How are winds aloft forecasts derived in the USA?
I took a tour of my local NWS office and spoke to the aviation weather unit people. My 8yo daughter even got to help launch that day's weather balloon, and then we went inside and monitored the balloon's data as it ascended. 



One of the data streams was the winds, which prompted me to ask: ok, so we know the winds right now for this location, but how does this knowledge allow you to forecast 24 hours into the future? And secondly, there are more FB locations that there are weather balloon sites, so how does that work?



Here is the answer:

Data from the balloons are just 1 of many factors entered into the weather forecasting computer models, and the FB products are actually the result of two different modeling systems. The first two products (the 6hr and 12hr forecasts) are generated by the Rapid Refresh (RAP) modeling system, and the third product (the 24hr forecast) is generated by the North American Mesoscale Forecast System (NAM) model.

Both models produce numerical forecast data on a grid, with each grid point being 13km apart in the RAP model and 12km in the NAM model, and each grid point containing 50 vertical data points up to the 10mb level. From these data points, each "FB Location" is interpolated (both horizontally and vertically) to the geographical location of the depicted airport, and the various forecast altitudes.

Additional reading:


FAA AC 00-45H Aviation Weather Services. (This document explains U.S. aviation weather products and services. It provides details when necessary for interpretation and to aid usage. Chapter 5.13 discusses the FB product)
NWS Instruction 10-812 Aviation Wind and Temperature Aloft Forecast. (This instruction provides guidance to NWS offices, outlining preparation and format of wind and temperature aloft forecasts in support of commercial and general aviation. It also lists each location where a forecast is provided)




Interesting Trivia:


All forecast locations correspond to an airport, except for 7 locations in the Gulf of Mexico, 1 location off the east coast of northern Florida, 5 locations in the Gulf of Alaska, and 1 location near the end of the Aleutian chain. These 14 additional forecast locations correspond to heavily traveled air routes.
The RAP model runs hourly, and provides forecast data 21 hours into the future. 
The NAM model runs every 6 hours, and provides forecast data 84 hours into the future