D.G. McMillan $^a$, C.G. Constable $^a$ and R.L. Parker $^a$
$^a$ Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, La Jolla, CA USA
Insight into the long term evolution and behavior of the geodynamo comes from two distinct sources, namely numerical modeling of dynamo action in the outer core and analyzing observations of the paleomagnetic field. Each method has its flaws and they are well known. For example, current computing power does not allow spatial resolution small enough to render viscous effects negligible in the main flow of the outer core, as they are thought to be. In fact, enhanced viscosity at smaller scales in dynamo simulations is frequently imposed to ensure numerical stability. And although the quality and quantity of paleomagnetic observations are steadily improving and increasing, respectively, existing data lack the spatial and temporal resolution to accurately characterize magnetic field morphology during specific epochs, such as during a reversal, or to reveal definitive local or global variations over a large range of timescales. An increasingly important branch of geomagnetism aims at bridging the gap between numerical modeling of the geodynamo and making observations of the paleofield. On the one hand, observations are needed to constrain numerical geodynamo solutions to be, at the very least, qualitatively Earth-like. The simulations receiving the most attention are those that have dominantly axial dipolar magnetic fields, secular variation similar (in at least some ways) to that of Earth's field and, in some cases, dipolar reversals. More detailed paleomagnetic observations are required to place firmer constraints on the geodynamo models. However, until the parameter space of the numerical models approaches that expected of Earth, it remains difficult to make direct comparisons of paleomagnetic data with the simulations. On the other hand, geodynamo simulations allow enormously detailed virtual observations of phenomena associated with a dynamo driven magnetic field, such as the spatial and temporal character of reversals. Because the accuracy of the simulations is very high compared to our knowledge of the geomagnetic field, we can assume that we know the `right' answer, in absence of the problems associated with filtering and uncertainty due to physical processes, measurement error and paucity of data. This type of analysis provides an important grounding for the interpretation of paleomagnetic observations. Further, the ability to model the processes, errors and sampling that contribute to a set of real data and applying the model to the magnetic fields of the geodynamo simulations gives a reasonable facsimile of geomagnetic observations. Thus the simulations provide a means of comparing realistic observations to an accurate model of the corresponding field. Herein, we use geodynamo simulations in this way to estimate the accuracy of results processed from imperfect paleomagnetic data and to explain the behavior of paleomagnetic data in terms of simple error models. Paleomagnetic studies of sediments are widely used to produce time series of the paleosecular variation: that is, measures of field direction and relative variations in paleointensity as a function of sediment age. The relative intensity variations can be subsequently calibrated by comparison with absolute paleointensity derived from other rock units using the Thellier method. In sedimentary paleomagnetism it has become commonplace to combine or stack records that span thousands to millions of years, a procedure that we consider here in the context of stacking relative paleointensity time series. Multiple records from a single locality are often averaged with the intention of increasing the signal of regional secular variation over the noise of recording and measurement processes. A recent example of a regional paleointensity stack is given by Laj {\it et al.} ({\it Phil. Trans. R. Soc. Lond.} A, {\bf 358} 1009-1025, 2000) for the North Atlantic region for the time period 10-80 ka. Records from geographically diverse locations are combined to eliminate local and short period variations, revealing signals that reflect global secular variation. Two well known examples of this are the SINT200 and SINT800 relative paleointensity stacks (Guyodo and Valet, {\it Earth Planet. Sci. Lett.}, {\bf 143}, 23-36, 1996; Guyodo and Valet, {\it Nature}, {\bf 399}, 249-252, 1999) designed to isolate variations in the strength of the geomagnetic axial dipole. In both cases, spectral analysis has been used to establish characteristic timescales of secular variations in Earth's core field that are thought to reflect geodynamo processes. Although it is generally acknowledged that stacking multiple records acts as a low pass filter on the data, it is not known precisely which spectral components are passed and which are filtered. Further, a number of uncertainties, such as those due to measurement and dating of core samples, affect the spectral content of a stacked record. Before drawing conclusions about secular variation from stacked paleomagnetic records and making inferences about the geodynamo, we must understand how the stacking process and the errors involved affect the result. We use SINT800 as an example to assess the spectral content of a stacked paleomagnetic record from globally distributed sites. We first generate intensity records from three different geodynamo simulations (Glatzmaier {\it et al.}, {\it Nature}, {\bf 401}, 885-890, 1999) corresponding to the geographic locations of the SINT800 records from marine sediment cores in set number one. We follow the stacking procedure that produced the SINT800 record and systematically impose errors from various sources. The stacking procedure yields an average relative paleointensity record that is thought to reflect global secular variation and is subsequently calibrated to virtual axial dipole moment (VADM) using absolute intensity data from volcanic sources. The VADM is a proxy for the Earth's dipole moment. We use statistical models of measurement errors in both the intensity records and the volcanic data, errors in the ages of control points that establish a depth to age relationship for each record, errors in the ages of control points due to ambiguities in their identification, and errors in the ages of magnetic samples due to interpolation of an assumed constant sedimentation rate between control points. We measure the geomagnetic content of the stacked record by computing the coherence and phase spectra between it and the axial dipole moment found directly from the simulation. Our results show that, with the largest errors, geomagnetic signal with periods longer than about 20 kyr are present in the stacked record of global secular variation. Modest errors in the ages of control points can account for most of the filtering effect, while those due to variations in sedimentation rate may have a similar effect if the mean sedimentation rate is small and the lengthscale of the variations is larger than the length separating paleomagnetic samples. These are robust results found by analyzing three statistically different geodynamo simulations. A spectral comparison of pairs of intensity records from the simulations suggests a method by which the quality of each record and its contribution to a stack may be assessed. Two error-free intensity records from nearby sites must have similar character over a wide spectral band. Increased distance between the sites and random errors, such as those discussed above, will act to decorrelate geomagnetic signals primarily at shorter periods. We show that error-free records separated by more than 20 degrees, in any direction, are likely to be uncorrelated at periods shorter than about 10 kyr. More importantly, it is possible for nearby records contaminated with random errors to be uncorrelated in most, if not all, period bands. Again, the main culprit is errors in the ages of control points. By varying the magnitudes of these errors, we are able to estimate critical periods below which the records will likely be uncorrelated. With real data, an examination of nearby and distant pairs of relative intensity records suggests that it is common to have poor correlation at long periods. Such pairs cannot be expected to yield representative signals when averaged together. By understanding the spectral behavior of pairs of accurate records, both locally and globally, we can determine if a particular record consistently fails to achieve significant correlation over a minimal period band and is consequently unlikely to contribute constructively to a stack.