Thinking Probabilistically

Meteorologist and MIT Professor Ed Lorenz is considered the father of chaos theory.  His "discovery" of chaos was accidental.  He was running a computer model on an early digital computer and decided do a rerun from the middle of a simulation.  The computer worked with 6-digit precision (e.g., numbers like .207689), but he had a printout based on 3-digit precision (e.g., .208).  When he input the 3-digit variables and did the rerun, he got dramatically different results, despite the fact that the difference between 6-digit and 3-digit precision is very small. 

This is the essence of chaos.  Small differences (sometimes very small) can evolve into differing outcomes.  It is one of the reasons why the atmosphere cannot be predicted precisely and why ensembles of computer model forecasts, which attempt to account for uncertainty that arises from chaos (and other sources) are so important for weather forecasting. 

Unfortunately, ensembles require computer time and, as a result, contemporary operational ensembles do not adequately resolve the western United States.  We provide products at weather.utah.edu that attempt to account for this shortcoming, albeit imperfectly.

The North American Ensemble Forecast System (NAEFS) is comprised of 42 forecasts produced by the U.S. National Weather Service and Environment and Climate Change Canada.  We take the NAEFS low resolution precipitation forecasts and we downscale them to 800 m grid spacing assuming seasonally varying climatological orographic precipitation gradients.  This gives us 42 high resolution precipitation forecasts from which we can derive statistics like mean, maximum, minimum and probabilities of exceedance for specific amounts.  An example is provided below. 

My colleagues here at the University of Utah, Court Strong and Lucas Bohne, are working on a better approach for the downscaling – one that would account for variations with the large-scale conditions.  I'm not sure if we'll be able to plug that in this winter, but it's something we hope to add soon.

Although the downscaling provides probabilistic water equivalent, we know you also want snowfall amount and perhaps even information on snow density.  Thus, we've developed algorithms for estimating snow level and snow-to-liquid ratio.  These are then applied to each ensemble member to provide high resolution snowfall forecasts, as illustrated below. 

Currently, we use the same algorithm across the western United States to estimate snow-to-liquid ratio.  We are currently working on regionally derived algorithms that we think will work a little better.  Those may be added incrementally over the next few months.

Another way to visualize these forecasts is using plume diagrams and violin plots, as presented below for Alta Collins.  On the left side are plume diagrams, graphs of the total accumulated water equivalent precipitation (top left) and snowfall (bottom left) produced by each ensemble member.  We identify members of the US (GEFS) and Canadian (CMCE) ensembles, their respective means, and the overall mean.  One can see, for example, that the Canadian ensemble is much wetter than the US ensemble. 

Also presented are violin plots of 6-h accumulated water equivalent precipitation (top right) and snowfall (bottom right).  These provide a statistical summary of the ensemble data.  The median for the total NAEFS ensemble is indicated by a horizontal red line, the middle 50% of the ensemble members by a vertical black bar, and the middle 90% of ensemble members by a vertical red line.  These take a while to get used to viewing, but provide a tremendous amount of information.  In the snowfall plot (bottom right) we also present the median snow-to-liquid ratio (dark grey line) and middle 50% of snow-to-liquid ratio estimates (grey fill area) predicted by our algorithm.  No grey shading prior to 06 UTC 29 September indicates that the precipitation falls as rain at this location.  After that, through 12 UTC 30 Sep, some ensemble members produce snow, whereas others produce rain.  After 12 UTC 30 Sep, all the members produce snow. 

I am a big fan of these ensembles as they provide estimates of the range of possibilities rather than a single outcome.  On the other hand, just like a single model forecast has capabilities and limitations, so does an ensemble.  Ensembles have biases and shortcomings as well.  We know, for example, that the US ensemble (GEFS) is underpredicts the full range of possible outcomes, especially at lead times less than 3 days.  Including the Canadian ensemble helps overcome this, but we're not sure yet to what degree or if it is overdone. 

We're spending quite a bit of time on these products this year, with one of my graduate students, Mike Wessler, leading the effort.  Expect to see better products coming in the future.