The day of opening is defined as the point when there are at least five open flowers on a given tree. Full bloom is when at least 80% of the flowers have opened.
They measure this by picking sample trees. The Yoshino cherry is used most often as it's the most widespread, but they also use others such as the Higan cherry and Sargent's cherry. There are 59 sample trees at any one time. When a sample tree gets too old or otherwise incapacitated, successor junior trees are chosen from among candidates.
The Japan Meteorological Agency actually gave a forecast for the cherry blossom front for many years, though they recently left this forecasting to the private sector. There is a publicly available formula that they use which is based on a formula for the temperature dependence of reaction rates.
