Atmospheric Precipitable Water and its Correlation with Clear Sky Infrared Temperature Observations

Total precipitable water (TPW) in the atmosphere is the vertically integrated amount of atmospheric water in all of its phases. TPW is a valuable predictor for weather forecasting, and it is routinely measured using radiosondes, ground-based global positioning systems (GPS), sun photometers, or microwave radiometers. The use of these sophisticated instruments limits the number of TPW measurement sites, which affects the accuracy of forecast models in regards to storm formation, strength, and the potential for precipitation. We have analyzed this relationship for the much drier climate zone found in the 5 Desert Southwest, specifically over Socorro, New Mexico (34◦N, 107◦W). Daily measurements of the ground and zenith sky temperatures have been made at Socorro for two complete annual cycles using infrared thermal sensors. Radiosonde TPW measurements from National Weather Service stations located in nearby Albuquerque, and Santa Theresa, New Mexico, are input into our dataset and analysed via a newly developed computational tool. Our results show that an exponential relationship between TPW and zenith sky temperature also holds for the Desert Southwest, but with parameters that are different than 10 those obtained for the Gulf Coast. Model simulations can accurately reproduce the observed relationship between TPW and temperature, and the results suggest that half of the signal in temperature is directly related to direct changes in opacity due to changes in TPW, while the other half is due to changes in air temperature that usually accompany changes in TPW.

niques could also be applied for higher elevation, arid and semiarid regions. A better understanding of this methodology may also demonstrate the feasibility of a citizen observer network, which could supply temperature data that would help monitor 25 the TPW variations across different locations in a region. Increasing the availability of TPW data will ensure more accurate forecasts; especially in higher elevation arid climate zones where there are large distances between existing TPW measurement sites (Maussion et al., 2014;Chen et al., 2018;Zhao et al., 2019).
TPW strongly influences atmospheric dynamics. This is most evident in the fact that when large amounts of TPW are observed, there is a greater probability for uplifting convection and cloud formation (Raj et al., 2004). This leads to applications 30 in numerical weather prediction (Wang et al., 2007), as well as climate change modeling and analysis (Gradinarsky et al., 2002). Higher amounts of TPW tend to be located near the equator and especially near Intertropical Convergence Zones, with a general decrease in TPW from low to high latitudes (Raj et al., 2004).
In this paper we use the standard definition of TPW (Salby, 1996), which is determined by the integrated amount of water that is contained in a vertical column of air extending from the Earth's surface to the top of the atmosphere, typically expressed where ρ is the mass density of liquid water, g is the acceleration of gravity, µ(p) is the mass mixing ratio of water vapor, and the integral is over pressure p from zero to some surface pressure p o .

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Typically, the water vapor mass density decreases quasi-exponentially with increasing altitude (decreasing pressure), such that the majority of the total column is near the surface. Previous studies have determined that 40% to 60% of the contribution to sea-level TPW occurs in the pressure layer between 1000 hPa and 850 hPa, with roughly 90% lying between the surface and 500 hPa (Ross and Elliott, 1996;Wang et al., 2007;Raj et al., 2004). Here we will emphasize the importance of surface elevation on TPW due to the high desert elevation of the Socorro measurement site. As most previous studies have focused on this instrument differs between two temperature ranges. The first is between -50 • C and 0 • C, with an error of ±3.9 • C (Harbor Freight Tools, 2017). The second range is 0.5 • C and 550 • C, the error for this range is ±2.2 • C (Harbor Freight Tools, 2017).
In our measurements we have found few instances where the temperature reading is below the -50 • C threshold. There is not a defined spectral range provided by the manufacturer for this sensor. However, we inferred that the spectral sensitivity of the sensor lies within the range 7µm -10µm by comparing to radiative model calculations as further discussed in Sect. 4. We 90 employed two sensors of this type: AMES 1, which was used starting on 22 January 2019, and AMES 2 which was put into service on 14 May 2019.

Measurement procedure
As discussed previously, the zenith sky temperature measurements are taken once a day, typically near 1700-1800 UTC or 2300-2400 UTC to avoid having the sun within the field of view of the sensors. Sky temperature is measured at the zenith to 95 facilitate the measurement of the vertical column air temperatures. This ensures the shortest optical path is used for infrared water vapor measurements (Smith and Toumi, 2008). A series of measurements were taken to investigate the impact of manual observations having small offsets from true zenith, where readings were taken at varying angles up to 30 • off of zenith over a week-long period. It was determined that with proper technique one can manually get within 5 • of true zenith, which introduces less than 1 • C variation in clear sky temperature measurements. We also measure the immediate ground temperature as a check 100 on instrument calibration and drift.
The presence of clouds, smoke, dust, or aerosol within the sensor field of view can have an impact on observed sky temperatures. Clouds, in particular, are capable of affecting the observations by providing an effective emission source at temperatures near cloud base. We screen and exclude any observations contaminated by clouds. Figure 1 shows the breakdown of sky conditions and sensor readings for the entire data record. We find that cloud screening results in the loss of data for about 25% 105 of daily readings. Additionally, there are occasions when a given sensor will not produce a reading (NaN) when the sky temperature falls below the calibrated range for that sensor. This occurs mostly under clear skies, and it varies from 2% of the measurements for AMES 2 to 50% for FLIR i3 (Fig. 1). The larger fraction of NaN days for the FLIR i3 instrument is likely due to a warmer low-temperature cutoff (-40 • C for FLIR i3 versus -50 • C for AMES), and a different spectral sensitivity that is closer to the transparent atmospheric window between 8 and 12 µm wavelength. We have not made any measurements in the 110 presence of noticeable smoke or dust. Surface solar radiation measurements at Socorro have shown that aerosol optical depths are typically very low, varying between 0.03 and 0.10 with maximum values during summer (Minschwaner et al., 2002). Variations in aerosol are not considered here, but they will contribute a small additional source of variability in sky temperature readings.

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As discussed above, the spectral sensitivity curves for each of our thermal sensors are not precisely known, but they are all assumed to have passbands that fall within relatively transparent atmospheric windows at wavelengths between ∼7 to ∼12 µm, corresponding roughly to the mid-IR spectral range. The downward mid-IR radiance observed at ground level with clear skies is primarily dependent on the vertical distribution of atmospheric temperature, and on the vertical distributions of greenhouse gases with mid-IR absorption signatures (e.g. Thomas et al. (1999)). The most important infrared-active gases at these wave-120 lengths are ozone, with a vibrational band at 9.6 µm, and water vapor, with a weak continuum between the 6.3 µm vibrational band and the far-IR rotational lines of H 2 O (Stephens, 1994). Although the 9.6 µm ozone feature is significant for transmission through the entire atmosphere, most of the ozone is located in the stratosphere and ozone generally has a negligible impact on mid-IR transmission for path lengths within the lowest few kilometers of the surface, except perhaps under highly polluted conditions. On the other hand, even though the H 2 O continuum absorption is considered weak (only 10%-20% absorption 125 through the entire atmosphere), the radiative effects can be significant for path lengths near the surface. The magnitude of this so-called e-type absorption varies as the square of the absorber amount (e.g. Burroughs (1979)). Furthermore, the scale height for the vertical distribution of water vapor (∼3 km) is much smaller than for the background atmosphere, so that most of the water vapor continuum effects are felt within the lowest few kilometers of the surface.
Figure2 shows instrument comparisons for clear sky temperatures and for ground temperatures, where the AMES 1 instru-130 ment is used as a standard due to its longevity and stability during the course of observations. We find that the AMES 1 and AMES 2 instruments agree to within ± 2 • C for both ground and sky temperatures, with no clear bias or offset. The FLIR i3 and AMES 1 instruments are also in good agreement for ground temperature, but they show a considerable difference in This pattern is consistent with the timing of the North American Monsoon in New Mexico. The corresponding zenith sky temperatures range from -40 • C in winter to -10 • C in late summer. The day-to-day variability is on the order of 2-5 mm for 160 TPW and roughly 2-5 • C for temperature. Note the difference in TPW between Spring-Summer 2019 and Spring-Summer 2020, which provides some measure of the interannual variability. A detailed analysis of seasonal and interannual variability is beyond the scope of this paper; however, Appendix A presents evidence indicating that some of these differences are related to large scale changes in relative humidity.

Analytical techniques 165
For the purpose of this experiment we developed a computational utility to analyze and visualize the collected data. Some of the visualizations used in the model include temperature and TPW measurements (as a function of time), direct sky temperature and TPW comparisons, and sensor performance comparisons. The tool implements common numerical methods to study the exponential relationship between the collected zenith sky temperature and TPW with ease . In the development of this computational model, we applied two common numerical methods: linearization of an exponential and 170 least-square linear regression (LSLR).
We begin the process of analyzing the collected data by purging data that is not viable; this includes out-of-range temperature readings in addition to incomplete precipitable water measurements. Sensor malfunctions on radiosondes contribute to the missing precipitable water measurements. Since the FLIR i3 may not produce reliable measurements below its temperature threshold, we have assigned these temperature measurements as not-a-number, and thus are not processed in the final analysis.

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As part of this procedure, 4 additional days were not included in the final analysis because the results from these days exceeded a 3σ limit of deviation from the rest of the entire dataset.
After the data has been pre-processed, the relationship between the zenith sky temperature and precipitable water is passed through a least-squares linear regression algorithm in (T b ,log(TPW)), resulting in the solid black exponential curve in Fig. 4. The exponential parameters for the best-fit function, physically defined as as noted in Fig. 4, are A = 20.86 mm and B = 0.036 • C −1 . The confidence interval, denoted as a dashed line, represents the variance of the parameters associated with the best-fit. The prediction interval, denoted as the shaded region, represents specific probability of future data points.
The goodness of fit in Fig. 4 confirms that a quasi-exponential relationship between the two variables provides a valid 185 description of these observations. The coefficient of determination (R 2 ) associated with this relationship is 0.661. Thus, based on the scheme defined by Schober et al. (2018), the correlation described by the model is considered to be strong. The scatter shown in Fig. 4 is dominated by the errors in TPW introduced by spatial/temporal averaging of radiosonde sounding from ABQ and EPZ, with additional contributions from the precision (2-4 • C) in the zenith sky temperature measurements and from variations in other atmospheric properties such as aerosol optical depth (estimated at ∼5% in radiance, or about 2 • C in 190 brightness temperature).

Error analysis
The primary sources of error that could have impacted our results include the errors of the sensors, both the thermometers and the radiosondes. As we have discussed in Sect. 2, we have stated that the error for the temperature range that closely resembles  the clear sky is ±2 • C and ±3.9 • C for FLIR i3 and AMES respectively.

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The residual standard deviation produced by the LSLR was calculated to be 0.335. We can make note that the data fits within roughly one factor of the standard deviation in Fig. 5.

Interpretation and comparison to model simulations
In this section, we interpret the observed relationship between TPW and zenith sky temperature using radiative transfer calculations with the MODTRAN6 (MODerate resolution atmospheric TRANsmission 6) model (Berk et al., 2014). This framework 200 inputs vertical profiles of temperature, density, and radiatively active trace gases, and computes atmospheric spectral transmittances and radiances over a wide spectral range from the ultraviolet through far infrared wavelengths. Here, we focus on vertical path lengths through a midlatitude summer atmosphere (Anderson, 1986), with a surface located at 1.42 km altitude, in order to simulate the zenith sky radiances at Socorro. In the base simulation, TPW = 11.4 mm and the temperature distribution is unchanged from the midlatitude summer case. Other model runs include changes to the TPW by uniformly scaling the water 205 vapor vertical profile by factors of 0.5 and 2 while keeping temperature fixed, and by uniform temperature changes of -5 K and +5 K while keeping TPW fixed.  taken for the sake of illustration because, as noted previously, the spectral passband of the AMES thermal sensor is expected to approximately correspond to this region. The radiances shown in Fig. 6 can be used to separately quantify the impact of 210 changes in temperature or water vapor on downward radiances. We find that changes in TPW have the largest relative impact on spectral radiances at 10 µm as compared to 7 µm, due largely to saturation effects closer to the edge of the strong H 2 O 6.7 µm band absorption. Changes in temperature, however, have a more uniform spectral effect.
For each case, we integrated the spectral radiances from 7 to 10 µm and determined the equivalent brightness temperature across this spectral range. The equivalent brightness temperature was found by integrating the Planck function over the 215 same spectral range, and solving for the blackbody temperature that provided the same integrated value as the MODTRAN6 downward radiances. The equivalent brightness temperature is intended to simulate our thermal sensor's zenith sky temperature reading, and as indicated in Fig. 6, we do find a relationship between TPW and equivalent brightness temperature that is somewhat consistent with the observations shown in Fig. 4. Higher TPW amount leads to higher effective temperatures, which can be interpreted as a simple lowering in altitude of the effective emission level due to increasing opacity from water 220 vapor, and lower altitudes generally correspond to higher temperatures. For atmospheric temperature, we find an expected increase/decrease in equivalent brightness temperature when atmospheric temperatures are respectively increased/decreased.
We developed a simple linearized model to further interpret our observations using the MODTRAN6 calculations above. If the equivalent brightness temperature, T b , is assumed to be a function primarily of TPW and atmospheric temperature, T air , The observed relationship between T b and TPW is clearly nonlinear, but for small changes about some basic state (T b −20 • C and TPW 11 mm) we assume that the observations can be represented by the left-hand side of Eq.
(3) and that the slope is approximately constant with a magnitude of about 1.9 • C mm −1 (Fig. 4). The MODTRAN6 simulations can be used to estimate the partial derivative terms, so that the first term on the right-hand side of Eq.
(3) has a magnitude of 1.04 • C mm −1 230 based on Fig. 6. This is the direct effect of changes in TPW on equivalent brightness temperature, and the results can be shown to capture some, but not all, of the variations in the observed relationship. The second term on the right side of Eq.
(3) accounts for changes in T b that may arise from any coupling between T b and TPW due to changes in atmospheric temperature, and it is composed of two factors. The first factor is 0.87 based on the MODTRAN6 calculations (Fig. 6). The second factor may be estimated by assuming that the atmosphere maintains a state of constant relative humidity, so that the water vapor partial 235 pressure at all levels (and hence TPW) is set by the Clausius-Clapeyron relation, where e s is the saturation vapor pressure, L v is the latent heat of vaporization, and R v is the specific gas constant for water vapor. If relative humidity is held fixed, then it can be shown that

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Evaluating this equation for T air = 273 K and TPW=11.4 mm, we find the second factor on the far right side of Eq.
(3) to be 1.21 • C mm −1 , hence the entire second term has a magnitude of 1.05 • C mm −1 . We conclude that the magnitudes of the two terms on the right side of Eq. (3) are nearly identical at about 1 • C mm −1 , implying an overall slope on the order of 2 • C mm −1 , which is in close agreement with the observed slope of 1.9 • C mm −1 . A comparison of the model results to the observations is shown in Fig. 7. Despite the use of a simple linearized model to describe a clearly nonlinear relationship seen 245 in the observations, we find a good level of agreement that confirms our hypothesis for the two primary influences on the relationship between TPW and zenith sky temperature.
In order to test the robustness of assumptions implicit in Eq. (5). we investigated the relationship between TPW and air temperatures near 3 km altitude using the Albuquerque sounding data spanning over one year. There was a considerable degree of scatter but TPW and air temperature were found to be well correlated, and a linear fit to the data (not shown) produced a slope 250 consistent with the value estimated using Eq. (5). Figure 7 also includes the temperature-TPW relationship fit to observations by Mims et al. (2011), which employs an exponential form somewhat similar to ours. While the overall patterns are similar and consistent with the model, there are differences between the two fits that are most likely due to different sensitivities between the sensors used, and to differences between climate regimes (e.g., mean relative humidities for our location are much lower than for the Mims et al. (2011) study

Conclusion and future directions
Our results demonstrate the feasibility of using low-cost (under $50 USD) sensors to measure TPW in less than five minutes using simple measurement protocols, confirming the findings by Mims et al. (2011), but our work extends the previous analysis by observing at colder zenith sky temperatures (down to -40 • C) and correspondingly lower TPW (down to ∼3 mm).
Our measurements also show that the exact T b -TPW relationship will be a function of instrument spectral sensitivity and 260 local conditions such as surface elevation and mean relative humidity. In addition, we developed a simple model that uses MODTRAN6 radiative transfer calculations to quantify how T b can be influenced by changes in TPW and in mean-column air temperature. The model analysis indicates that the observed relationship between zenith sky temperature and TPW can be explained primarily by two dominant influences. First, an increase in TPW leads to increasing atmospheric opacity and a lower altitude for the effective emission height as viewed from the surface. Under typical conditions a lower height corresponds to a 265 higher temperature. Second, an increase in TPW is typically correlated with higher air temperatures; although relative humidity is not perfectly constant, the climatology is such that positive relationships between temperature and humidity are generally to analyze the relationship between zenith sky temperature and precipitable water in different climate zones. We are also developing plans to work with schools to continue manual data collection in different parts of the American West to help advance science learning while collecting data from regions with different elevations and precipitation profiles. Current efforts are fo-285 cused on testing and optimizing a machine learning algorithm (more specifically a Support Vector Machine) to predict a binary set of weather conditions, clear sky or overcast, based on zenith sky temperature and TPW data. These predictive models will have the capabilities to further quantify the aforementioned relationship by applying common statistical metrics, and will be the subject of a future paper.

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This Appendix presents two supplementary figures that support discussions about variability in TPW and spectral passbands of our instruments. Figure A1 shows a comparison of surface relative humidity measured at Socorro, NM for the first halves of 2019 and 2020, along with the corresponding T b and TPW measurements analyzed over same two time periods. We find that RH values in late spring and early summer of 2020 were much lower than those observed in 2019. Similarly, TPW values in Spring-Summer 295 2019 were lower in 2020 compared with 2019. However, measured values of T b did not undergo a proportional change so that the 2019 and 2020 relationships show small differences that can be seen in the fits. The reductions in RH and TPW appear to be consistent with the La Nina pattern seen in 2020, although a more complete analysis would require more years of T b and TPW measurements.