JOHN ALROY
Department of Paleobiology, Smithsonian Institution,
MRC 121, Washington, DC 20560, USA

tel: (202) 357-4030
FAX: (202) 786-2832
e-mail: jack@homebrew.si.edu

ABSTRACT--Recently developed algorithms have made it possible to redefine the North American mammalian time scale. These algorithms are fully quantitative, employ all the relevant paleontological and geochronologic data, and are governed by a single optimality criterion. The overall procedure has two basic steps. First, a sequence of 6196 taxonomic first and last appearance events is generated and the sequence is calibrated to numerical time. To do this, 4105 faunal inventories and 186 stratigraphic sections are converted into "first/last" statements. The appearance event sequence is made to fit the F/L statements in the most parsimonious way: it minimizes the number of unproven, but implied F/L statements, which is just the same as minimizing temporal overlaps between taxa. The sequence is highly consistent with 152 radioisotopic and paleomagnetic age estimates for fossil localities, and the observed relationship is used to calibrate the sequence to numerical time. The procedure's second major step is to impose discrete time intervals on the faunal ordination. The location of the interval boundaries is set to maximize the number of implied F/L statements - the same criterion that governed the ordination. However, the need for each boundary in the first place is itself evaluated statistically. This is done by showing that an interval is bounded by a significant amount of faunal turnover at each end. The test supports a division of the sequence into 50 intervals. Finally, the names of these statistically recognized intervals are tied to reference localities that take on the role of "type specimens" in preserving priority. The resulting time scale agrees closely with the standard system, identifying all of the widely-accepted land-mammal ages. However, a few nominal "zones" cannot be distinguished (e.g., in the Paleocene and Pliocene), there are many reassignments of individual faunal assemblages, and the "Arikareean" is subdivided into two ages. Because the existence of each interval is treated as a testable hypothesis in this scheme, the tasks of diagnosing and naming intervals have been fully separated. Therefore, the new approach is similar to the Wood committee's original system. Importantly, it avoids both the nominally chronostratigraphic "immigrant first appearance datum" scheme and the heavy reliance upon geochronology that has come to characterize mammalian biostratigraphy. The new analysis does, however, generate independent numerical age estimates for every one of the fossil localities, with the analytical errors on individual faunal "dates" being comparable to geochronological errors. Therefore, it is suggested that the land-mammal age system may now be redundant, and that the international chronostratigraphic time scale may eventually become so.

..........The North American Land-Mammal Age system is one of the most robust and long-standing paleontological time scales. It was first formalized by Wood et al. (1941) as a series of "provincial land-mammal ages," which in turn were based on faunal correlations that had been known since the 19th century (Osborn and Matthew, 1909). The system has become the benchmark for correlations in the terrestrial Cenozoic record of North America, and a source of inspiration for analogous time scales dealing with other continents (e.g., Mein, 1975). Few paleontological time scales are better understood, more widely applied, or more precise.
..........In recent years the system has been revised heavily (Woodburne, 1987; Prothero, 1995; Woodburne and Swisher, 1995), not just because of continuing empirical discoveries, but because mammalian paleontologists have shifted to an incommensurate paradigm. After a century of allowing the time scale to be flexible, informal, and grounded on faunal correlations that took all the evidence into consideration, efforts have been made to transform it into a formal chronostratigraphic system. This approach is nominally justified by international stratigraphic codes (e.g., Hedberg, 1976); it holds that first appearances of individual species in particular sections are the only valid basis for naming and defining the land-mammal ages.
..........This paper proposes a radically different approach to the problem. It argues that the traditional, faunally-justified correlations were adequate, and merely needed to be made more robust with quantitative methods and explicit optimality criteria. It argues that new methods make this goal fully attainable, and shows that the entire fossil record late Cretaceous and Cenozoic North American mammals can be brought together into one objective chronological scheme. Many features of the traditional time scale are robust, but better methods resolve several conflicts and focus attention on a few persistent shortcomings in the data. Finally, this paper argues that the tasks of defining and naming such entities as "ages" need to be separated methodologically. Breaking up the flow of time into discrete intervals is a scientific process of discovering natural units, and so these intervals must be treated as testable hypotheses. Permanently fixing a boundary would make it impossible to test the hypothesis that the boundary is naturally placed. Therefore, the widely accepted procedure of fixing "golden spikes" is not appropriate in mammalian biochronology.
..........Despite this, formal rules for choosing names still are needed to maintain priority, regardless of the fact that the task of naming is itself a secondary consideration. I follow the Wood committee and many later European workers (Mein, 1975) in tying the names of these biochrons to selected reference localities. For example, the Barstovian Land-Mammal Age is known as such because it includes the Barstow fauna from southern California. All of this is akin to biological alpha taxonomy: species are recognized and characterized as independent entities using objective criteria, but their names depend on arbitrarily designated type specimens. The task of naming still involves the laws of priority, but all other biochronological problems are dealt with in a scientific manner. Because this does not follow chronostratigraphic conventions, the resulting time intervals are paleontologically-based biochrons instead of formal, stratigraphically-based units such as stages and zones.
..........Each step of the current approach has advantages. First, it employs virtually all relevant temporal information instead of focusing on a handpicked set of taxa, sections, or geochronological results. Second, it involves fully algorithmic computations that can be repeated by other workers and are based on a simple and universal optimality criterion. Third, it derives its hypothesized ordering of appearance events from the data, instead of trying to shoehorn the data into an a priori set of time intervals. Fourth, the hypothesized event sequence is calibrated geochronologically, yielding explicit estimates of prediction error and allowing every fossil locality to be assigned its own numerical age estimate. The errors in these biochronological age estimates are comparable to geochronological errors. Fifth, the same optimality criterion that governs the event sequence is used to determine the best locations for boundaries between biochrons; the number of biochron boundaries is fixed by an additional statistical test. Finally, the designation of reference localities fixes the names of biochrons using strict rules of priority, which largely prevents nomenclatural ambiguity and instability. The rest of this paper is concerned with describing the available data, the quantitative biochronological methods, the content of the North American mammalian time scale, and the status of alternative approaches.

..........The algorithms used here were designed to synthesize virtually all available faunal, stratigraphic, and geochronologic data. They are robust even when abundance data are not available, many or most fossil localities have not been placed in a stratigraphic section, and taxonomic lists are individually brief but numerous. Unlike some popular methods including graphic correlation (Shaw, 1964), the algorithms ignore absolute distances between fossil horizons in stratigraphic sections, focusing instead on the relative order of appearance of fossils. This admitted shortcoming is made unavoidable by several facts: most terrestrial stratigraphic sections are very short, containing just one or a few fossiliferous horizons; most terrestrial fossil remains are concentrated in short stratigraphic intervals instead of being spread uniformly throughout sections; and many terrestrial paleontologists do not report exact section measurements in any case. Thus, these methods are best applied to macrofossils, including not just mammals but potentially such groups as plants, echinoderms, and molluscs.

Data collection

..........The North American mammalian data set also has figured in analyses of diversity dynamics (Alroy, 1996, in press A; Wing et al., 1995). It consists of 4015 taxonomically standardized faunal lists that include 27,241 occurrence records of 1195 genera and 3182 species, and are sorted into 186 stratigraphic sections (37.8% of the lists are not tied to a section). The data set spans the Campanian (late Cretaceous) through last interglacial (late Pleistocene), and includes localities from as far south as Panama and as far north and east as Ellesmere Island. Sampling is heaviest in the western U.S.A., and in certain temporal intervals like the early Eocene and late Neogene.
..........The lists, taxonomy, and stratigraphy are referenced to 2415 publications. Every effort was made to include all relevant literature published during the 20th century. Earlier publications by such workers as Leidy, Cope, Marsh, Osborn, and Scott were not heavily consulted due to the lack of precise stratigraphic and geographic information. Unpublished sources were avoided wherever possible. However, a small number of important, yet unpublished master's and Ph.D. dissertations were consulted (e.g., Kihm, 1984).
..........The lists include references, stratigraphic and geographic information, and both original and corrected identifications. Meanwhile, the genus- and species-level taxonomic data set includes all original and subsequent combinations and proposed synonymies, with original authors and lists of references employing each combination or synonymy. The North American Paleofaunal Database and North American Fossil Mammal Taxonomy Database can be searched and viewed on the World Wide Web at http://homebrew.si.edu/nampfd.html and http://homebrew.si.edu/nafmtd.html.
..........Although no effort was made to compile faunal information for groups other than mammals, I agree with Lucas (1993) that the land-mammal ages eventually should be transformed into "land-vertebrate" ages by taking such groups as turtles, archosaurs, and lepidosaurs into consideration. Such a synthesis also might include paleobotanical data (e.g., Wing et al., 1995), opening up the possibility of "terrestrial ages" that incorporate all available paleontological data.

Ordination

..........The basic unit of currency in the analysis is the first/last (F/L) statement, which is a record showing that the first appearance event (FAE) of one taxon is known to predate (<) the last appearance event (LAE) of another (Alroy, 1994). If two taxa i and j are found in the same fossil quarry or at the same stratigraphic horizon, then their age-range zones overlap: FAEi < LAEj and FAEj < LAEi. Similarly, if a fossil of taxon i is found lower in a section than one of taxon j, FAEi < LAEj. If the fossil record could be sampled infinitely, then it would be possible to compute a matrix of all F/L statements, and from this directly infer a perfectly accurate appearance event sequence.
..........Because age-ranges only can be extended with greater sampling, the number of known F/L statements does inexorably increase. However, the currently available set of statements may be small, and virtually any event sequence therefore must imply some F/L statements that are not yet known to be true. These unproven, hypothesized statements are unparsimonious because each of them can only be explained with an ad hoc argument that the raw data are inadequate. Therefore, the optimal sequence will minimize the number of such implied statements. A multivariate ordination algorithm known as appearance event ordination (AEO: Alroy, 1994) is used to find the event sequence that best meets this criterion. AEO is related to correspondence analysis (Hill, 1973), but it employs a swapping algorithm to optimize its solution after the usual round of reciprocal averaging (Alroy, 1996).
..........Appearance event sequences, which are relatively ordered strings of FAEs and LAEs derived from F/L statements, capture most of the temporal information presented by the fossil record. Most importantly, taxonomic age-ranges are directly equivalent to the intervals between appropriate pairs of FAEs and LAEs. This has several implications. For example, the concurrent range zone of a fossil assemblage is just the youngest FAE of any taxon in that assemblage paired with the oldest LAE of any taxon in the assemblage. Meanwhile, taxonomic diversity curves can be computed by summing up the number of taxa whose age-ranges span each part of the sequence.
..........After excluding 1156 singleton taxa (those known from only a single locality and therefore presenting no temporal information), the lists and sections were used to compute 796,505 F/L statements. The "square graph" method (Alroy, 1996) was used to factor out biogeographic patterns by inferring an additional 61,596 statements, for a total of 858,101. Ordination of these data generated a sequence of 6196 events: 3221 FAEs and 2975 LAEs, 246 taxa having survived into the last glacial. Note that fewer than 180 immigrant genera are currently recognized (Woodburne and Swisher, 1995); thus, there are 34 times as many informative events in general as there are immigration events.
..........An event sequence can be turned into a time scale with three further steps: calibrating the sequence to numerical time; placing boundaries, or "breaks," into the sequence, with the number of breaks being determined by a statistical test of the null hypothesis that apparent faunal turnover is an artifact of undersampling; and applying names to the biochrons lying between the breaks by examining the distribution of reference localities through the sequence.

Criteria for consideration of calibration points

..........The calibration employed 152 geochronologic age estimates, of which 127 were radioisotopic and 25 paleomagnetic (Table 1). Most of these estimates are discussed in Woodburne (1987) , Prothero and Swisher (1992), Swisher et al. (1993), and Fremd et al. (1994), and references given therein. Estimates were accepted if 1) their analytical errors were less than both 10% and 2.5 m.y., 2) they were considered analytically reliable by the authors, and 3) they were demonstrably within the same stratigraphic unit, or an immediately over- or underlying unit, as the relevant fossil localities, and were not separated from the fossils by an obvious unconformity. Following Naeser et al. (1980), fission-track dates on glass were rejected, but fission-track dates on other materials (especially zircon) were considered.
..........Paleomagnetic estimates presented three difficult problems. First, paleomagnetic chron assignments could in theory be treated both as numerical age estimates and as virtual stratigraphic assignments; the global chrons could be translated into a single "section" for the purpose of calculating F/L statements. But using a single source of data both as the basis for the ordination and for a numerical calibration of that ordination would be circular. Because the paleomagnetic age estimates were essential for calibrating certain intervals, such as the Paleocene, I chose to employ these data strictly for calibration.
..........The second problem concerns the reliability and independence of chron assignments. Many of these assignments are justified in the first place by a priori biochronological hypotheses regarding mammalian faunas in the same sections. Using such "dates" to calibrate the event sequence would again be circular. Furthermore, many correlations are said to be "secure" only to be contradicted by later literature. These problems are discussed later. For the moment, it will suffice to say that there is active debate on several important paleomagnetic correlations in the middle Tertiary, and that the more certain correlations in this interval are only certain because there are numerous high-precision 40Ar/39Ar age estimates in the same sections, which however render them superfluous.
..........The third problem is exactly this: chron assignments often are redundant with radioisotopic age estimates. The latter may be much more analytically precise, particular when 40Ar/39Ar dating is used. On the other hand, a single well-constrained radioisotopic tie point may lead to faunas throughout a section being assigned correctly to multiple chrons, thereby providing numerous, separate age estimates. As a compromise, I generally preferred radioisotopic estimates when these fell in the same chron as the fossil localities, but chron assignments when the radioisotopic tie points occurred elsewhere in the section.
..........These strict criteria led to many paleomagnetic assignments being rejected. However, for early Paleocene assignments the unambiguous presence of the K-T boundary allows secure identification; and in the late Neogene, the frequent reversals often allow a chron assignment to capture different information than a radioisotopic tie point elsewhere in the same section. Therefore, most of the 25 paleomagnetic estimates in Table 1 clustered in these two intervals.

Interpolation procedure

..........The problem of calibrating event sequences is best dealt with using linear interpolation methods (Alroy, 1992, 1994, 1996, in press A; Wing et al., 1995). The event sequence is numbered from oldest to youngest; concurrent range zones are computed for all localities that are tied to geochronologic age estimates; the concurrent range zones are assigned numerical values based on the numbered event sequence; and the resulting values are regressed against the age estimates. Because the pattern is patently non-linear (Fig. 1), it might be reasonable to use polynomial regression instead of linear interpolation. However, the changes in slope are so abrupt (e.g., at the K-T boundary) that an interpolation-based method seems preferable (Alroy, 1996, in press A). This method uses the smallest statistically significant set of "hinge points" to split the calibration into a series of linear segments. Significance is tested using an F-ratio that compares the sum of squares explained by progressively more complex solutions.

Figure 1 Calibration of the appearance event sequence. Events are numbered from oldest to youngest. Data points represent fossil localities that have been dated using radioisotopic or paleomagnetic methods; their positions in the event sequence are computed by finding the concurrent age-range zones of the taxa they include. Analytical errors are small enough that most of them would be smaller than the symbols used to show the data points. The calibration function (solid line) is based on linear interpolation among eight hinge points (including a point for the Recent) selected to form a monotonic series and to maximize the explained sum of squares; the 145 additional points also are shown.

..........Finding the optimal solution is difficult, and the current algorithm does not search through all possible sets of hinge points. Instead, for each possible number of segments it starts with each possible pair of initial, internal hinge points, and then adds hinges one at a time so as to maximize the explained sum of squares. Apart from the possibility that it will find suboptimal solutions, this general approach has other minor weaknesses: with the present data set it generates very weakly serially correlated residuals, showing that it is somewhat unresponsive to short-term changes in the slope of the underlying curve. However, these problems are trivial in light of the observed scatter around the interpolation lines. Thus, it does not seem that any more optimal solution, or less linear calibration function, would improve the fit substantially.
..........The monotonic interpolation algorithm selected five calibration points as statistically significant internal hinges, thereby defining six linear segments (Table 2). However, increasing the number of hinges by just one would break up an extraordinarily long Paleogene segment by imposing an additional hinge at 48.32 Ma, and would not disrupt the solution in any other way. Therefore, this slightly more complex arrangement is favored (Fig. 1; Table 3). One more hinge point after this one could be justified because its explained sum of squares implies a p value of just about 0.01, but switching to this very different eight-segment solution would put three of the hinges in the middle Miocene, a complication that would add very little explanatory power while rendering the explanatory framework much less elegant. In any case, all of these solutions explain more than 99.7% of the variance, showing that the event sequence is extraordinarily consistent as a time indicator.

Prediction error

..........One important side-benefit of the interpolation procedure is that it makes computing errors relatively easy; they are inferred separately for each segment based on the sum of squares around the line, and on the number of calibration points falling in that part of the event sequence. These errors can be applied both to age estimates for boundaries between biochrons, and to age estimates for individual localities.
..........However, there is an important complication related to the fact that both the independent and dependent variables contribute to the error. Should the sum of squares about the line of interpolation be converted to a standard deviation of the population (s) or a standard error of the mean ([s/n]^0.5)? If all the error was due to the geochronological age estimates, the interpolation line would best be analogized with the mean of multiple geochronological age estimates. Therefore, the standard error of the mean would be the appropriate estimate of the prediction error. This is essentially the method used in standard least squares regression to compute prediction error, although that equation uses an additional correction term not needed for linear interpolation.
..........In this case, however, much of the variation is due to the inaccuracy of the event sequence. This ordination error is independent of the number of calibration points. Even if two localities were of exactly the same age and the calibration was perfectly known, the two localities still would be given age estimates that would differ non-trivially as a predictable function of ordination error. This error must behave as within-population standard deviation.
..........Because the error structure of the event sequence is unknown, the balance between these two extremes cannot be estimated exactly. However, a worst-case prediction can be obtained by estimating the error due to calibration, which has two major subcomponents. These are analytical errors in the geochronological age estimates, and differences between the true ages of geochronologically analyzed stratigraphic horizons and fossil horizons ("stratigraphic error").
..........The analytical subcomponent is estimated by comparing geochronologic estimates with very low analytical errors, i.e., those obtained using the 40Ar/39Ar method, to the others. Although the "precise" estimates still do have analytical error, it is roughly an order of magnitude less, so the discrepancy serves as a reasonable minimum estimate. In the current analysis, the average sum of squares for the 48 "precise" estimates is 50.5% as great as the average for the 104 "imprecise" estimates.
..........The stratigraphic subcomponent is estimated by comparing the geochronological age estimates for horizons underlying fossil horizons to those for overlying fossil horizons. On average, the former should fall slightly to the "old" side of the interpolation line, and the latter to the "young" side. In most cases the relative stratigraphic positions are either not known or not determinable; some dated horizons are bracketed by multiple fossil horizons, and some fossil horizons are bracketed by multiple dated horizons. Nonetheless, there are 56 cases where an unambiguous relationship can be determined: 24 "underlying" dates and 32 "overlying" dates. The underlying dates average 0.241 m.y. older than predicted, whereas the overlying dates average 0.103 m.y. younger; the difference is 0.344 m.y., so the average error due to stratigraphy should be 0.172 m.y. The overall standard deviation around the prediction lines is 1.220 m.y. So if the stratigraphic error could be removed, the sum of squares would be ([1.220 - 0.172]/1.220)2 = 73.8% as great.
..........Now that separate error components have been estimated, the prediction error around each line segment can be estimated as follows:
..........1) The sum of squares around each point falling within the segment is divided into stratigraphic (26.2%) and non-stratigraphic (73.8%) components.
..........2) The remaining, non-stratigraphic sum of squares around "imprecise" age estimates is divided into analytical (49.5%) and ordination (50.5%) components. Sums of squares for "precise" estimates are treated as entirely due to the ordination.
..........3) The stratigraphic component is divided by the total number of points falling within the segment, and non-stratigraphic, analytical component is divided by the number of "imprecise" points falling within the segment. This is needed to convert the components from standard deviations to standard errors.
..........4) The ordination sum of squares that was set aside in step 2 is added back to the calibration components computed in step 3. This sum is the estimated prediction sum of squares.
..........5) The prediction sum of squares is converted to a standard deviation: first it is divided by one less than the number of points falling within the segment, and then the square root is taken. This yields the estimates given in Table 4.
..........The errors computed by this method are comparable to the analytical errors of the geochronological estimates that went into the calibration, i.e., on the order of 1 m.y.; the majority of radioisotopic estimates are based on fission track or 40K-39Ar analyses with errors this large, and paleomagnetic chrons are often 1 - 2 m.y. long. Thus, the calibration has essentially "bootstrapped" a partial set of age determinations for a handful of localities into a complete set of comparably precise estimates for all of the localities.

Break selection

..........In an earlier paper I proposed an iterative algorithm for defining discrete time intervals by setting boundaries, or "breaks," in event sequences (Alroy, 1992). Breaks must fall after a run of LAEs and before a run of FAEs. The criterion used to select breaks is consistent with the parsimony-based idea of minimizing implied, ad hoc F/L statements: each break is set at the point that maximizes the product of preceding LAEs and succeeding FAEs. LAEs are counted only if they fall before the current break point B and after the next-oldest previously defined break point. Likewise, FAEs are counted if they fall after B and before the next-youngest break. "Preceding" LAEs define the end of the biochron that extends up to B, and "succeeding" FAEs contribute to the definition of the biochron starting with B.
..........In theory, there could be as many breaks as there are runs of FAEs in the event sequence. However, most of these breaks would be uninformative. The current approach to the previously unsolved problem of determining whether a break is statistically significant involves two complementary tests, one for last appearances and one for first appearances (Fig. 2). For LAEs, the null hypothesis is that undersampling has created all of the apparent extinction within a biochron that one wishes to break up. Sampling is estimated using the "Lazarus" or "faunal completeness" method (Paul, 1982; Holman, 1985; Maas et al., 1995), which contrasts taxa known to be present before, after, and during an interval (sampled range-through taxa) with those known before and after, but not during an interval (Lazarus taxa: Jablonski, 1986). These counts are used to formulate a standard G test for independence in a two-by-two table, the entries being the numbers of sampled range-through taxa (Fig. 2A) and Lazarus taxa (Fig. 2B), and the numbers of LAEs falling at the ends of the current biochron (Fig. 2C) and of the preceding biochron (Fig. 2D). If the ratios in this table show no interaction, then apparent extinction could be explained as a sampling artifact. The comparable test for FAEs contrasts the former two counts with the number of FAEs at the beginning of the current (Fig. 2E) and succeeding (Fig. 2F) biochrons. In other words, it tests the complementary hypothesis that apparent originations are a sampling artifact.

Figure 2 G tests of LAE and FAE counts. Six different distributions are examined, four in each test. A = Sampled range-through; B = Lazarus; C = LAE at end of current biochron; D = LAE at end of preceding biochron; E = FAE at beginning of current biochron; F = FAE at beginning of succeeding biochron. Null hypotheses are illustrated at bottom; if these hypotheses cannot be rejected, apparent turnover within the current biochron can be explained as a sampling artifact.

..........A p < 0.01 cutoff was used to select 49 breaks that delineate the time scale (Table 4; Fig. 3). Breaks were accepted if G tests of either FAE or LAE counts met this cutoff. As it happened, breaks supported by one test at the p < 0.05 level were never supported at the same level by the other test. The very first break (Wh-Ge1) had to be accepted even though testing at this point was mathematically impossible because only one, undivided biochron had been defined. The five oldest breaks were accepted without support because 1) all five are conventionally recognized, including La-Pu, which corresponds to the K-T boundary; 2) their location early in the sequence meant that only the LAE test could be performed, so the absence of results from the FAE test cannot be held against them; and 3) the Pu-To1 break is antecedent to a series of later, Tiffanian breaks that do have significant support, but depend on accepting this early, less significant break. Similarly, five breaks only having support at the p < 0.05 level were accepted because more significant, but later breaks depended on them: To1-To3, depended upon by To3-Ti1; Br2B-Br3, by Br2A-Br2B; Ch1B-Ch3, by Ch1A-Ch1B; Ba2-Ba3, by Ba1-Ba2; and Bl1A-BlB, by BlB-Bl2. Four significant middle-Tertiary breaks were rejected because the resulting "biochrons" would be very short, would lack diverse and well-studied reference localities, and would be characterized by only a handful of common species. These fell within Du2, Wh, Ha1, and He1, all of which are understudied.

Figure 3 Revised North American mammalian time scale. "Epoch" and "Polarity" = global marine epochs and magnetic polarity chrons (Berggren et al., 1995); chron numbers are indicated at side of polarity column. "1987," "1995," and "Revised" = time scales of Woodburne (1987), Woodburne and Swisher (1995), and the current paper. Equivalent boundaries are indicated with dashed lines. Full names of mammalian biochrons are given in Table 4, except for "Ar" (Arikareean); due to space limitations, the names of some land-mammal subages are omitted. Arrows indicate age estimates used as interpolation points in the calibration (Fig. 1).

Reference localities

..........Although the number and empirical content of the biochrons can be determined entirely by the preceding algorithms, the names of these intervals have to be set using a criterion that will promote stability and ease of interpretation. There are only three possible approaches: tying the biochrons to individual appearance events, to joint sets of events, or to individual faunal assemblages. The first approach is widely favored (see Woodburne, 1977, 1987, 1996), but clear reasons for rejecting it are discussed later. The second approach is dangerous: if each biochron is tied to a set of FAEs and/or LAEs, the resulting time scale implies a substantive empirical hypothesis - i.e., an appearance event sequence. In other words, joint event definitions are hypothetical constructs instead of unambiguous inferences from raw data. Such hypotheses may or may not be overturned by range extensions and therefore are unstable even when the underlying faunal sequence and stratigraphy are not. Because nomenclature should refer to known entities instead of hypotheses whenever this is possible, joint event definitions should be an option of last resort.
..........The final approach is tying names to reference localities. This system captures many of the information summarized by joint event definitions because a large number of F/L statements are implied by any faunal list (Alroy, 1992, 1994), and even more are implied by a temporal sequence of lists. Reference localities not only equate with sets of F/L statements, but have a distinct, built-in advantage over arbitrary sets of events: as more fossils are collected from a reference locality, more taxa are recognized, improving the locality's correlation and narrowing the duration of the assemblage's concurrent range-zone. While individual events always are pushed in one direction or another by additional fossil discoveries, there is no expectation that these discoveries will push the concurrent range-zone forward or backward in time in addition to narrowing it. All of this makes it clear that reference localities offer the firmest basis for nomenclature.
..........The rules of priority are applied easily in the reference locality system. For example, if a biochron is split, its reference locality becomes the reference locality for one of the new biochrons, and new reference localities are designated for the others; and if multiple named biochrons within an age are merged into one, the age becomes equivalent to the first-named of these biochrons.
..........Normally, the reference locality for a land-mammal age also is the reference locality for one of the biochrons falling within that age. But in some cases, a previously designated reference locality for an age is not diagnostic of any particular biochron. In such cases the reference locality, and therefore the age, can still be retained; but specific reference localities for each biochron must be designated.
..........If a reference locality has a poor fauna it may "move" from one apparently homogenous biochron to another. In these cases, synonymized biochrons should take on the designation of either the first-named or geologically oldest among them, and "abandoned" biochrons might be assigned new reference localities. So if the Ba2 reference locality is ever placed in the last biochron of the Barstovian, "Ba3" will cease to exist, and Ba1 may have to be split into "Ba1A" and "Ba1B." However, renaming biochrons should be avoided whenever possible.
..........The name-bearing reference localities listed in Table 5 mostly have been informally recognized as such ever since the original time scale was described (Wood et al., 1941). Others were selected to meet the following criteria: certainty of correlation; affiliation with independent geochronologic data; placement in a well-understood stratigraphic section; historical importance; and, primarily, preservation of a diverse and well-described fauna that balances small and large mammals.

Characterization

..........Because the names of land-mammal ages are tied to reference localities in this paper, in theory there is no need to list "index fossils," "immigrant first appearance datum," or other such "defining criteria." In fact, I believe that listing immigration events would be positively misleading because it would foster the demonstrably false impression that these events are more informative than others (Alroy, in press B). However, it also is useful to lend some flavor to the biochrons by listing some of the most common species appearing for the first or last time within them, or as "index" species not found in other intervals. I therefore provide such lists in the Appendix. "Common" species were defined arbitrarily as those found in the largest number of the lists; the five most common species are listed in each category. I break with tradition (Wood et al., 1941; Woodburne, 1987) in listing species instead of genera because the appearances of species are much less diachronous (Alroy, in press B).
..........It also is helpful to list some of the important localities referred to each interval. Because there are 4015 faunal lists, an arbitrary criterion again is needed to select a few as "characteristic." I take an algorithmic approach to the problem, iteratively choosing lists to maximize the number of different genera and species that they contain. The most diverse list in the biochron is selected first, then the list having the most taxa not found in the first, and so on until five lists are found. The results are given in the Appendix.

Biochrons and land-mammal ages

..........Although in principle one simply could use the names of reference localities as the names of their respective biochrons (e.g., "the Thomson Quarry biochron"), in practice a smaller number of broadly-defined land-mammal age names have been used, modifying them when it is necessary to refer to individual biochrons (e.g., "the late Hemingfordian subage"). The nomenclature of land-mammal ages is muddied by the vague hierarchical distinctions that this system creates. Instances of arbitrarily erected ages and subages are numerous in the literature, and there even are cases of subages being arbitrarily reassigned from one age to another. In order to avoid such problems in the future, rigid distinctions among these categories must be drawn. I offer the following definitions in the hope that they will promote stability:
..........1) A unit biochron is an analytically indivisible biochron including a designated reference locality. Example: the interval between appearance events 4372 and 4459 is a unit biochron whose reference locality is Barstow.
..........2) A joint biochron is a temporally continuous set of two or more unit biochrons. Example: the interval between appearance events 4212 and 4686 is a joint biochron including three unit biochrons.
..........3) A land-mammal age is a name for a biochron or an arbitrary, but continuous set of biochrons that is exclusive relative to other land-mammal ages. It is a joint biochron if it spans more than one unit biochron. Example: the joint biochron including the early, middle, and late Barstovian unit biochrons is the Barstovian land-mammal age.
..........4) A land-mammal subage is a name for a biochron or an arbitrary, but continuous set of biochrons that is exclusive relative to other land-mammal subages and falls entirely within part of a single land-mammal age. Examples: the unit biochron including the Barstow reference locality is the middle Barstovian; the joint biochron including the early middle and late middle Bridgerian biochrons is the middle Bridgerian.
..........Naming conventions for the smallest biochronological units have varied. Most workers have given them subage designations, modifying these terms appropriately when biochrons are split (e.g., "early late Barstovian": Tedford et al., 1987). Some other workers have avoided subage designations and preferred only to define ages and "zones" (e.g., Archibald et al., 1987). Both sets of workers seem to agree on two points: separate names for ages and their included subages, such as "Barstovian" and its subset the "Valentinian," are confusing and unnecessary (e.g., Tedford et al., 1987), whereas the system of numbered age subdivisions introduced by Gingerich (1983; see also Woodburne, 1987; Woodburne and Swisher, 1995) is clearcut and economical.
..........I agree with these assessments, believing that terms like "Graybullian" are best left undefined, and should be avoided where possible. Additionally, the term "zone" cannot be used because the biochrons are not biostratigraphic units and are tied to reference localities instead of individual events. This leaves two possible sets of terms, both of which seem acceptable. The "early" and "late" designations do cause difficulties because many ages can be divided into three or even four biochrons, and terms like "early late," "late early," etc., are confusing. However, these designations can be made more clear by using "early," "middle," and "late" when there are three biochrons, and splitting "middle" into "early middle" and "late middle" when there are four.
..........The numbering system also is useful, but it has its own shortcoming. By numbering biochrons successively from oldest to youngest within each age (e.g., "Ti1" through "Ti6"), it follows inevitably that future revisions will demand radical changes in the numbering scheme. Just such a problem already has occurred: Gingerich (1983) proposed seven "faunal zones" for the Wasatchian; the total was cut to four by Woodburne (1987); expanded back to eight by Gingerich (1989); and then set at five by Woodburne and Swisher (1995). In the course of all these changes, fundamental disagreements over the meaning of terms like "Wa4" have followed necessarily.
..........I would propose that when biochrons are split the new subdivisions should be given lettered designations, in analogy with the nomenclature of geomagnetic polarity chrons. For example, if the middle Wasatchian (Wa3) proves to be subdivisible, the new designations should be "early middle Wasatchian (Wa3A)" and "late middle Wasatchian (WA3B)." Similar ideas have been aired by Woodburne and Swisher (1995) in their discussion of the "Arikareean."
..........Using two naming conventions at once may seem burdensome, and if there is any confusion one must be given priority over the other. Because the numbering system is more precise and flexible than the "early/middle/late" scheme, I propose that only the numbered abbreviations be treated as the formal names of the biochrons, and therefore fixed by the laws of priority. The "early/middle/late" terms should be open to modification whenever the number of biochrons within each age is changed.
..........It goes almost without saying that subages should never be transferred from one age to another. However, there are cases to the contrary; for example, the "Gardnerbuttean" was proposed as the final subage of the Wasatchian (Robinson, 1966), but then shifted to the earliest Bridgerian (Stucky, 1984; Krishtalka et al., 1987). Although I agree with this choice, it has no basis in priority and is acceptable only because other workers have agreed to ignore what should have been an unambiguous criterion. In the future, such arbitrary actions should be avoided.

Semantics

..........Some of the terms used in this paper are unorthodox and require explanation. First, subdivisions of global marine epochs are not capitalized (e.g., "late Eocene"). This is due to uncertainties in correlating terrestrial localities to global subepochs, and to the fact that the intervals of time being referred to in a particular context may transgress subepoch boundaries (e.g., "late Eocene" sensu lato may refer to more than just the Priabonian).
..........Second, the term "land-mammal age" is presented without quotes. Quotes would create needless confusion and foster the impression that mammalian biochrons are somehow less well-defined or understood than their marine analogs. As discussed later, the marine ages are untestable legalistic formalities, whereas the current system is not just precisely defined but fully testable. Furthermore, the widespread argument that the rigidly chronostratigraphic stage-age system has priority over the more informal land-mammal age system is historically inaccurate; Wood et al. (1941) greatly predates the first modern attempts to fix "golden spikes" for epochs and stages in the 1960's (Harland et al., 1990).
..........Third, I retain the term "reference locality" (Mein, 1975) because it is widely used. However, I believe it is needlessly confusing because it suggests a geographic and stratigraphic basis (i.e., a particular horizon) for what is unquestionably a biochronological concept (i.e., a time interval tied to a faunal assemblage). The term "reference fauna" would be preferable, and perhaps should be used in the future.
..........Finally, I retain use of the term "land-mammal age" while recognizing that the term "biochron" would suffice. "Biochron" is not only less controversial, but more flexible because it is not necessarily tied to a particular hierarchical level. My intention in doing so is merely to facilitate communication with the large majority of paleontologists still using this term.

NORTH AMERICAN LAND-MAMMAL AGES

..........All of the 23 widely recognized land-mammal ages (Lillegraven and McKenna, 1986; Woodburne, 1987; Woodburne and Swisher, 1995) can be identified with one or more of the 50 computer-generated biochrons. Almost all of the ages have been subdivided by other authors, either into "zones," named subages, or "early" and "late" intervals. Most of these fine-scale biochrons are preserved in the current analysis, making it easy to designate reference localities. For example, the two early Eocene ages have been split into seven subages (Krishtalka et al., 1987). All of these biochrons save the first ("Wa1," the "Sandcouleean") are recovered by the analysis. Similar patterns are seen throughout the time scale. However, it is necessary to discuss a few nomenclatural changes, new and unexpected correlations, and cases in which proposed "zones" and "subages" cannot be supported. Reference localities are indicated by bold lettering.

Cretaceous

..........The Aquilan is known from only two field areas but seems clearly distinct. The Judithian is even less problematic. Reference localities for these two ages (Verdigris Coulee [UA-MR-6]; Clambank Hollow) are unambiguous. Only a handful of faunas have ever been assigned to the "Edmontonian" (Lillegraven and McKenna, 1986), including Scabby Butte, the reference locality. The current analysis recovers these correlations, but the interval is so poorly documented that its status must be considered uncertain. The Lancian is well-established, the reference locality (Lull 2 Quarry) being the best and longest-studied assemblage in the Lance Formation. However, some of the Lancian faunas falling close to the K-T transition have been defined as the "Bugcreekian" (Sloan, 1987). This is hotly disputed (Archibald et al, 1987), with recent studies showing that the Bug Creek fauna itself is composed of mixed Cretaceous and Paleocene fossils (Lofgren, 1991). Only one similar fauna is known from elsewhere, namely, at the MHBT Quarry in Saskatchewan (Johnston and Fox, 1984). Like the others, this one occurs at the base of a channel fill sequence cutting into the highest level of a Cretaceous formation. The current analysis treated all of these "transitional" faunas as taphonomic composites, with separate Cretaceous and Paleocene lists being defined for the purpose of ordination. After making this conservative adjustment, no evidence for a "Bugcreekian" or any other subdivision of the Lancian could be found.

Paleocene

..........Earlier time scales included 15 nominal "interval-zones" in the Paleocene (Archibald et al., 1987), each of which was based on just one or two appearance events (e.g., "Pu3," defined by the first appearance of Taeniolabis taoensis). Here, the 10 m.y.-long Paleocene is split into nine biochrons whose durations are more typical of the overall time scale. No support is found for any of the three Puercan "zones," although earliest Puercan faunas like Mantua Lentil do fall out first in the Puercan part of the event sequence, well before Mammalon Hill. Similarly, zones "To1" through "To3" are barely recognizable. "To2" has not been clearly defined outside of the "type" Torrejonian sediments of the San Juan Basin, and its best faunas are early Torrejonian. I therefore synonymize To1 and "To2" until the faunal distinction can be shown more clearly. To1 includes Dragon Canyon, the reference locality of the widely rejected "Dragonian" age (Wood et al., 1941).
..........Supposed "To3" faunas are split between early Torrejonian (Swain Quarry, Little Pocket) and late Torrejonian (Gidley Quarry, Rock Bench Quarry). Some of these recorrelations may be due to compression of the "type" Torrejonian faunal sequence into a single biochron, which points to inadequate faunal information. Unfortunately, Archibald et al. (1987) implied that their To3 was based primarily on the uppermost Torrejonian faunas in Torreon Wash, but these faunas fall just short of the relevant boundary. Nonetheless, I retain the term To3 for the late Torrejonian biochron, designating East Flank Torreon Wash (Lower) and Gidley Quarry as the respective reference localities in the hope that this will promote stability.
..........The six Tiffanian "zones," all of which were defined on presumed ranges of Plesiadapis species (Gingerich, 1983; Archibald et al., 1987), are reduced to four. These earlier biochronological assumptions were unparsimonious. Although all of the Plesiadapis species appear in the event sequence in the predicted order, the first appears in the late Torrejonian (P. praecursor), and the last in the early Clarkforkian (P. gingerichi); another species (P. simonsi) only has been reported from two undescribed localities of unstated stratigraphic position, so its age-range is indeterminable. Like P. praecursor, each of the remaining three species appears one biochron earlier than predicted. This is due both to minor disagreements over faunal correlation and to the inevitable age-range extensions that come with better sampling.
..........Regardless of such details, it is quite easy to designate reference localities because the most important assemblages that were listed for the Tiffanian "zones" generally fall where predicted by Archibald et al. (1987). The four key Ti1 localities all are assigned to that interval (Bangtail, Cochrane 2, Douglass Quarry, Shotgun). Cochrane 2 is easily the most diverse, but its correlation is insecure because of its marginal position at the very last point in the event sequence before the next biochron. Shotgun presents no such difficulties.
..........The two key Ti2 localities (Saddle; Scarritt Quarry) again fall in the predicted biochron. However, so does Mason Pocket, the reference locality for the Tiffanian (Wood et al., 1941) and therefore for this biochron. Ti3 was most closely tied to Cedar Point Quarry, which again falls where predicted. Ti4 was believed to include the inadequately known Mason Pocket assemblage, as well as several samples that are assigned here to Ti3 (e.g., Malcolm's Locality; Olive Locality). Although "Ti5" was nominally based on the first appearance of the elusive P. simonsi, it clearly was characterized by the Princeton Quarry fauna. Because this and the preceding "zone" are indistinguishable, and because Princeton Quarry is the best-known assemblage from Ti4 sensu lato, I designate it as reference locality. As admitted by the authors (Archibald et al., 1987), "Ti6" cannot be characterized clearly by any faunal assemblage. I believe it is indistinguishable from Cf1.
..........The extraordinary changes that would be required by a literal interpretation of Archibald et al.'s (1987) Tiffanian Plesiadapis zones, despite the remarkable stability of their hypothesized faunal correlations, is a good demonstration of the zone system's impractical nature.
..........The Clarkforkian is modified in one major way: almost all of the old "Cf1" through "Cf3" localities from the classic Polecat Bench section in northern Wyoming are placed in the late Clarkforkian. This is supported by the existence of a long, poorly fossiliferous interval between the unambiguously Tiffanian and Clarkforkian parts of the section (Rose, 1981). Nonetheless, major Cf1 localities from other areas (e.g., Big Multi, Bear Creek) do form the core of a robust early Clarkforkian. Cf2 sensu lato includes virtually all of the Clark's Fork Basin assemblages discussed by Rose (1981), the best-known being Holly's Microsite.
..........A final issue is calibration. The Torrejonian has been thought to be much shorter than the succeeding Tiffanian (Archibald et al., 1987). This is still true in the current scheme, but the date for the Torrejonian-Tiffanian boundary may be too young (63.0 +/- 0.5 Ma vs. about 64.0 Ma in Swisher et al., 1993). The current correlations, when matched with the paleomagnetic stratigraphy of Williamson (1996), also suggest that the early-late Torrejonian boundary (61.7 +/- 0.5 Ma) is within chron C27n (60.9 - 61.3 Ma: Cande and Kent, 1995) or even younger. Because the former discrepancy is about twice the standard error, it is marginally significant and troubling: the early Torrejonian may actually be much longer than the current estimate of 1.3 m.y. If any explanation is called for, it is that higher rates of turnover during the Puercan than later in the Paleocene may have distorted the calibration in this interval.

Eocene

..........The Eocene yields few surprises. A suggested "Sandcouleean" or "Wa1" subage in the earliest Wasatchian (Krishtalka et al., 1987; Gingerich, 1989) is not recovered, although the relevant faunas are in the correct order within the early Wasatchian biochron. This is unremarkable given the fact that for decades, most workers have avoided subdividing the early Wasatchian ("Graybullian" sensu lato); only in recent years has more and more research on the Wasatchian made it desirable to further subdivide it. Because the first Wasatchian biochron is dominated by traditionally "Graybullian" faunas, Wa2 is the preferred term for "Wa1" and "Wa2" of Krishtalka et al. (1987) and "Wa0" through "Wa4" of Gingerich (1983, 1989). Wa3 and Wa4 ("Lysitean" and "Lostcabinian") are used in the sense of Krishtalka et al. 1987, although the Wa2-Wa3 boundary is slightly older than in earlier schemes, with Wa3 now including the "Bunophorus Interval Zone" of Schankler (1980). Elk Creek (YPM 356) is the best fauna from the Graybull area that falls unambiguously in Wa2. The remaining Wasatchian reference localities are uncontroversial (Lysite, Buck Spring Dark Red Stratum), as is that of the early Bridgerian or "Gardnerbuttean" (Huerfano II).
..........Subdivision of the middle Bridgerian (Br2) is a novel feature, but it is compatible both with stratigraphy and with the distinct features of the Bridger A fauna (Gunnell and Bartels, 1994). Grizzly Buttes and Henry's Fork Hill (Bridger D) are the classic Br2B and Br3 faunas. Definition of the Bridgerian-Uintan boundary has long been problematic because it has been believed that the Bridger and Uinta formations are separated by a gap in time. However, I find no support for an intervening "Shoshonian" subage (Flynn, 1986; Krishtalka et al., 1987; McCarroll et al., 1996). This biochron was based on several faunas that now appear to be much older (Washakie A, Br2B-Br3 boundary and C21n or C21r; Sand Wash and Tepee Trail Bonebed A, Br3 and older C20r); only the Poway and Whistler Squat faunas seem to be early Uintan (younger C20r). The relatively young date of the Bridgerian-Uintan boundary (here 46.4 +/- 1.0 Ma, but > 48 Ma in Krishtalka et al., 1987) is not the source of the discrepancy, because it is due to revisions of paleomagnetic and radioisotopic data by other workers (Woodburne and Swisher, 1995; Walton, 1992). Because the term "Shoshonian" is directly tied to a locality that appears to be Bridgerian (Tepee Trail Bonebed A), and because the Uintan only includes two clearly separated biochrons to start with, I conclude that recognition of the term "Shoshonian" would be premature and unnecessary. In any case, the nominally "Shoshonian" Poway fauna appears to provide the most balanced early Uintan fauna with unambiguous paleomagnetic dating, Poway Pipeline Two being one of its best quarry assemblages.
..........A side-issue is that the current analysis places both the uppermost Bridger Formation faunas and a key fauna from low in the Uinta Formation (Uinta B1) in the late Bridgerian (Br3). However, both the Uinta B1 fauna and the late Uintan Uinta B2 fauna are in the same reversed chron, identified by Prothero and Swisher (1992) as C20r. Chron C20r is 2.5 m.y. long in the latest paleomagnetic time scale (Cande and Kent, 1995), whereas the gap between Uinta B1 (46.9 +/- 1.0 Ma) and Uinta B2 (43.1 +/- 1.0 Ma) is 3.8 m.y. Worse, the early Uintan includes several faunas that that seem to straddle the C20r-C21n transition (e.g., Lower Washakie B: McCarroll et al., 1996). Unless there is an overlooked normal interval in Uinta B that can be equated with C21n, it seems most likely that the out-of-date and large mammal-dominated Uinta B1 faunal list has been miscorrelated.
..........The Duchesnean has long been controversial, but I agree with other workers (Lucas, 1992) that it is useful and recognizable and that it can be subdivided. Unfortunately, the type Uintan fauna (Myton Pocket) falls just before the Uintan-Duchesnean boundary, whereas the type Duchesnean fauna (LaPoint Member) falls just before the early-late Duchesnean boundary. Hence, at least three alternative sets of nomenclatural changes could be made necessary by future analyses. This instability cannot be avoided because the faunas involved were implicitly, but unambiguously designated as reference localities by Wood et al. (1941). The precarious locations of these two reference localities explains why the date of 42.4 +/- 1.0 Ma for the beginning of the Duchesnean is older than the chron C18r (40.1 - 41.3 Ma) estimate implied by Prothero and Swisher (1992) and Prothero (1996); their concept of "late Uintan" in fact includes some early Duchesnean faunas.
..........The Duchesnean-Chadronian boundary falls at 37.7 +/- 1.0 Ma, older than the estimate of Prothero and Swisher (1992) but not significantly so. Diamond O Ranch is the most diverse of the securely-dated Du2 faunas. The Chadronian-Orellan boundary falls at 33.6 +/- 1.0 Ma, which is indistinguishable from the 33.91 +/- 0.06 40Ar/39Ar age estimate used as a proxy for this boundary by Swisher and Prothero (1990). Therefore, the results confirm confirm their placement of the Chadronian entirely in the Eocene.
..........The current analysis agrees with earlier studies that recognized three distinct intervals within the Chadronian (Emry et al, 1987); however, the content of these subdivisions is greatly modified, with Ch1 being split and "Ch2" being merged into Ch3. A case could be made that Ch1B is the same as the old "Ch2," but all the key "Ch2" faunas are placed in the early part of Ch3: the medial faunal zone from the type area of the Chadron Formation in South Dakota, all but the very oldest of the Flagstaff Rim faunas, and Pipestone Springs. Other, less crucial "medial" Chadronian faunas are removed to Ch1A (McCarty's Mountain, Pilgrim Creek, Raben Ranch) and Ch1B (Airstrip, Ash Spring, Calf Creek, Thompson Creek). Ch3 is preferred over "Ch2" because all of the key faunas originally referred to this biochron still are in the current analysis. Ch1 was tied to the lower Chadron Formation fauna; Emry et al. (1987) indicated that this was best typified by the Yoder assemblage, which falls in Ch1A. Ch1B is best represented by Calf Creek, and Brecht Ranch is designated the reference locality of Ch3 because the Chadron section and fauna are better-described in this part of Nebraska than in South Dakota.

Oligocene

..........The Orellan and Whitneyan are recovered with little modification; Cottonwood Pass (Lower Nodular Zone) and Indian Stronghold (Protoceras Channel) are typical, if understudied assemblages from the classic Big Badlands field area. Both of these intervals are very short and cannot be subdivided meaningfully. There are no major surprises in correlation other than that the diverse, but poorly described Slim Buttes sequence (South Dakota) and Orella D small mammal faunas (Nebraska) appear to be Whitneyan. This may be due to incomplete faunal accounts. The controversial Cedar Ridge fauna (Wyoming) is placed in the Orellan, contra Emry et al. (1987) but in accord with several recent taxonomic studies.
..........The widely accepted "Arikareean" Land-Mammal Age is split into four biochrons, each with a clear stratigraphic basis. This 11.0 m.y. interval is more than twice as long as any other named age (none last more than 4.8 m.y.), and it spans the Oligo-Miocene boundary (Steininger et al., 1997). I therefore follow some earlier authors (Martin, 1980) by splitting the "Arikareean" into Geringian (roughly late Oligocene) and Harrisonian (early Miocene) ages. This is the largest nomenclatural change suggested by the present analysis. Because I believe that hierarchical distinctions such as those separating "subages" and "ages" are artificial, I allow that the "Arikareean" might be retained as a "superage" or biochron of unnamed rank that encompasses the Geringian and Harrisonian.
..........Although none of the middle Tertiary paleomagnetic correlations were used in the current analysis, the Whitneyan-Geringian boundary estimate of 30.0 +/- 1.0 Ma still accords well with its placement in chron C11r (Tedford et al., 1996; this is 30.1 - 30.5 Ma: Cande and Kent, 1995). The Gering Formation of western Nebraska is entirely early Geringian and probably does span a very short interval of time, as indicated by Tedford et al. (1996); Durnal Ranch Quarry is its best fauna. On the other hand, the partially correlative Sharps fauna of South Dakota appears to span two biochrons. This is unsurprising because the very best early and late Geringian localities in the Wounded Knee section (SDSM V-5360; SDSM V-5362) are also in separate stratigraphic units (Macdonald, 1963). The superposed Monroe Creek Formation yields the best late Geringian fauna (Monroe Creek [SDSM V-6229]).
..........The John Day Formation in central Oregon presents major difficulties. Faunal lists and high-quality geochronology are available, but major alpha taxonomic problems have been left unresolved for eight decades. As a result, four distinct faunal horizons dated securely at between 30.0 and 27.2 Ma (Fremd et al. 1994) are compressed into a 0.5 m.y. window between 28.6 and 28.1 Ma. All of these dates, however, fall within the early Geringian, which shows that the statistical tests for distinguishing biochron boundaries are sensitive to shortcomings in the data such as these. The underdescribed Cabbage Patch sequence (Montana) presents an opportunity to resolve some of these difficulties because its early and late Geringian faunas seem to be a temporally correlative geographic intermediary between the Sharps and John Day sequences.

Miocene

..........Both the Harrisonian and Hemingfordian are poorly known, with much room for improvement in correlation and calibration. However, the data suggest that each interval can be subdivided. The early-late Harrisonian boundary may correspond to the hiatus between the Harrison Formation sensu stricto and the Upper Harrison Beds (Tedford et al., 1987), because the best-known faunas from each unit fall at either side of the boundary (AMNH Rosebud 7, Keeline, and Pine Ridge Escarpment vs. Morava Ranch Quarry and Agate Springs Quarries). This split corresponds with the intentions expressed by Tedford et al. (1987) in their discussion of "Ar3" and "Ar4" biochrons, making the designation of reference localities unproblematic.
..........The late Harrisonian includes several faunas previously assigned to the early Hemingfordian (Flint Hill North, Martin Canyon Quarry A, Thomas Farm), leaving the early Hemingfordian sensu stricto quite poorly known. Tedford et al. (1987) were correct that it spans the time represented by the understudied Runningwater Formation, best known from Cottonwood Creek (UNSM), but it may also continue into superposed the Box Butte Formation. The late Hemingfordian and later intervals are less problematic and include classic localities from across the continent. Changes are few. He2 includes Thomson Quarry, the best-known assemblage within the "Sheep Creek fauna" that was central to the original definition of the age (Wood et al., 1941). Ba1 is poorly known; the redescribed Anceney assemblaged is diverse and well-balanced. Ba2 is tied to the Barstow fauna by the age's original definition (Wood et al., 1941).
..........Although the first appearance of proboscideans substantially predates the early-middle Barstovian boundary, contra Tedford et al. (1987), assignments of Barstovian faunas to subages are quite conservative. The major change is the transferral of the basal Valentine Formation faunas, notably Norden Bridge Quarry, to the late Barstovian. Norden Bridge Quarry is the best-known Neogene faunal assemblage in North America. The Valentine Formation appears to be younger in general than believed by Tedford et al. (1987); for example, the Burge Quarry fauna from the upper part of the formation is early Clarendonian, not late Barstovian. Another interesting change is that the Fleming Formation (Texas Gulf Coastal Plain) spans the early Hemingfordian through middle Barstovian, not the early through late Barstovian.
..........The Barstovian-Clarendonian boundary is given the same age estimate as in earlier studies (11.5 Ma: see Tedford et al., 1987; Woodburne and Swisher, 1995). As noted, however, the Clarendonian includes a few faunas often said to be Barstovian. This is a minor discrepancy relative to the standard error of time estimates in this span of the calibration (1.1 m.y.).
..........The three-way division of the Clarendonian is new. These intervals have almost equal durations, but only the last corresponds clearly with the late Clarendonian Cl2 biochron sensu Tedford et al. (1987). Therefore, I treat the first two biochrons as subdivisions of Cl1. The best-known fauna from the classic Clarendon Beds is MacAdams Quarry, which is early Clarendonian as argued by Tedford et al. (1987). Middle Clarendonian faunas are best-known on the West Coast, although the important Big Spring Canyon fauna (South Dakota) also is middle Clarendonian. The Ricardo fauna (California) is one of the most diverse, most taphonomically balanced, and best-dated Clarendonian faunas known.
..........The late Clarendonian is slightly modified in that the Love Bone Bed fauna is removed to the early Hemphillian, joining several other faunas from Florida that are very similar if apparently slightly younger (e.g., McGehee Farm; Mixson's Bone Bed). Although the "late Clarendonian" Xmas Quarry fauna is seemingly middle Clarendonian, all other quarries from this channel system are retained in the late Clarendonian (e.g., Hans Johnson Quarry). Of the remaining classic Cl2 faunas, all are either relatively depauperate or dominated either by large or small mammals. Although described after Tedford et al. (1987), Lemoyne Quarry (Leite, 1990) contains a diverse, well-balanced fauna from a stratigraphic unit (the upper Ash Hollow Formation) that always has been identified with this time interval.
..........The Hemphillian was divided into three subages by Tedford et al. (1987), but their "early early" and "late early" Hemphillian cannot be distinguished. Cambridge, also the reference locality for the spurious "Kimballian" Land-Mammal Age (Schultz et al., 1970), is also the most diverse early Hemphillian assemblage. Because Tedford et al. (1987) implicitly based their Hh2 on the Cambridge fauna, "Hh1" is treated as an invalid synonym. Meanwhile, the "late Hemphillian" of Tedford et al. (1987) is split. Hh3A includes Coffee Ranch, the reference locality of the Hemphillian Land-Mammal Age. True late Hemphillian, Hh3B assemblages are known from coast to coast; the best of them is Yepomera (Chihuahua). The distinction between Hh3A and Hh3b was foreshadowed by Tedford et al. (1987).

Plio-Pleistocene

..........The Blancan traditionally has been divided into early ("Rexroadian"; Bl1) and late ("Senecan"; Bl2) subages. This arrangement is supported strongly by the analysis, which furthermore allows subdivision of Bl1. Interestingly, Bl1A conforms very closely to the informal earliest Blancan interval discussed by Lundelius et al. (1987); as implied by these authors, the most well-balanced fauna in this biochron is White Bluffs. Bl1B includes the reference locality of the Blancan (Red Quarry, the largest individual assemblage in the Blanco fauna) in addition to that of the "Rexroadian" (Rexroad KU Locality 3) and other correlative faunas discussed by Lundelius et al. (1987). The late Blancan is unproblematic, and Seneca, the reference locality for the "Sencan," does fall within it. A much more elaborate system of "zones" within the Blancan (4.3 - 1.4 Ma), based on first appearances of microtine rodents, also has been proposed (Repenning, 1987). This chronology cannot be replicated, providing yet another case in which zones based on a handful of species fail to yield stable and reproducible definitions.
..........The Irvingtonian has been divided into either two or three conventional subages (Lundelius et al., 1987). The first of these ("Sappan"; Ir1) appears to be divisible. The Irvington fauna falls in Ir1A and becomes its reference fauna. The best-known Ir1B fauna is probably Leisey Shell Pit 1A; geochronological data including paleomagnetic stratigraphy suggest that it falls in chron C1r.1r (Jones et al., 1995; this is 1.07 - 1.77 Ma: Cande and Kent, 1995), which would make it the only diverse fauna in Ir1B with any kind of dating that agrees with the biochron's suggested span. However, Lundelius et al. (1987) indicated that the well-known Gilliland fauna represented a younger phase of their "early Irvingtonian," and this generalized fauna's best individual quarry sample (Burnett Quarry) therefore has priority as reference locality.
..........The later "Irvingtonian" ("Cudahyan" and "Sheridanian"; "Ir2") shares the same biochron as the classical Rancholabrean, based on Rancho la Brea. Although the relevant faunas do fall out within the event sequence in roughly the predicted order, the amount of turnover between these intervals is not significant. Therefore, "Ir2" must for the moment be treated as a synonym of Rlb.

..........ALTERNATIVE APPROACHES

..........Although some workers agree that mammalian time scales should be treated strictly as biochronological entities (e.g., Lucas, 1993), many North American mammalian paleontologists have advocated a completely different, nominally "chronostratigraphic" approach. This methodological disagreement is clear and profound. In explicitly chronostratigraphic schemes, a "golden spike" that represents the base of an interval must be set at a fixed point in a measured section (Hedberg, 1976). Mammalian paleontologists have taken two broad approaches to the perceived problem of transforming land-mammal ages into chronostratigraphic units like these: one involves biostratigraphy, and the other focuses more on geochronology. Both of these approaches are brought together by certain workers (e.g., Woodburne and Swisher, 1995), but they involve logically distinct claims.
..........Before discussing this literature, it is important to note that none of these efforts has followed the international chronostratigraphic codes exactly. True "golden spikes" correspond to stratigraphic levels whose relative order within any particular section is set by definition and cannot be modified by future discoveries. Although normally selected after careful consideration of paleontological and other data, at their very core they are not paleontological, have no necessary connection to data or hypotheses, and cannot be restricted to particular geographic regions and taxonomic groups.
..........None of this forbids recasting the land-mammal ages as formal stages whose bases would be tied to particular stratigraphic horizons in particular stratotype sections. However, this would create an intractable problem: the new land-mammal "stages" would be entirely redundant with the international chronostratigraphic units. Because their defining criteria would be purely stratigraphic and not paleontologic, nothing about these criteria would distinguish land-mammal "stages" from the standard stages. The systems would merely slice up the same global intervals of time in slightly different ways. Although the characterization of land-mammal "stages" of course would involve fossils, the international Cenozoic stages already do make use of the land-mammal record for the purpose of characterization (e.g., Steininger et al., 1997). In light of this, it is not surprising that does all North American workers seem to prefer not using purely stratigraphic "golden spikes" to define land-mammal ages. Instead, workers who favor a "chronostratigraphic" approach have favored slightly broader approaches.

Immigrant first appearance datums

..........In the most popular nominally chronostratigraphic approach, "golden spikes" are replaced with "immigrant first appearance datums" (IFADs: Woodburne, 1977, 1987, 1996; Woodburne and Swisher, 1995). Although they are not true "golden spikes," IFADs are believed to maintain all the advantages of formal chronostratigraphic markers: stability, objectivity, and synchrony. This belief is based on empirical claims, namely, that immigrant taxa are easily recognized as such and that immigrants disperse instantaneously across continents.
..........The IFAD system suffers from several shortcomings. The first is logical. Unlike "golden spikes," IFADs are not stratigraphic horizons; they are hypothesized correlations of biological events to stratigraphic horizons (Walsh, 1996). Therefore, they are subject to all the vagaries of sampling that make paleontological age-ranges subject to revision. And because new discoveries of fossils always can extend age-ranges back in time, IFADs are neither stable within sections nor necessarily isochronous among sections. Such discoveries have forced workers to let land-mammal ages swallow up faunas usually referred to preceding biochrons ("early Barstovian" based on Copemys: Lindsay, 1995), or to substitute the species making up the "stable" IFAD definitions while preserving the traditional faunal correlations ("late Barstovian" based on Proboscidea: Woodburne and Swisher, 1995; or compare early Paleocene "zones" of Archibald et al., 1987 and Williamson, 1996). Cases like these show that if anyone took IFADs seriously, the relative order of biochrons would frequently have to be reversed.
..........A second problem is methodological. Claims that IFADs are broadly synchronous across the continent are not supported by quantitative analyses that compare appearances in the mid-continent to those to the west of the Rockies (Alroy, in press B). In brief, the diachrony of genus-level IFADs averages about 3.0 m.y., which is even greater than that of native first appearances (2.6 m.y.). Not only that, but last appearances of both native and migrant genera (2.1, 1.8 m.y.) are less diachronous than first appearances of either kind.
..........A final problem is practical. IFADs constitute only about 2% of all North American mammalian appearance events, which means that a pure IFAD system would ignore the vast bulk of the data. Worse, the North American mammalian record includes thousands of scattered fossil localities, 62.2% of which cannot be placed in even a crude stratigraphic section. Together, these facts mean that IFADs by themselves only can be used to correlate a handful of localities. Therefore, quantitative biochronological analyses of entire faunas would be helpful in any correlational scheme.
..........In practice, the IFAD system's difficulties have resulted in IFADs being recognized only in light of broader studies that have characterized biochrons with well-known faunas (e.g., Woodburne, 1987). Thus, IFADs actually have been treated as secondary biochronological hypotheses, not as free-standing biochronological definitions. For this reason, for practical reasons, and because the IFAD system conflates the issues of naming and hypothesis testing, it is not employed here.

Geochronology

..........The alternative approach to the supposed "chronostratigraphy" problem is to focus on geochronological analyses. With enough radioisotopic and paleomagnetic data, one could hope to tie every faunal assemblage to a geochronological age estimate, thereby sidestepping the paleontological issues that can get in the way of a pure, inflexible chronostratigraphic nomenclature. In such a system, the bases of mammalian "stages" would not even need to be tied to biological events. For example, the Chadronian-Orellan boundary might simply be set at the geochronologically dated Purple White Layer (e.g., Swisher and Prothero, 1990). Although no worker has unambiguously advocated such an approach, it very nearly has been put to work by Prothero (1995) as a de facto solution to correlational problems in the mid-Tertiary.
..........The geochronological approach suffers from the same three general failings. Logically, a time scale based only on numerical values for arbitrarily defined time unit boundaries says nothing at all about biological events within a particular taxonomic group on a particular continent. Employing this strategy would lead to a reductio ad absurdum in which the land-mammal age nomenclature would lack any independent information content; one could simply abandon terms like "Orellan" without changing any of the age estimates, such as "chron C13r," for individual fossil assemblages.
..........Methodologically, there has been excessive confidence in the robustness of geochronological results. It is well known that the vast majority of fossil localities are not stratigraphically proximate to radioisotopically dateable horizons, which means that the geochronological approach would have to focus instead on paleomagnetic stratigraphy. But paleomagnetic correlations are remarkably fluid; they have been repeatedly overturned on the basis of new biochronological assumptions and improvements in the global paleomagnetic time scale. This is seen clearly in the middle Tertiary record, where paleomagnetic correlations are numerous. Is the Uinta B interval mostly in chron C20r (Prothero and Swisher, 1992) or C19r (McCarroll et al., 1996)? Does the Flagstaff Rim section, which has long had the most extensive radioisotopic dating of any Chadronian sequence, end with chron C11r (Prothero et al., 1983; Prothero, 1985), C12r (Swisher and Prothero, 1990), or even C13r (Prothero and Swisher, 1992)? Is the Chadronian-Orellan boundary within chron C11r (Prothero et al., 1983; Prothero, 1985) or C13r (Swisher and Prothero, 1990)? Is the base of the Gering Formation within chron C9n (Prothero et al., 1982), C10r (Prothero and Swisher, 1990), or C9r (Tedford et al., 1996)? Does the Deep Creek Tuff level in the classic John Day sequence correlate with chron C7n (Prothero and Rensberger, 1985) or C9n (Woodburne and Swisher, 1995)? Given this host of uncertainties, it seems premature to declare that paleomagnetic stratigraphy should become the primary method for correlating terrestrial sections.
..........The final, practical problem is the unlikelihood that even a large minority of fossil assemblages could ever be dated geochronologically. This is an insurmountable hurdle that has nothing to do with the reliability of the methods themselves. At present, there are 4015 assemblages and 152 separate geochronological age estimates, and only 281 of the former (7.0%) are tied to any of the latter. Again, this leaves biochronology as the only way to place the vast majority of assemblages and appearance events into a temporal scheme. Without using explicit and testable biochronological methods to tie faunal patterns into "chronostratigraphic" definitions and geochronological data, workers who favor those methods will have little to show for their efforts.

CONCLUSION

..........The limited number of discrepancies between traditional, subjective correlations and the current, almost completely automated analysis has several important implications. First, it shows that faunally-derived paleontological time scales are generally robust, even when methodological disputes later cloud their definition. Second, it shows that considerable effort can be spared by directly computing time scales from raw faunal, stratigraphic, and geochronologic data. Computed time scales have such side-benefits as generating diversity data (Alroy, 1994, 1996, in press A; Wing et al., 1995). Finally, the example shows that time scales of long duration and wide geographic utility may now be viewed as explicitly testable scientific hypotheses instead of legalistic formalisms.
..........Two remaining issues have been avoided up to this point. First, the argument for separating nomenclature and temporal hypothesis testing does not merely apply to the mammalian fossil record. In theory, the entire paleontological record could be brought together in a single analysis like the one presented here. Unfortunately, the results would be incompatible with the international system of chronostratigraphy: the formally defined stages could not be equated with the most parsimonious boundaries among analytically-derived biochrons, because these boundaries often would not correlate exactly with the locations of "golden spikes" in particular sections. Conceivably, new names could be given to the biochrons that would not be tied to the old stages, series, etc., and the chronostratigraphic system therefore could be maintained in parallel. But because virtually all of the paleontological, stratigraphic, and geochronological data in the geological record could be brought together in such a system, the chronostratigraphic time scale itself would retain very little useful, independent information, and abandoning might would present a more rational alternative.
..........Second, because all of the faunal lists were used in the ordination and therefore have positions in the event sequence, all of them have been given independent age estimates with error terms. This fact has an important corollary. The land-mammal age system per se has now been rendered a mere appendange of methods that use fossils to define the flow of time: any conceivable scientific problem that might require paleontological dating could proceed directly from the locality-specific age estimates, and thereby avoid the age system. All of this raises the question of whether said system should simply be abandoned. I believe that the question is best left to the entire paleontological community. For the present, I offer a revision of this admittedly typological scheme for the benefit of those workers who still find it useful.

ACKNOWLEDGMENTS

..........I thank K. Roy and P. Wilf for comments on the manuscript, W. DiMichele, D. Erwin, B. Huber, S. Suter, and S. Wing for reviews of earlier drafts, and numerous correspondents, including B. Albright, T. Fremd, D. Garcia, G. Gunnell, D. Prothero, and R. Stucky, for contributing to the database. This research was supported by the University of Chicago, the University of Arizona's Research Training Group in the Analysis of Biological Diversification, and the Smithsonian Institution.

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