Mean Element Theory
Mean element theory for astrodynamics and celestial mechanics began with the work of Lagrange himself, and has been developed further by many people over many years. It is a formal mathematical theory for approximating motion by separating the effects of fast motions (e.g. the motion of the true anomaly) from those that are more slowly varying (e.g. the slow motion of the right ascension of the ascending node caused by the J2 gravity term).
The most widely used form of mean element theory is based upon averaging the differential equations of motion over a fast-moving angular variable, and then using these averaged equations to predict the motion of the slowly varying elements. (Indeed, this was Lagrange's original idea.) The term "mean element" does not refer to a numerical average (i.e. mean) of a sampling of the element and is not related to statistics at all.
STK currently implements two mean element theories.
Kozai-Izsak Mean Elements
The Kozai-Izsak (KI) mean elements are based upon the paper "The Motion of a Close earth satellite," Y. Kozai, The Astronomical Journal, Nov 1959, pp.367-377. Only the short period terms (i.e. those involving averaging over the period of the orbit) are considered. The only perturbation force considered is the oblateness arising from the J2 gravity term.
Brouwer-Lyddane Mean Elements
The Brouwer-Lyddane (BL) mean elements are based upon the papers "Solution of the Problem of Artificial Satellite Theory Without Drag," D. Brouwer, The Astronomical Journal, Nov. 1959, pp.378-396, and "Small Eccentricities or Inclinations in the Brouwer Theory of the Artificial Satellite," R. H. Lyddane, The Astronomical Journal, Oct. 1963, pp.555-558. Two versions have been implemented:
Types of Brouwer-Lyddane Elements
Brouwer-Lyddane Short Mean Elements
The difference between the KI and BL Short mean elements is very small, and arise solely from the variable formulation itself, as they intend to model the same effects. Both are equally accurate even when producing slightly different results. Each can be considered a proxy for performing numerical averages of the Keplerian elements over short durations (small number of revolutions).
Brouwer-Lyddane Long Mean Elements
The BL Long mean elements consider the long period effects and thus do not act as a proxy for a numerical average of the keplerian elements over short durations. The most pronounced difference between the BL short and long mean elements can be seen in the mean eccentricity itself: the long period mean eccentricity can be significantly different from the short period mean eccentricity because the amplitude of the eccentricity oscillation can be quite significant. Thus, the BL long period elements would be more appropriate when performing numerical averages of the Keplerian elements over the time period needed for several apsides rotations to have occurred (usually quite a long time depending upon the orbit itself). During such long durations, other unconsidered force effects (other gravity terms, third-body gravity, solar radiation pressure, drag, etc.) may have a dominating influence on the numerical average of the Keplerian elements, so that the BL long mean elements do not act as an accurate proxy.