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# Enhancing Data Assimilation and Rainfall Prediction in the Tropics

The term “data assimilation” typically refers to researchers’ use of algorithms to create best estimates of initial conditions for a forecast, such as in the context of weather or climate. Essentially, it seeks to optimally combine information from two sources: yesterday’s forecast and today’s observations. Scientists usually begin by generating a forecast based on a general idea of the system state. When observations arise from satellite readings, weather balloons, and the like, an algorithm combines those observations with the initial forecast to yield an updated estimate of the system state. Sequential repetition of this method occurs each time a new observation enters the system and assimilates into the model. A wide variety of algorithms are available to conduct this process.

A particularly challenging application area is the prediction of tropical weather and climate patterns. During a scientific session at the 9th International Congress on Industrial and Applied Mathematics, currently taking place in Valencia, Spain, Samuel Stechmann of the University of Wisconsin-Madison introduced multiple techniques to forecast rainfall in the tropics. He began by comparing the tropics with the extratropics (the middle latitudes beyond the tropics). While this mid-latitude region is subject to fronts and cyclones, the tropics instead experience scattered thunderstorms nearly every day. This is mathematically due in part to the Coriolis force; with the absence of rotational effects to shape things, scattered thunderstorms occur consistently.

Stechmann compared weather and climate patterns in the tropics versus the extratropics (mid-latitudes) to get a sense of predictability.
Stechmann then introduced a time series of data—ranging from January 2016 to March 2016—from a particular latitude belt to study the different longitudes for both the tropics and extratropics. Relative to the tropics, the mid-latitude patterns are fairly regular; every few days, a new system propagates through. In contrast, the tropics endure irregular wave patterns of all types with various signals going both eastward and westward. These waves are called convectively coupled equatorial waves. Stechmann used the correlation coefficient to calculate the skill of rainfall forecasts in the tropics and extratropics. He found a skill level of two-four days in the tropics, and one of five-six days in the mid-latitudes. In short, the prediction skill is lower in the tropics, likely due to tropical rainfall’s coupling with irregular wave patterns (and the fact that the tropics are notoriously difficult to properly model). This finding inspires the following three questions.

1. Is low skill due to model error, or intrinsic limits of predictability?
2. What is the upper limit of predictability for tropical rainfall?
3. How predictable are convectively coupled equatorial waves?

The third question in particular, which enquires about the waves themselves, hints at a type of predictability that researchers have only begun to explore in the last several years.

Scientists have traditionally used operational forecast models to test rainfall predictability in the tropics. However, Stechmann and his colleagues sought to estimate predictability almost solely on observational data. To do so, they conducted wave decompositions in Fourier space to yield a power structure with wavenumber on the x-axis and temporal frequency on the y-axis. Stechmann then split the data into different wave types. For instance, repeatedly smoothing out raw data results in a smoothed background dataset that ultimately identifies the bulging residual waves in the ensuing diagram. Conducting this splitting process in Fourier space isolates the individual waves. When isolating each wave, two indices in the accompanying equation identify the wavenumber and wave type. Stechmann modeled each wave type as a damped oscillator.

Next, he presented an example of the Madden-Julian Oscillation (MJO), an intraseasonal, eastward-moving atmospheric wave disturbance in the tropics comprised of clouds, wind, rainfall, and pressure. Stechmann conducted a parameter fit over the model’s parameters. “This is a relatively crude method, but it does reasonably well,” he said, adding that he and his colleagues are testing a large quantity of waves and thus do not want to spend extraneous time trying to fit additional parameters. Stechmann found that the MJO has a forecast scale of 25 to 30 days for each wavenumber.

However, the team wanted to return to the original question pertaining to overall predictability of tropical rainfall. Over the Pacific and Indian Ocean regions—where much tropical rainfall occurs—overall predictability is four to six days. But when one removes the predictability of the MJO, overall predictability drops to only two to four days. This decrease indicates that the MJO wave is crucial in climate-based study, and allows for prediction of behavior that goes well beyond basic thunderstorms. It also reaffirms the fact that waves contribute significantly to predictability. Even so, there is still room for improvement in model simulation of waves and overall data assimilation methods.

Stechmann proceeded to reveal a new potential test model that is currently in the works — a stochastic partial differential equations (SPDE) model for tropical locations. “It’s definitely not perfect, but it’s a good test model for thinking about tropical data assimilation,” he said. He then introduced the concept of multi-model communication for mitigating model errors, refining data assimilation, and improving overall forecast. Multi-model communication involves using multiple models—each with their own pros and cons—to tighten forecasting efforts. “What if you let each model run for a while, then allowed them to share information during the forecast?” Stechmann asked. For example, if a researcher knows that one model is bad at MJO prediction, he/she can assimilate data into the forecast from another model that skillfully addresses the MJO.

A large part of motivation for this PDE model is prediction of the MJO itself. A hierarchy of model options exists for making such predictions, ranging from complex global climate models (GCMs) to skeleton models and low-order models. Past results have indicated that GCMs and low-order models are skillful for only 15-20 days, which is a bit lower than one might have hoped. To combat this dearth, Stechmann established a deficient tropical climate model and a low-order MJO model that employs principal component analysis. He then compared observations from the SPDE model and the stochastic differential equations (SDE) model, with the goal of assimilating the SDE model—which contains valuable information about the MJO—into the SPDE model to ultimately improve the forecast.

In summation, Stechmann acknowledged the difficulty of modeling weather and climate conditions in the tropics due to the sheer number of open questions; this is especially true of rainfall predictability. However, his SPDE model as a tropical test for data assimilation, prediction, and uncertainty quantification—combined with a multi-model communication strategy for mitigating model error and improving forecasts—has the potential to advance rainfall prediction around the Equator.

 Lina Sorg is the associative editor of SIAM News.