The purpose of this note is to explain the derivations of the mathematical results in the paper. Much of the reasoning follows similar lines and uses similar techniques to those presented in full detail in the earlier paper [1]. Here we will draw freely upon results and arguments from that paper, and some of those arguments will be sketched in a less rigorous way here.One of the principal conceptual messages of the results discussed here about the simple graph models is that a seemingly rather severe form of population subdivision can still be compatible with recent common ancestors. In these simple models, the population is divided up into subpopulations that exchange migrants very infrequently. We assume some small fixed number of migrant individuals per generation; for example, that number could be just one migrant per generation, of even smaller.The model begins with a connected graph G consisting of G nodes, which we will refer to here as "islands," with a constant population size of / n G on each island. This is a discrete-time model with time measured in generations. We could choose to call an arbitrary generation 0 t = , and then t increases by 1 whenever time proceeds forward by one generation. Each individual lives on a particular island (the individual's "home island") in a particular generation. We will use the notation ( , , ) I t i m to refer to individual number m on island i in generation t .

Global mapped data of reflected radiation in the visible (0.63 /im) and near-infrared (0.85^m) wavebands of the Advanced Very High Resolution Radiometer (AVHRR) onboard National Oceanic and Atmospheric Administration satellites have been collected as the global vegetation index (GVI) dataset since 1982. Its primary objective has been vegetation studies (hence its title) using the normalized difference vegetation index (NDVI) calculated from the visible and near-IR data. The second-generation GVI, which started in April 1985, has also included brightness temperatures in the thermal IR (11 and 12/um) and the associated observation-illumination geometry. This multiyear, multispectral, multisatellite dataset is a unique tool for global land studies. At the same time, it raises challenging remote sensing and data management problems with respect to uniformity in time, enhancement of signal-to-noise ratio, retrieval of geophysical parameters from satellite radiances, and large data volumes. The authors explored a four-level generic structure for processing AVHRR data-the first two levels being remote sensing oriented and the other two directed at environmental studies-and will describe the present status of each level. The uniformity of GVI data was improved by applying an updated calibration, and noise was reduced by applying a more accurate cloud-screening procedure. In addition to the enhanced weekly data (recalibrated with appended quality/cloud flags), the available land environmental products include monthly 0.15°-resolution global maps of top-of-theatmosphere visible and near-IR reflectances, NDVI, brightness temperatures, and a precipitable water index for April 1985-September 1994. For the first time, a 5-yr monthly climatology (means and standard deviations) of each quantity was produced. These products show strong potential for detecting and analyzing largescale spatial and seasonal land variability. The data can also be used for educational purposes to illustrate the annual global dynamics of vegetation cover, albedo, temperature, and water vapor. Development of the GVI data product contributes to the activities of the International Geosphere-Biosphere Programme and Global Energy and Water Cycle Experiment and, in particular, to the International Satellite Land Surface Climatology Project. Monthly standardized anomalies of the GVI variables have been calculated for April 1985-present and are routinely produced on UN IX worksta

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led the forest impact modeling and contributed writing and expertise to much of the assessment. All modeling teams coordinated their efforts impressively.

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