2 h in length or 2) when the estimated hourly averaged precipitation $\widehat{P}(t)$ exceeds 5 mm h⁻¹. In the second case, it is useful to note that successful identification of a rainfall event results when the dip and recovery behavior of NSS is well-approximated by the model (dashed orange line). The lower horizontal stripe summarizes the performance of the proposed classification method in terms of TP (blue), TN (green), FP (red), and FN (yellow) outcomes. Relative to the performance of the quantile-based method in Fig. 3b, the number and length of false positive detections (red stripe segments) has been reduced substantially, though a small amount of false negative outcomes (yellow stripe segments) was also introduced where the parametric method failed to detect the beginning of a rain event. We also note that the interval identified by the quantile method on 17 September as potentially containing rainfall, has been correctly ruled out by the second-stage classifier, resulting in a true negative (green) detection under each parameter regime shown. This is because the dip in NSS identified by the quantile method does not resemble the type of dip that we expect to see due to dilution by a rainfall event as predicted by the salinity balance equation.

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Illustration of rainfall detection via the two-stage classifier. Rainfall detections are shown as gray bands. The lower stripe represents instances of TP, TN, FP, and FN rainfall detection. Orange curve and right axis: NSS. Blue curve and left axis: measured precipitation. The solid lines represent measured variables at the SPURS-2 mooring. The dashed orange line represents predicted NSS. Four different parameter settings are considered: quantile thresholds α = (left) 0.1 and (right) 0.3 and penalty term values λ = (top) 5 and (bottom) 10.

Citation: Artificial Intelligence for the Earth Systems 2, 3; 10.1175/AIES-D-22-0009.1

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4. Results

Performance metrics for the proposed approach at the SPURS-2 mooring location are summarized using ROC curves, which illustrate the trade-off between the true positive rate (TPR) and the false positive rate (FPR). In general, the steeper an ROC curve is, the better the classifier, since this indicates that the increase in the false positive rate required for an increase in the true positive rate is small.

The left panel of Fig. 6 shows ROC curves for the quantile-based method without the model-based correction. Each curve corresponds to a different half-width n of the sliding window used to compute the empirical distribution of NSS. Each point on an individual curve corresponds to a different value of the quantile threshold α used to determine whether to assign a rainfall event to a given NSS observation. Individual values of α that generate a specific pair of coordinates (FPR, TPR) are not shown in the plot, but can easily be obtained from the model output. As an example, consider the curve corresponding to n = 2 (blue solid line). A value of α = 0.23 produces a point on the curve with coordinates (FPR, TPR) = (0.15, 0.78). The characteristic trade-off in the TPR for a decrease in the FPR can be observed when considering α = 0.12, which generates the pair (FPR, TPR) = (0.072, 0.62).

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(left) ROC curves for the quantile-based classifier only. Each curve shown is generated using a different half-width n of the sliding window used to compute empirical NSS quantiles upon which the classification is based. Each point on a single curve corresponds to a different value of the quantile threshold α below which an NSS measurement is considered to be associated with rainfall. The closer an ROC curve is to the top and left sides of the box, the better the classifier is. (right) The dashed line is provided for comparison. It represents the ROC curve for the quantile-based method with a sliding window half-width of n = 2 days (it is the same as the solid blue line in the plot on the left). The solid lines are ROC curves for the two-stage classifier corresponding to different values of the penalty parameter λ. Each point on a single ROC curve corresponds to different length of the local interval Δt.

Citation: Artificial Intelligence for the Earth Systems 2, 3; 10.1175/AIES-D-22-0009.1

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The right panel of Fig. 6 shows ROC curves generated from the full two-stage rainfall classification method (solid lines). The dashed blue line, provided for comparison, represents the ROC curve generated with the quantile classifier with a sliding window half-width of n = 2 days, which is shown as a solid blue line in the left panel. Here, each solid curve is obtained under a different value of the penalty parameter λ. The smaller the λ, the more conservative the parametric classifier is (in terms of number of resulting rain events). Therefore, the choice of λ also drives the trade-off between TPR and FPR for the two-stage classifier. Each point of the solid lines is again associated with a different choice of α for the first stage of the method. To aid interpretation, we consider the following example. The following values can be read off the plot directly by finding the TPR coordinates for each curve under a fixed FPR of 0.1. Thus, we find that a FPR of 0.10 is associated with a TPR of 0.73 when λ = 5 (solid blue curve), 0.7675 when λ = 10 (solid red curve), and 0.80 when λ = 35 (solid orange curve). In contrast, the quantile method yields a TPR of only 0.67 (dotted blue curve).

5. Conclusions

We have proposed a two-stage approach to detect hourly averaged precipitation occurrence based on NSS data measured at a high-resolution research mooring. An empirical model first identifies candidate time intervals for rainfall, and then fits a salinity balance equation model to hourly averaged precipitation. The estimated parameters, which correspond to the number and temporal location of rainfall events within each interval predict the presence or absence of rainfall on time scales between one and 20 h. This second step corrects frequent overestimation of the duration of rainfall from the first stage. Because the salinity balance equation can reproduce the qualitative response of NSS to precipitation locally, it can be incorporated into a statistical model for the identification and quantification of rainfall. By fitting this parametric model to the SPURS-2 NSS data, we are able to predict rainfall occurrence on hourly time scales. Validation is performed when rainfall is measured along with NSS and allows the tuning of parameters defining the classification method. The results for the SPURS-2 data, illustrated in Fig. 6, are encouraging. Under different classifier settings, we are able to achieve substantial true positive rates (for instance, 0.60–0.90) at the cost of relatively low false positive rates (ranging from 0.02 to 0.30).

The idea we have explored, of using upper-ocean salinity as a rainfall detector, has been articulated in the past (Lagerloef et al. 2008; Terray et al. 2012; Hosoda et al. 2009; Supply et al. 2018; Durack 2015). Here we have restricted the scope to detecting rainfall occurrence and have put this principle on a much shorter time (hourly) and smaller space (a single point) scale than previous work. As a preliminary approach to the rainfall detector idea, we have developed a technique that simply classifies hourly average rainfall occurrence. We have used a stripped down salinity balance equation model and an empirical approach, which avoids the complications of quantifying upper-ocean processes such as vertical entrainment, horizontal mixing, and advection. All of these are difficult to determine from a single mooring. Because the proposed procedure monitors the change in NSS over 1-min time scales (which are averaged hourly for our procedure), we do not need to use a complicated procedure to determine a background value, nor is the absolute accuracy of the sensor that important. The NSS anomalies we use are simply located in the lower tail of the observed empirical distribution within a fixed window. Figure 6 explores the trade-off involved in choosing a longer or shorter time period from which to determine anomalies.

In the future, our approach can allow us to apply the same method to a much larger set of mooring data that is available either from the Global Tropical Moored Buoy Array (GTMBA), or from the many mooring time series available through the OceanSites network (Send et al. 2010). The GTMBA has ∼123 moorings, each of which measured NSS, some having 20+-year-long records (Bingham et al. 2021). In addition, there are 88 or so of them with precipitation records for validation. From this large dataset we can further refine and articulate the method we have developed.

Data availability statement.

Data used in this publication may be found online (SPURS-2 Central mooring rainfall and NSS: https://doi.org/10.5067/SPUR2-MOOR1; IMERG 1-day precipitation: https://doi.org/10.5067/GPM/IMERGDF/DAY/06; SMOS LOCEAN NSS: https://doi.org/10.17882/52804).