### Abstract:

The paleomagnetic assumption that the Earth's magnetic field is reduced to a geocentric axial dipole (GAD) when sufficiently sampled has been called into question for Mesozoic and earlier times. It has been suggested, for example, that modest contributions from axial quadrupolar (10%) and octupolar (25%) fields are resolvable using inclination only data from paleomagnetic studies. The underlying assumption in inclination only studies is that considerable continental drift has occurred over a sufficiently long period of time to render paleomagnetic sampling random in a paleogeographic sense. This assumption was stated in all previous studies dating back to 1976, but was never tested. We have developed a random walk model designed to test this assumption. Our model uses three different configurations for the continents in the random walk and allows the user to vary parameters such as maximum velocity, sampling distribution, sampling frequency and frequency of directional change. The model generates large sample sizes that cannot be adequately evaluated using the standard χ2 statistical test and therefore we introduce two statistical parameters used in structural equation models. Our models indicate that the 'random paleogeographic sampling' assumption used in the previous studies is not valid due primarily to the lack of an adequate sample size and temporal distribution. We show, for example, that even the most robust dataset compiled in 1998 is severely undersampled. A series of model runs on a GAD earth with sampling over a 600 Myr period demonstrates that detailed sampling will, on average, produce a GAD-like distribution only 30% of the time. Other model runs demonstrate that inadequate sampling can produce false quadrupolar and octupolar effects. It is our conclusion that time-averaged inclination only studies using the extant paleomagnetic database should be viewed with extreme caution.