1. Introduction
Global tropospheric water vapor amounts are widely expected to increase in a warming climate, leading to a positive feedback arising from the enhanced greenhouse effect (Held and Soden 2000). The water vapor feedback is the strongest global feedback in general circulation models (GCMs) (Soden and Held 2006), and observations, model results, and theory show a consistent and strong link between increases in surface temperature and increases in water vapor mixing ratio (WVMR) (e.g., Dessler and Davis 2010).
Our understanding of the water vapor feedback and, more generally, the humidity of the free troposphere has been substantially clarified by the so-called last-saturation concept, which approximates the WVMR of air by its saturation value when it was last in a cloud (Pierrehumbert et al. 2006; Hurley and Galewsky 2010b). This concept has proved very useful, especially for understanding the controls on subtropical humidity, which is influenced by far-field processes and nonlocal mixing (Galewsky et al. 2005; Cau et al. 2007; Wright et al. 2010). To first order, the humidity of the subtropics can be understood in terms of drying via condensation to the coldest point encountered by air parcels traveling to the subtropics and the subsequent moistening that may occur. Changes in the balance between cold-point drying and subsequent moistening can arise from a variety of processes and are difficult to disentangle.
Many studies that make use of the last-saturation concept rely in part on reanalysis or GCM output (Pierrehumbert 1998; Dessler and Sherwood 2000; Galewsky et al. 2005; Cau et al. 2007). Given the importance of the water vapor feedback, it would be useful to identify an independent dataset that could be used to constrain the processes that influence free-tropospheric humidity. In principle, measurements of the stable isotopic composition of water vapor could provide such a dataset. The most readily measured isotopologues of water vapor are HDO and 
Since 2012, we have been making measurements of water vapor isotopic composition at the Atacama Large Millimeter Array (ALMA) on the Chajnantor Plateau (Fig. 1), an extremely arid site at an elevation of 5 km in the subtropical Chilean Andes (latitude 23°S). We now have nearly continuous data for two dry seasons, from the years 2012 and 2014, providing us with the first dataset suitable for more fully realizing the potential of water vapor isotopologues to constrain the dynamics of subtropical water vapor.
Location map showing the Chajnantor Plateau (black star) and topography (gray shaded). Longitude in degrees west.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0160.1
Here we present a new stochastic model of water vapor isotopic composition that can be applied to large datasets. The best-fitting model parameters for a given dataset are found using a genetic algorithm approach. We will show that the model can generate an excellent fit to the large dataset we have collected on the Chajnantor Plateau and that it can provide constraints on the changes in the relative balance between drying and moistening processes that govern the interannual variability of humidity.
2. Methods
For background material on the stable isotopic composition of water vapor and on the Chajnantor Plateau site, the reader is referred to our previous publications based on this dataset (Samuels-Crow et al. 2014; Galewsky and Samuels-Crow 2015, 2014) and the citations therein.
Mixing ratio and isotopic composition were measured using a Picarro L2130 cavity ringdown spectroscopy (CRDS) analyzer. Details on the instrument and the standardization techniques used are provided in Samuels-Crow et al. (2014) and Galewsky and Samuels-Crow (2015, 2014).
The 1σ uncertainty in 5-min averages of measurements increased at lower mixing ratio but was 5‰, 0.4‰, and 6‰ in δD, δ18O, and deuterium excess (d-excess), respectively, at 110 ppmv and 0.4‰, 0.1‰, and 0.5‰ in δD, δ18O, and d-excess, respectively, at 2750 ppmv.
The water vapor isotopic analyzer we originally deployed on Chajnantor suffered a hardware failure in February of 2013 and was replaced in July of 2014 with an identical model, but with an updated water vapor retrieval to correct for the bias in water vapor concentration measurements reported in Samuels-Crow et al. (2014). We compared the mixing ratio reported by this analyzer with the humidity measurements from the ALMA operations site and confirmed that the mixing ratio bias had been corrected. The Picarro mixing ratios were within about 2% of those measured at the array operations site, a difference that we determined to be insignificant for the analysis presented below. The replacement isotopic analyzer was subjected to the same concentration dependence correction and standardization as our original system, so we believe that the data obtained by the two instruments are comparable.
We used the ERA-Interim dataset (Dee et al. 2011) from the European Centre for Medium-Range Weather Forecasts (ECMWF) for the temperature profile used in the Rayleigh distillation calculation and for diagnosis of interannual variability in upper-tropospheric temperature, middle-tropospheric subsidence, and storm-track activity.
3. The stochastic model
In this section, we present the stochastic model and describe the techniques we use for fitting the model parameters to data. In section 4a, we then use the model in a forward modeling context to explore the model’s sensitivities and then in section 4b and beyond, we actually fit the model parameters to the data. Our guiding philosophy in developing this model was to find the simplest set of processes that fits the data. Most of the process representations we use are fairly simple, and one can easily envision more complex formulations. While such formulations may be fruitful for future study, here we focus on the most parsimonious model we could identify that fits the data.
a. Parameters
Our model generates a large number of synthetic measurements by subjecting each synthetic measurement to Rayleigh distillation to a random cold-point temperature and ice supersaturation (e.g., Jouzel and Merlivat 1984). Each point is then moistened by a random mixing fraction with a moisture source. Additional parameters are the boundary layer isotopic composition prior to ascent and the isotopic composition of the moisture source. The physical parameters and their associated model parameters are summarized in Table 1, and the model is illustrated in a cartoon in Fig. 2.
Physical and model parameters.
Cartoon illustrating the physical parameters of the model and the sequence of events that generate a synthetic measurement at the CRDS site.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0160.1
The initial water vapor isotopic composition prior to ascent is parameterized as an initial water vapor δD and d-excess (“A” in Fig. 2). Explicitly parameterizing the d-excess allows us to directly restrict the d-excess to plausible values. The 
Rayleigh distillation (“B” in Fig. 2) is initiated at a random lifting condensation level (LCL) between 900 and 775 hPa. Any fixed LCL in this range can fit the data, and no variability in the LCL is required. Rayleigh distillation occurs along the sounding shown in Fig. 3, which is a tropical sounding retrieved from ERA-Interim averaged over August–November 2012 between longitudes −130° and −120° and latitudes 0° and −10°. The most important characteristics of the sounding are the initial condensation temperature (set by the LCL) and final condensation temperature (set by the final saturation mixing ratio 
Sounding used for the Rayleigh distillation component of the stochastic model. Average tropical Pacific sounding from August through November 2012 from ERA-Interim.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0160.1
While the last-saturation point is explicitly parameterized as a mixing ratio 
Studies of the global distribution of supersaturation have shown that an exponential distribution fits satellite data (Gettelman et al. 2006). We found that no exponential distribution of supersaturation could fit our isotopic data. The d-excess values were especially sensitive to the supersaturation, and we found that a normal distribution for 
The moistening after last saturation (“D” in Fig. 2d) is as important as the last-saturation process itself for fitting the observations. The moist source (“E” in Fig. 2) is parameterized by normal distributions of the humidity 
b. Fitting the model to data
We used genetic algorithm techniques to seek the set of model parameters that minimized the mismatch between the observations of mixing ratio and isotopic composition and the simulated dataset. Genetic algorithms are a class of optimization techniques based on a natural selection–like process inspired by biological evolution. The algorithm modifies an initial population of individual solutions and at each step the algorithm selects individuals from the current population as parents to produce the children for the next generation. Over successive generations, the population evolves toward an optimal solution as determined by a fitness function (Beasley et al. 1993a,b).
A candidate solution consists of a genome that spans the parameters of the problem. The initial population is generated by a random-number generator and can be constrained as required by the problem. The fitness of each individual is determined by a fitness function, and individuals with higher fitness are assigned a greater probability of reproducing. A mating pool is generated and offspring are generated from each mating pair by swapping randomly selected portions of each parent’s genome. The offspring are then subjected to random mutation. The average fitness of the next generation will be higher than previous generations, and the process is repeated for as many generations as are necessary for the population to converge to a steady-state population set.
Other techniques for global optimization include particle swarm (Kennedy 1995) and simulated annealing (Kirkpatrick et al. 1983), which we tested and found yield similar results to the genetic algorithm approach.
The selection of a fitness function is key to finding the optimal solution. We explored several different test statistics for comparing two-dimensional datasets, including simple metrics like the root-mean-squared (RMS) difference between the observed and simulated probability densities and the mean absolute error (MAE). We also tested more sophisticated techniques including the 2D kernel-based test of Duong et al. (2012), the nonparametric test of equality between two copulas described by Rémillard and Scaillet (2009), and the paired-sample Kolmogorov–Smirnoff test as implemented by Peacock (1983). The different techniques yielded consistent and comparable results for our application, but the results presented below are based on the Kolmogorov–Smirnoff test. This test also returns a p value for testing the null hypothesis that the simulated and observed datasets were generated from the same underlying distribution. The fitness function was the average Kolmogorov–Smirnoff statistic from the joint distribution of δD and mixing ratio and the joint distribution of d-excess and mixing ratio.
To keep the problem computationally tractable, we explored the number of simulated points required to obtain reliable results and found that 1000 simulated points yielded the same best-fitting model parameters as simulations with many more (10 000) points, and the results presented below are for 1000 simulated points.
There are several potential sources of uncertainty in the fitted parameter estimates. First, the measurements of mixing ratio and isotopic composition are subject to uncertainties that may influence the results. As described above, the reported mixing ratios have uncertainties of up to 4%, δD has uncertainties of up to 5‰, and d-excess has uncertainties of up to 6‰. These are the most extreme measurement uncertainties, which occur only at the lowest mixing ratios. Experiments in which the mixing ratios and isotopic measurements were randomly perturbed by up to these maximum uncertainties yielded model parameters that were nearly identical to those presented below (not shown), suggesting that measurement error is not biasing the results.
We also considered the influence of small perturbations to the best-fitting parameters. We systematically subjected each parameter to progressively larger perturbations until the results were statistically significantly different from the observations at the 95% confidence level. This technique was used to determine the upper and lower bounds for each parameter reported below.
4. Results
In the previous section, we presented an overview of the stochastic model and the techniques used for fitting the model parameters to the data. In this section we apply the model in two ways. First, we use it in a forward modeling context to illustrate some of its sensitivities and then we apply it to the data from Chajnantor.
a. Idealized cases
To explore the behavior of the model before trying to fit the model parameters to the data, we first present several forward modeling results. Figure 4 (red lines) shows some results of the model with parameters that yield results representative of isotopic data from subtropical settings (Noone et al. 2011; Galewsky et al. 2011; Hurley et al. 2012; Samuels-Crow et al. 2014) and illustrates some of the links between the modeled parameters and the resulting water vapor isotopic composition. The contours show the estimated probability densities for the water vapor δD and d-excess. In this example, the average 
Model output for reference simulation (red; 
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0160.1
With the blue lines in Fig. 4, we perturbed the reference case to lower the average 
This framework allows us to explore one of our goals for the use of water vapor isotopic measurements: can they be used to constrain the processes that govern moistening or drying of the subtropics? We used two idealized cases that moistened the reference case in Fig. 4 to identical average mixing ratios (2220 ppmv) but used different processes to reach that state. In the first case (Fig. 5, red lines), the average mixing fraction is increased from 7% to 9.5%, while the last-saturation parameters were held constant. In the second case (Fig. 5, blue lines), the mixing fraction distribution from the control case was held constant, but the average last-saturation mixing ratio was increased to 575 ppmv.
As in Fig. 4, but moistened to 2200 ppmv by an increase in mixing fraction (red) and by an increase in saturation mixing ratio (blue).
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0160.1
When moistening is effected by enhanced mixing, the δD values are moved along a mixing line toward moister values and correspondingly higher δ values (Fig. 5a). Conversely, when the moistening is effected by increasing the last-saturation temperature (Fig. 5e), the δ values remain closer to the Rayleigh curve. If one only had measurements of mixing ratio, the processes governing the humidity would be difficult to disentangle. The addition of isotopic measurements, however, makes it clear that two different processes are at play. The joint distributions of mixing ratio and isotopic composition are clearly different between the two cases and are formally different at the 99% confidence level using the Kolmogorov–Smirnoff test. These idealized results show that there is some potential for using water vapor isotopologues for disentangling the mechanisms that govern changes in free-tropospheric humidity.
b. The 2012 dry season
In the previous section, we used forward modeling to illustrate some of the features of the stochastic model. In this section, we turn to the problem of fitting model parameters to the data and assessing the uncertainties in the fitted parameters. Figure 6 shows the probability density estimates for the water vapor δD (Fig. 6a) and d-excess measurements (Fig. 6b) from August through November 2012 plotted against the water vapor mixing ratio. The mean observed mixing ratio r for the period was 1680 ppmv. The mean δD was −234‰, and the mean d-excess was 21‰. Consistent with previous studies, the data are suggestive of condensation to a relatively low last-saturation temperature, followed by mixing with a moister source. The d-excess data were presented in Samuels-Crow et al. (2014) and are consistent with last saturation under ice-supersaturated conditions.
(a),(b) Probability density estimates for the August–November 2012 measurements and (c),(d) the best-fitting model output for water vapor δD and d-excess. Contour interval is 
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0160.1
The genetic algorithm found a best-fitting set of parameters that yielded simulated data points consistent with the observations (Figs. 6c and 6d), with average simulated δD of −241‰, average simulated d-excess of 20‰, and an average simulated mixing ratio of 1677 ppmv. The last-saturation and moistening distributions (Fig. 8) are skewed, with a mean 
Modeled parameters for 2012 and 2014.
About 1% of 
c. The 2014 dry season and comparison with 2012
The 2014 dry season (Figs. 7a and 7b) was moister than 2012 (2210 ppmv in 2014 versus 1680 ppmv in 2012), with higher average δD (−220‰ in 2014 versus −234‰ in 2012) and lower average d-excess (14‰ in 2014 versus 21‰ in 2012).
As in Fig. 6, but for August–November 2014.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0160.1
The best-fitting model (Figs. 7c, 7d, and 8) captured the observed moistening and changes in isotopic composition well, with simulated values for the average mixing ratio, δD, and d-excess of 2234 ppmv, −224‰, and 13‰, respectively. The last-saturation distribution for 2014 was less skewed than for 2012, with an average 
Estimated probability densities for (a) the last-saturation mixing ratio 
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0160.1
To further explore the processes that governed the moistening between 2012 and 2014, we performed idealized experiments in which we held 
The modeling results suggest that the moistening in 2014 required an increase in the average 
(a) Difference in August–November average 650-hPa vertical velocity between 2014 and 2012, derived from ERA-Interim. Contour interval is 0.01 hPa s−1; only negative contours (indicating weaker subsidence) are shown. (b) As in (a), but for temperature at 225 hPa; contour interval is 0.2 K and negative contours are dashed. Star indicates location of Chajnantor.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0160.1
The upper troposphere over the South Pacific was, on average, slightly warmer during 2014 than 2012 (Fig. 9b). While the average increase in 
Changes in August–November storm-track activity between 2014 and 2012 computed from ERA-Interim with a 24-h difference filter applied to meridional winds (υυ, m2 s−2). Gray contours indicate an average of the two years; red and blue contours show 2014 increases and decreases, respectively, relative to 2012.
Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0160.1
5. Discussion
Most previous studies that use the last-saturation framework rely, in one way or another, on some sort of air parcel trajectory analysis (Pierrehumbert 1998; Dessler and Sherwood 2000; Galewsky et al. 2005; Cau et al. 2007; Hurley et al. 2012). While there is no explicit representation of air parcel trajectories in our model, the changes in last-saturation PDFs described above do implicitly account for any circulation changes, in the sense that the PDFs are fitted to the data. Thus, the method does not ignore circulation changes despite the fact that circulation is not explicitly included. The potential for circulation changes to affect humidity is important and an understanding of such changes can provide a coherent framework for interpreting changes in humidity. For example, Hurley and Galewsky (2010a) showed how a shift in last-saturation positions to higher and colder regions during El Niño years, relative to La Niña years, could explain the increased aridity in the North Pacific subtropics during El Niño years. In their diagnosis of global warming GCM output, Hurley and Galewsky (2010b) showed that the moistening associated with the water vapor feedback may be partially offset by a shift of extratropical last-saturation sites upward and farther poleward. In the current modeling framework, condensation takes place entirely within a Rayleigh distillation framework along a single temperature profile, and such changes in last-saturation locale would all be mapped onto the single tropical sounding used here. Because we use a profile derived from ERA, we can extract the distribution of last-saturation pressures and diagnose their changes, but those are tied to the tropical sounding. In principle, one could extend this modeling framework to allow for some fraction of Rayleigh distillation to occur along different soundings, and that may allow for improved resolution of changes in last-saturation location; however, because Rayleigh distillation primarily depends on the initial and last-saturation temperatures and only weakly depends on pressure, the potential for fully diagnosing the influence of changes in circulation on humidity using this technique is probably limited.
Postcondensation moistening is modeled here as a mixing process between the freeze-dried air from the upper troposphere with a moist air source (rm = 20 000 ppmv or 12 g kg−1). The selection of this value of 
We have shown how measurements of water vapor isotopic composition can be used to constrain some key processes that govern free-tropospheric humidity. In particular, the method presented here can be used to constrain the distributions of last-saturation mixing ratio and postcondensation moistening. These are important parameters for diagnosing changes in humidity associated with ENSO and with the water vapor feedback, and we suggest that ongoing monitoring of water vapor isotopic composition may thus complement reanalysis and GCM-based diagnoses and provide a largely independent way to constrain the processes governing free-tropospheric humidity.
6. Conclusions
Our goal in this study was to use measurements of water vapor isotopic composition to constrain the last-saturation and mixing parameters that govern subtropical humidity. We applied a new stochastic model to two dry seasons of data and found that the model can reproduce the observations with reasonable fidelity and that the fitted parameters provide a coherent, physically consistent picture of the differences between the two seasons. For the period August–November 2012, we found that the data were best fit by an asymmetric last-saturation distribution with mean 
Acknowledgments
This study is based on data collected at the Atacama Large Millimeter/Submillimeter Array (ALMA), an international astronomy facility which is a partnership of Europe, North America, and East Asia in cooperation with the Republic of Chile. We thank the staff of ALMA for their generous support of this project, especially Richard Hills, Joaquin Penroz, and Jim Murray. We also thank Kimberly Samuels-Crow, Dylan Ward, Alex Lechler, Alec Tunner, and Lauren Vargo for field assistance and Chris Rella, Danthu Vu, and Kate Dennis from Picarro, Inc. for their technical support. The data used in this study are available at https://repository.unm.edu/handle/1928/151. This project was supported by NSF-AGS Award 1158582 to JG.
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