## Gavin Collins and Dr. Matthew Heaton, Department of Statistics

### Introduction

Satellite remote-sensing is often used to collect atmospheric data, providing insight into climate variability over large regions of the earth. Common issues with such data include (i) missing information due to cloud cover at the time of a satellite passing, and (ii) large blocks of time for which measurements are unavailable due to the infrequent passing of satellites. While methods are available to estimate missing data in space and time, in the case of land surface temperature (LST), these approaches generally ignore the temporal pattern called the βdiurnal cycleβ which constrains temperatures to peak in the early afternoon and reach a minimum at sunrise. We directly incorporate the diurnal cycle in a statistical model to obtain realistic estimations of LST. Our methods are demonstrated using a remote sensing dataset of LST for 19,103 locations in the Houston area (Harris County, Texas USA) collected by NASA’s Aqua and Terra satellites during four satellite passings on July 1-2, 2014 (Figure 1).

### Methodology

We utilize the normal distribution to model LST as π¦_{i}(π‘) ~ π© (π(πΌ_{i}, π_{i}, π½_{i}),π^{2}), where π¦_{i}(π‘) is LST at location π and time π‘, where π(β) is the diurnal cycle of LST. The parameter πΌ_{i} is LST at the time of sunrise at location π, π_{i} is maximum LST at location π, and π½_{i} is LST at the time of sunrise the next day. Finally, π_{i} is a variance parameter, which is constant across time for all locations in the region. We will estimate these parameters using the Bayesian statistical method known as Markov chain Monte Carlo (MCMC).

This three-parameter version of the diurnal cycle π(β) is simplified from its original form to allow for estimation in the presence of sparse data. Because we have only a maximum of four satellite measurements per location per day (depending on cloud cover), this simplification is necessary to obtain an identifiable model. We will see in the following section that this three-parameter approximation is sufficient to obtain an accurate temporal representation of LST.

Finally, to further reduce the number of parameters, and to ensure that predictions follow a realistic spatial form, we constrain the diurnal cycle parameters to vary smoothly across space via multi-resolution basis functions.

### Results

Using this model, we obtain diurnal cycle estimations for every location in the Houston region. A visualization of this cycle for various values of πΌ, π, and π½ is depicted in Figure 2. Also shown in Figure 2 is an estimate of the diurnal cycle of LST for one location in the Houston region, along with 95% prediction intervals around the predictions β note that the predictions are quite accurate, that the prediction intervals include the measured LST value in all three cases, and that the diurnal cycle approximation is smooth and realistic. With only three parameters, the diurnal cycle is still flexible enough to fit the measured data.

In Figure 3, estimations are given for the measured data shown in Figure 1. Note that estimations are accurate for locations where we have measured data β the root mean square error for predictions across the four satellite passings is 0.902, meaning that predictions were within 1Β° C, on average. Also note that estimations are realistic for locations where cloud cover is obscuring the view of the satellite.

### Discussion & Conclusions

In conclusion, we were able to successfully infill a complete diurnal cycle of LST for each of the 19,103 locations in the Houston region. Additionally, we developed methodology that may be applied to any general region of interest, and may easily be extended across multiple days of LST data, which we are working to implement in the near future.

These accomplishments will allow atmospheric scientists to draw inference on various aspects of the climate, as LST is strongly related to several pertinent factors, including the concentration of various pollutants, environmental factors leading to severe weather events, and the identification of urban heat islands. Because of its importance, we are working to disperse our work widely via a peer-reviewed publication, as well as through various presentations.