In coastal ocean, estuary, and lake systems, there is much interest in understanding, detecting, and predicting biological events such as harmful algal blooms. This requires a combination of numerical modeling and observation from both the ground and the air.
Mathematical modeling of the biology and chemistry, both simplified and complex, is a rich and active area of research, but the predictive accuracy of these models is often degraded by the accuracy of hydrodynamic inputs such as temperature, salinity, and currents. These fields can come from both numerical models and from observations. Models that numerically solve the (incompressible, typically hydrostatic) Navier-Stokes equations have the advantage of providing all variables at any specified time and location. The downside in the ocean is the same as it is in the atmosphere: all models have errors and those errors can lead to significant errors in the model results. Observations, either in situ or remote, sample the real system, but typically only at infrequent times and over a fraction of the domain of interest. Satellite images, for example, can only observe certain surface variables, are limited by cloud cover, and are inaccurate near the coast.
Some of these issues can be addressed through a process that has been mentioned a few times in this blog: data assimilation. At a simple level, data assimilation can be thought of as an interpolation of both observational data and model predictions, where each piece is weighted by its uncertainty. Where no observations exist the model can give reasonable estimates, while observations can be used to move the model fields closer to the true state. This process is used to improve initial conditions at all of the major operational weather centers in the world for the purpose of forecasting. In the ocean, error in the boundary conditions is often a larger source of error than chaos, but data assimilation can still improve results through improving the initial conditions. This is something that I have worked on in the Chesapeake Bay  to produce more accurate flow fields that can be used for driving biological and chemical models.
Data assimilation can incorporate satellite observations, but the satellite observations themselves warrant some consideration. Satellite temperature observations are available, but satellites do not actually observe temperature directly. Instead, radiation given off by the ocean and passed through the atmosphere is recorded (the observation is known as a radiance). These values are turned in to temperatures through what is mathematically an inverse problem. This inverse modeling requires some knowledge of the physical relationship between the desired quantity (in this case temperature) and the emitted radiation. This relationship is less clear when the desired quantity is something like an abundance of algae.
As an alternative to the inverse modeling, statistical models can be developed by using satellite radiances as predictors for some quantity observed in situ. One example is using this to estimate salinity values in Chesapeake Bay , but this type of statistical modeling can also be used to predict other quantities, such as biology. This is done experimentally in the Chesapeake Bay for sea nettles.
What’s Next? Mathematicians are working on enhancing the utility of these types of models with the long-term goal of guiding environmental and public health officials making policy decisions.
 Hoffman, M.J., T. Miyoshi, T. Haine, K. Ide, R. Murtugudde, and C.W. Brown. 2012. Advanced data assimilation system for the Chesapeake Bay. J. Atmos. and Oceanic Tech., 29, 1542-1557, 10.1175/JTECH-D-11-00126.1.
 Urquhart, E, M.J. Hoffman, B.F. Zaitchik, S. Guikema, and E.F. Geiger. 2012. Remotely Sensed Estimates of Surface Salinity in the Chesapeake Bay. Remote Sensing of the Environment., 123, 522-531, doi: 10.1016/j.rse.2012.04.008.
Matthew J. Hoffman and Kara L. Maki
School of Mathematical Sciences
Rochester Institute of Technology