SEDIMENT ACCUMULATION RATES AND THE COMPLETENESS OF STRATIGRAPHIC SECTIONS' PETER M. SADLER Dept. of Earth Sciences, University of California, Riverside, CA 92521, USA ABSTRACT A compilation of nearly 25,000 rates of sediment accumulation shows that they are extremely variable, spanning at least 11 orders of magnitude. Much of this variation results from compiling rates determined for different time spans: there is a systematic trend of falling mean rate with increasing time span. The gradients of such trends vary with environment of deposition. Although measurement error and compaction contribute to these regressions, they are primarily the consequence of unsteady, discontinuous sedimentation. The essential character of the unsteadiness may be cyclic or random, but net accumulation is characterized by fluctuations whose magnitudes increase with increasing recurrence interval. Ratios of median long- to short-term accumulation rates provide a measure of the expected completeness of sedimentary stratigraphic sections, at the time scale of the short-term rate. Expected completeness deteriorates as finer time scales are considered. INTRODUCTION Continuity and steadiness of sedimentation are troublesome notions. Sediment comprises discrete particles, so its deposition is inevitably discontinuous and unsteady. Clearly the discontinuity due to the particulate nature of sediment is appreciable only in terms of rather precise time and thickness scales. At coarser time scales it might seem appropriate to adopt a continuum convention as in fluid mechanics (Batchelor 1967) and thus ignore the particulate nature of sediment. But there are other sources of discontinuity in sedimentation. Sedimentation is a result of unsteady geomorphic processes: fluctuations in erosion, transport and subsidence can produce discontinuities at coarser, geologically pertinent time scales. At such scales, the convenient assumption that sedimentation has been steady could prove to be as inappropriate as applying the continuum convention to the entire space beneath a dripping faucet. Continuity of sedimentation is directly ' Manuscript received March 12, 1981; revised May 18, 1981. [JOURNAL OF GEOLOGY, 1981, vol. 89, p. 569-584] © 1981 by The University of Chicago. 0022-1376/81/8905-011$1.00 linked to the notion of completeness in stratigraphic sections, and it follows that all sedimentary stratigraphic sections must contain hiatuses at some scale. Although incompleteness is readily acknowledged (e.g., Barrell 1917; Reineck 1960; Miller 1965; Newell 1972; Ager 1973; Schumm 1977), it is less easily quantified. Incompleteness is often summarily or tacitly assumed to be either very significant or negligible. Eldredge and Gould (1972, p. 27) exposed this arbitrariness in discussions of biological evolution. Completeness of stratigraphic sections is a critical limiting factor in the precision of many geological scales and correlations. It is thus evidently desirable to be able to quantify the continuity of sedimentation and the completeness of stratigraphic sections. Reineck (1960) and Newell (1972) have used the observation that sedimentary sequences are much thinner than would be inferred from modern sedimentation rates as a means to estimate some local values of completeness. The burgeoning number of radiometric age determinations now provides a means of evaluating rates of sedimentation for a wide range of time spans in most depositional environments. In this paper I shall attempt a more general quantification of completeness using Reineck's (1960) approach with a more substantial data base of sedimentation rates. The 569 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). PETER M. SADLER attempt will face three difficulties outlined below. Single stratigraphic sections which have been intensively dated without the assumption of steady accumulation are generally unavailable. I shall, therefore, combine information from separate sections and must be limited to probabilistic, rather than deterministic, statements of completeness. I shall attempt to elucidate the degree of dependence of sedimentation rates upon time span and thus provide the means of estimating the probable completeness of stratigraphic sections. This will be a preliminary analysis of nearly 25,000 rates compiled from over 700 references which either quoted rates or provided observations, radiometric age determinations, and descriptions of fossiliferous sedimentary sections from which rates could be calculated. Biostratigraphic units were assigned approximate durations after Cohee, Glaessner and Hedberg (1978), Harland and Francis (1971), Harland, Smith and Wilcock (1964), and Van Eysinga (1975), in that order of decreasing priority. These data are positive, measured rates of sediment accumulation. They can be representative only of measurable rates. If our completeness estimates are to be of value in assessing the precision and potentials of stratigraphy, there is a danger in deriving those estimates from the data of chronostratigraphy itself. Ages that have been interpolated between radiometric determinations in single sections give a false sense of steadiness. They can and have been avoided here, but there is also an element of interpolation in the geochronometric calibration of many biostratigraphic units. The thicknesses and the durations of these units had to be used to generate an adequate number of long-term accumulation rates. Fortunately, the effect of incomplete sections is to reduce the precision of interpolated ages, not to introduce a systematic error. When the calibration of biostratigraphic boundaries involves averaging the results of interpolation in several sections, precision is regained by cancelling out some of the unsystematic error. Finally, if stratigraphic completeness is to be estimated from the discrepancy between short-term and long-term accumulation rates, then it will be necessary to identify suitable short and long time spans. This third difficulty is really an important principle: since sedimentation is unsteady, its continuity can be meaningfully specified only in terms of some stated time scale. Reineck (1960) and Schindel (1980) have presented limited data that show a systematic inverse relationship between accumulation rates and the time spans for which they are determined. We shall substantiate their findings and establish the form of the relationship between expected completeness and time scale. MEASURED RATES OF SEDIMENT ACCUMULATION As a first step in their analysis, all measured rates of sediment accumulation are plotted together against the time spans for which they were determined (fig. la). Logarithmic rate and time scales are used, not only for convenience, but also to acknowledge the nature of the variables. A log-normal distribution of sedimentation rates will be demonstrated and the logarithmic time scale reflects the reduced precision achieved in estimating the duration of long time intervals for which ancient rocks must be dated. Three features of figure la should be discussed: the four distinct modes, the very wide range of rates, and the negative regression of rate on time span. The origin of the modes in figure la is straightforward. They correspond to the time spans most often involved in the four most popular means of estimating absolute accumulation rates: (1) continuous observation, (2) reoccupation of survey stations, (3) radiocarbon dating, and (4) biostratigraphy calibrated with K/Ar radiometry. The extreme variability of accumulation rates is striking. Measurable rates alone range over at least 11 orders of magnitude. Before addressing the question of the inverse relationship to time span it is useful to try to identify other sources of variability in the rates. The data can then be divided into meaningful sub-samples, for which the dependence of 570 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). SEDIMENT ACCUMULATION RATES FIG. 1.-Cluster density of positive rates of sediment accumulation plotted against the time span for which they were determined, a: All depositional environments. Numbered circles indicate modes produced by most popular means of determining rates, as explained in text. b: Fluvial environments. Circles indicate median rates for given time span. c: Key to density scale in number of rates per counting square. Key boxes give size of counting square. rate upon time span can be evaluated separately. Sedimentation rates may be expected to vary with place (non-uniformity) and time (unsteadiness), and the latter variation may lead to a dependence of rate upon time span. We should regroup the data into place categories that may be expected to reflect patterns and scales of unsteadiness. Since the unsteadiness in sediment accumulation will be mostly attributable to climatic and tectonic fluctuations, and to the dynamics of the depositing processes, rates may be usefully differentiated according to environment. Schwab (1976) has attempted to relate accumulation rates to tectonic environment. Schwab's selected data may suggest general relationships between sediment accumulation and tectonic setting, but the analysis was not focused on the influence of time span, and thus obscured a strong coincidence of faster rates with shorter measurement periods. Recontruction of former tectonic environments was found here to be too often uncertain, and the data are regrouped according to the less speculative interpretation of depositional environment. Figures lb-5 depict the relationships of accumulation rate to time span for those environments for which sufficient data has been amassed to reveal a stable trend: rivers (lb), lakes (2), carbonate platforms and reefs (3a), terrigenous shelves (3b), small 571 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). PETER M. SADLER TIMESPAN (yr.) FIG. 2.-a: Accumulation rates in lacustrine environments. Symbols as in figure 1. b: Trend that could be produced by compaction in steadily accumulating muds. This trend is free to "slide" parallel to lines of constant thickness to reflect different accumulation rates. Data after von Engelhardt (1977). basinal seas (4), and abyssal ocean floors accumulating oozes (5). This categorization is admittedly gross, but the data base is inadequate to establish reliable trends after more discriminating subdivision. For each depositional environment there is a consistent trend of falling mean sediment accumulation rates with increasing time span. The slope of these trends varies with environment, but there is a characteristic frequency distribution about the mean that is present in all plots: taking measured rates from a single environmental category, and with time spans (in years) that vary only within one order of magnitude, the frequency distributions are found to be nearly lognormal (figs. 6 and 7 give examples). Since the logarithm of the mean value of the original variables is not the same as the mean value of their logarithms, it is simpler to represent the central tendency by median values. RELATION OF ACCUMULATION RATE TO TIME SPAN The consistent observation of falling median accumulation rates with increasing time span may be explained in terms of measurement error, post-depositional compaction, long-term evolution of geomorphic systems, or episodic sedimentation. Greater errors will be expected to accompany age determination than the measurement of thickness. This will tend to reinforce the negative regression gradients, but this factor alone cannot explain the whole trends or the differences between environments. None of the rates amassed here is corrected for the effects of compaction, but the short term rates are measured, for the most part, in young, uncompacted sediments. Compaction, therefore, certainly contributes to the relative diminution of long-term rates. Figures 2b and 5b indicate the regressions that could be produced by this factor alone. Clearly, thickness reduction by compaction is too limited, and is significant over too short a range of time spans, to be the primary factor responsible for the trends in figures lb-5. Since the precision of geological age determination generally decreases with increasing antiquity, the data for short time spans are taken primarily from relatively young deposits. Measurement of rates for very long time spans necessarily involves ancient deposits. The time span axes of figures 1-5 are, therefore, very roughly interpretable as age axes, and it could be concluded that sedimentation has simply been accelerating during the evolution of the crust. This idea, which is nearly as old as the 20th century, arose from the observation that, at the scale of whole periods and epochs, the maximum known sedimentary thicknesses per unit time generally increase exponentially with decreasing age. Barrell's (1917) landmark treatise on rhythms in sedimentation includes a thorough discussion of that observation. He attributed it to "the very nature of sedimentation implying the presence of breaks of all 572 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). SEDIMENT ACCUMULATION RATES FIG. 3.-Accumulation rates. Symbols as in figure 1. a: Carbonate platform and reef environments. b: Terrigenous shelf environments. TIMESPAN lyr.) - FIG. 4.-Accumulation rates in small basinal seas. Symbols as in figure 1. orders of magnitude, the lost intervals increasing with more distant eras" (p. 891). He listed additional factors including the more imperfect preservation of older sections, and he finally mentioned the possibility of an increased frequency of diastrophism. This last notion, often extended to increasing frequency and magnitude of tectonism, has since been adopted by some writers as the sole cause of the apparently accelerating sedimentation (Gilluly 1949-a review of early examples; Salop 1977-a more recent example). The concept is not straightforward, since a "speeding up of the tectonic pulse of the earth" (Salop 1977, p. 321) should be expected to increase the discontinuity of sedimentation as well as its rate. Gilluly (1949) opposed the notion by presenting good evidence for Barrell's assertion that the exponential increase in maximum accumulation 573 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). PETER M. SADLER THICKNESS (m.). TIMESPAN (yr.) FIG. 5.-a: Accumulation rates for abyssal calcareous oozes. Symbols as in figure 1. b: Trend due to compaction. Data after Hamilton (1976), otherwise as explained in figure 2b. rates reflects the better preservation of the younger parts of the geologic record. Gilluly demonstrated a similar exponential increase, with decreasing age, of the currently exposed area, per unit time, for the same periods and epochs. Veizer and Jansen (1979, p. 342) extend the data into the Pre-Cambrian and show that "the described exponential relationship is a fundamental law of any present day age distribution of geologic entities. Our compiled rates are a fuller sample of the geological record than the selected eras, epochs and periods and can be used to support a return to Barrell's first assertion that the older sequences contain more discontinuities. But first it must be realized that age itself is not the crucial factor in the maximum thickness data. The epochs and periods traditionally used show an exponential increase in duration with age (fig. 8) and it is duration that shows a better exponential relationship to the maximum accumulation rates (fig. 9). Thus we may use the concept that discontinuities exist at all scales and realize that FIG. 6.-Probability graphs for frequency distribution of rates of accumulation of fluvial sediments. Each line represents the cumulative frequency of rates in a sub-sample with time spans (in years) ranging across only one order of magnitude. The mid-point of range (in years) is given by the number which labels the line. A straight line on this plot indicates a perfectly log-normal distribution. sections which span longer time intervals have the opportunity to incorporate more and longer hiatuses. Gilluly's (1949) treatment of the data was sound, but incomplete: the processes that recycle sediments and leave a poorer record of older sequences also operate throughout deposition and tend to build more FIG. 7.-Probability graphs for frequency distribution of rates of accumulation, as explained in figure 6. a: Carbonate platform and reef sediments. b: Abyssal calcareous oozes. 574 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). SEDIMENT ACCUMULATION RATES FIG. 8.-Relationship between the duration and the age of the mid-point of eras, periods and epochs used in analyses by Gilluly (1949) and/or Salop (1977). pp: 3500-2600 m.y. (Paleoprotozoic of Salop), mp: 2600-1900 m.y. (Mesoprotozoic), np: 1900-1000 m.y. (Neoprotozoic), ep: 1000-650 m.y. (Epiprotozoic), ec: 650-570 m.y. (Eocambrian), c: Cambrian, o: Ordovician, s: Silurian, d: Devonian, c: Carboniferous, p: Permian, t: Triassic, j: Jurassic, k: Cretaceous, pa: Paleocene, eo: Eocene, ol: Oligocene, mi: Miocene, pl: Pliocene, q: Quaternary. primary discontinuity into sediment sequences of longer duration. The best record of long, ancient periods is not only poorly preserved now, it was also probably originally much less complete than that of a short, more recept epoch. If we do not accept this extension of Gilluly's (1949) explanation, we can account for the dependency of maximum accumulation rates on time span, but we have no reason to expect the trends of median rates presented here. Certainly the notion of a long-term acceleration of sediment accumulation can be regarded as an unwarranted complication. NUMERICAL SIMULATION OF UNSTEADY ACCUMULATION We can examine best the adequacy of unsteady accumulation to explain the observed FIG. 9.-Relationship between duration and maximum accumulation rate for eras, periods and epochs shown in figure 8. relationships of accumulation rate and time span by reference to simple numerical models. A record of the net thickness of sediment accumulating unsteadily is a complex time series. Such a time series can be simulated by the form of a summation of simpler wave functions, or cycles, of different periodicities and wave heights. Unfortunately, our simulation task is complicated by three factors. First, the natural time series are not directly available for comparison; they are known only by the relationship of their chord gradients to chord time span. Figures 1-5 express this relationship, and it is the purpose of the simulation to explore the kinds of unsteadiness that can explain the range of relationships shown in these figures. It is instructive to complicate the simulation by progressively adding simple wave functions and expedient to use simple sine waves as shown in figure 10. 575 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). PETER M. SADLER b. FIG. 10.-A simple time series model for unsteady sedimentation, a: a linear secular trend added to a sine wave. The dotted portion of the curve represents erosion, the solid portion sedimentation, and the thick, solid portion permanent sedimentation. Dashed lines are chords whose gradients represent accumulation rates. The range and mean of the gradients are forced to decline as the chords span longer time intervals, b: The resulting sedimentary section "exploded" along its time axis. Permanent sediment increments are shown solid; sediment increments that are later eroded are ruled, non-depositional intervals are open. The open and ruled portions together are the total gaps. Second, as figure 10 illustrates, the stratigraphic record is a very selective sample of the time series and it is only for this sampled part that chord gradients, i.e., accumulation rates, are generally available. The samplable portion of the full time series will usually be a lower amplitude step function (the heavy risers in figure 10, linked by horizontal treads) which can also be simulated by cyclic and secular components. This does mean, however, that it will be very difficult to extend the simulation process to attempt to reconstruct any primary cyclicity in deposition. Third, any time series which can generate figures 1-5 must represent accumulation in a hypothetical, median, sedimentary basin, since the data have been composited from numerous real basins. This carries the advantage that we will simulate only the generally significant patterns of unsteadiness, since the local noise should be eliminated in the compositing process. We must return to this point. Consider first figure 1 la, which shows the chord gradients generated by a simple sine wave (360 yr period, 2 m amplitude) added to a constant secular trend (1 m in 10 yr). FIG. 11.-Rates of accumulation (open circles) and their relation to time span as simulated by the positive gradients of chords to simple wave functions and summations. Dotted lines give the long-term limit of negative gradients; the horizontal line is the secular trend; oblique lines are lines of constant thickness (or wave height), a: Summation of a sine curve (360 yr period and 4 m wave height) and a linear secular trend (1 m in 10,000 yr). h: Numerical model used in a with the addition of a second sine wave (360,000 yr period and 400 m wave height). 576 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). SEDIMENT ACCUMULATION RATES Geologically the sine wave may simulate short-term alternation of erosion and deposition, while the secular trend models longterm subsidence that allows some permanent sediment accumulation. We see that it is an intermediate range of time spans which generates maximum chord gradients that fall systematically as time span increases. The resulting trend follows a line of constant thickness: the wave height. The trend extends from the time span of the rising portion of the wave to the time span at which the secular trend attains a thickness equal to wave height. Beyond these limits the maximum gradients approach constant values: the gradient of the secular trend and its sum with the maximum gradient of the sinusoidal wave. Note that the density of plotted gradients (accumulation rates) for any time span is highest very close to the maximum limit. Thus the median values are sufficiently close to the maximum limit that this limit can be expected to be approximated well, even by a relatively small sample of rates. To provide a better simulation of real geological situations, the model must be complicated in three ways: (1) the wave function must be allowed to assume a variety of forms, (2) several wave functions must be summed, and (3) the trends for different basins must be compiled together. The most geologically realistic modifications to the wave function will be those that simulate intervals without erosion or deposition, i.e., plateaus in the wave form. The effect on a plot such as figure 1 la is to increase the concentration of gradients close to the maximum limit and to allow the segment in which the maximum limit is defined by wave height to extend to shorter time spans, i.e., it improves the simulation of figures 1-5. Figure 1 lb illustrates the result of adding a second sine wave to our simple model (360,000 yr period, 200 m amplitude). It shows that the trend of the maximum gradients for a summation will clearly depend upon the relationship of wavelength to wave height in the summed set. Less steep trends are produced where wave height increases more rapidly with increasing wavelength. This is the key to simulating the variation in figures 1-5. Summation of waves of increasing period, but constant or falling wave height, produces trends of maximum chord gradient that are close to lines of constant thickness, and convex up. The real geological data in figures la-5 are combined for many depositional basins which have their surface environments in common. As a consequence, the distribution of rates for a given time span is log-normal, in contrast to the negatively skewed distributions simulated from single time series. Since even limited data for a single basin will most probably plot close to its maximum limit, the median trends seen in figures lb-5 approximate the regression of maximum accumulation rate against time span for a hypothetical "median basin." Klemes (1974, p. 675) has warned that "an ability to simulate and even predict a specific phenomenon does not necessarily imply an ability to explain it correctly." We have used cyclic functions for convenience. It is not necessary to assume a rigid cyclicity in sedimentation events, simply the more palatable notion that there is some general relationship between the magnitudes and recurrence intervals for those events. We can also simulate the data of this paper using an essentially random model. A time series showing sediment accumulation presents the cumulative effect of a sequence of depositional and erosional increments. An intuitively unexpected consequence is that a long-term, quasi-periodic fluctuation in net thickness will be generated even if erosion and sedimentation increments are equally probable and of equal thickness. A simple coin-tossing analogue illustrates this effect if heads represent one positive increment and tails one negative increment and the cumulative "score" is plotted against time. In this analogue the expected maximum range of cumulative scores about the mean, and the mean wavelength of the fluctuations will increase with the length of the time series (Feller 1968). This means that a model of random, non-cyclic sedimentary processes can generate greater hiatuses in longer sedimentary se577 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). PETER M. SADLER quences and, thus, simulate the trends of falling net accumulation rate with increasing time span. There is some evidence that geological and geomorphic time series are not generated by random and short-memory processes. The evidence is a deviation of the dependence of the "rescaled range" (range of cumulative departure from the mean, standardized by division by the standard deviation) upon the time span of the series from that dependence predicted for random processes. This is called the Hurst phenomenon. Mandelbrot and Wallis' (1969) catalogue of natural time series showing this phenomenon includes lithology, bedding type, bed thickness, and varve thickness. They conclude that "for practical purposes" these records "must be considered to have an infinite span of statistical interdependence" (p. 321), i.e., infinite memory. There is a viable alternative generator for the Hurst phenomenon, however, in which memory is short, but the mean is allowed to fluctuate (Klemes 1974). ENVIRONMENTAL CONTROLS OF UNSTEADY ACCUMULATION The trends plotted in figures lb-5 are generally less steep than lines of constant thickness. The simulations presented above show that this is consistent with unsteady accumulation in which the magnitude of episodic or quasi-periodic fluctuations increases with their recurrence interval. This is also in keeping with the findings of Wolman and Miller (1960) that the magnitudes of geomorphic processes approximate a log-normal frequency distribution, i.e., larger fluctuations are increasingly rare. Detailed analysis of the trends for each environment are beyond the scope of this account, but it is instructive to consider the environments treated here as two groups: those at or above wave base, and those below. Those above wave base-fluvial, terrigenous shelf and carbonate platform-are dependent upon crustal subsidence for long-term accumulation. Their accumulation rate trends are convergent and become largely coincident (within the level of internal variation) at time spans longer than 1,000 yr, suggesting a common, large scale controlling influence. Although climate may be influential, the dominant control is more likely to be the behavior of the crust during long-term loading and subsidence. Accordingly, figure 12 has been prepared to combine the data for these environments in a comparison with the median rates for vertical crustal motion (based on nearly 2,300 rates from 70 references). The vertical motion data does not separate uplift and subsidence, but the bulk of published subsidence rates had to be discarded since they were either derived directly from sedimentation rates or failed to consider eustatic changes. Thus, the data for comparison are properly independent, but describe the general vertical mobility of the crust, not merely its subsidence. There is good agreement between the patterns of vertical crustal mobility and sediment accumulation rates for periods longer than 1,000 yr. Both trends remain moderately steep, and may even steepen, at time spans longer than 1,000,000 yr. Certainly no long-term secular trend is seen as in the simulations. The long-term trend is very similar to that generated by data from loaded continental margins and collapsing oceanic rises, where prolonged subsidence has been found to wane exponentially (Hays and Pitman 1973; Watts and Ryan 1976). With a larger data base we may be able to relate subsidence and accumulation patterns to tectonic setting-particularly crust type and position within a plate. The rates of accumulation for lake sediments do not fit the above pattern unambiguously. The scanty information from longlived lake basins is also compatible with a linear secular trend. Such is the pattern seen in biogenic and terrigenous sediments accumulating below marine wave base (fig. 12b). In such a situation sediment accumulation is largely free of the influence of subsidence, and the secular trend most probably represents the long-term, background rate of sediment supply. Abyssal manganese nodule accumulation is a more complex situation and has been analysed elsewhere (Sadler 1980). 578 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). SEDIMENT ACCUMULATION RATES FIG. 12. -Comparison of accumulation rates and rates of vertical crustal motion, a: Fluvial (Fluv.), terrigenous shelf (Terr.), carbonate platform (Carb.), and lacustrian (Lac.) sediments represented by the range of their accumulation rates between median and maximum values, for each time span. These ranges are superimposed and stippled to indicate the number of overlapping ranges. Circles show the median rates of vertical crustal motion for each time span. h: Outline of trends for environments at or above wave base (taken from a), compared with trends for muds and calcareous and siliceous oozes accumulating below wave base. COMPLETENESS OF SEDIMENTARY SEQUENCES Sedimentation is an essentially discontinuous process, and we have examined evidence for its long-term unsteadiness. Not only is active sedimentation intermittent, but previously deposited sediment may also be eroded. Sedimentary sequences, therefore, record the passage of geological time as an alternating set of sedimentary increments and gaps. The ratio of these two components is the completeness of the section. Since each gap may comprise a non-depositional portion and some sediment increment lost during concurrent erosion, the completeness of a section will generally be less than the continuity of the sedimentation that produced it. It is not useful to consider a complete section to be one without gaps. Such sections do not exist, and the concept leaves completeness without the possibility of practical quantification. A complete section must be one in which no gap is of longer duration than one time unit at a specified scale, i.e., it will contain some sediment that is representative of each time unit at that scale. The information compiled here allows us to quantify completeness in the terms set out above. Consider, for example, an interval of 7.5 m.y.-the approximate duration of a horse genus (Simpson 1949). Referring to the median accumulation rates of fluviatile sediments, one would expect such an interval to be represented, on average, by about 500 m of a sedimentary sequence. Taking the median rate of accumulation for time spans of one year, it appears that this thickness could represent a cumulative, active sedimentation time of as little as 5,000 yr or 5,000 annual flood increments. Dividing the time represented as sediment by the total time elapsed we determine a completeness of 1 in 1500 (0.07%) at a scale of one year. Taking the median accumulation rate for one hour, an interval in which sedimentation might be more nearly continuous, the cumulative sedimentation time falls to 9.5 months and the completeness to one in 10 million, at a scale of hours. It is clear that completeness of sedimentary sections can become very low if we concern ourselves with fine time scales. The main usefulness of such estimates of completeness is to enable us to restrict our endeavours to time scales and precisions for which the completeness of the sedimentary record is acceptably high. Before formulating the procedures for completeness estimates, we must recall that the data base comprises only mea579 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). PETER M. SADLER surable accumulation rates. This means that an increment of sediment which is too thin to measure is, for practical purposes, a gap. Moore and Heath (1977, p. 71) advanced a similar proposition when they defined a hiatus in deep sea sediments as "the absence of one or more biostratigraphic zones or an estimated accumulation rate of less than 1 m/Myr." ESTIMATES OF EXPECTED COMPLETENESS The completeness of a sedimentary section is given by the ratio of the long-term rate of accumulation (S) of the whole section to its average rate of accumulation at the given time scale (S,). For a real single section, S may be known, but S, is almost never available. The expected value of S, must then be the average for all comparable environments of deposition (S ), which may be used to estimate the mean or expected completeness. S, may be read from figures 1-5, which give S as a function of time span (t). The gradient (m) of the trends in figures 1-5 is related to the expected completeness (S/S,): In.S/S, = In.S - In.S, (1) = m(ln.t - In.t,), (2) where d In.S d ln.t and t is the time span for the whole section and t, is the time unit of the given time scale. We may now write a formula for expected completeness in the convenient terms of the chosen precision (t,/t) and the gradient (m) of the regression of S on t: S/S, = (t/t)-m. (3) For a single section, the long term rate of accumulation cannot exceed the mean short term rate, and the total section thickness must be at least the average short-term increment. Thus, for a real stratigraphic section, m is constrained to vary between -1 and 0, and the most pessimistic estimate for completeness will be t /t, the chosen precision. Estimated frequency distributions of m are given in figure 13 as a function of environment and t,/t. For sections in which S is known, a better estimate of m will be given by the gradient of a line tying a plot of this S value to the desired S,. PROBABILITY OF A GIVEN COMPLETENESS It is desirable to be able to estimate the probability of a given completeness. This requires use of the standard deviation (cr) of the logarithms of the values of accumulation rates about the mean, for a given time span. These standard deviations range from 0.2 to 2.4 but have a strong mode at 0.9. The logarithm of completeness is given by (In.S - In.S,), which is the difference of two variables of known, normal distribution. The probability distribution for such a difference can be readily described: its mean is simply the difference of the means of the two variables, and its variance (o-2) is given by x_- - OxJ + -,- - 2p-x-7, (4) where the variable x is In.S and the variable y is In.S and p is their correlation coefficient (Snedecor and Cochran 1967, p. 190). In order to describe the probability distribution of the logarithms of completeness using the mean and equation 4, it is necessary to have some estimate of the variance of short-term rates for single sections, and their covariance with the long term rates. The data compiled in this account include all the temporal and spatial variability of rates, which is appropriate for o but may over-estimate the appropriate value for T,. Further, if there is a tendency toward a positive correlation between long- and short-term accumulation rates, at a given site, the last term in equation 4 cannot be assumed to vanish, but must be positive. The covariance of long- and shortterm rates in single sections is the subject of continuing investigations. APPLICATIONS Applications of completeness estimates will be concerned with time correlation, since 580 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). FIG. 13.-Frequency distributions of m values by environment of deposition. Area under total curve is divided by heavy lines to indicate distributions of m values for t,/t values greater than that indicated on line. Heavy stipple shows the portion of the distribution in which the combination of m and t,/t values give a completeness greater than 50%. Light stipple indicates completeness between 10% and 50%. The distributions are derived by comparing all pairs of median values in figures 1-5. Geologically unreal m values arose where the data base was too small, and were cut off. This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). PETER M. SADLER our knowledge of geological time can only be as complete as the rock record of it. It is now evident that the assumption of steady sedimentation, when interpolating radiometric ages, uses an improbable model, which becomes less acceptable as finer subdivision is attempted. To interpolate directly between known ages is to apply a measured rate to a shorter time interval, and this is clearly to some extent improper. Steady sedimentation means that m is zero and subsumes a complete section as defined here. To allow linear interpolation, it is only necessary to assume that, at the level of subdivision, all time units are represented by equal sediment increments. The joint probability of a complete section and uniform sediment increments will be low. It is a strength of radiometric ages that they are quoted with a statistical error factor. An interpolated age inherits not only the uncertainties of the original determinations, but also an additional uncertainty due to the probable incompleteness of the section. Given an estimated completeness, at an appropriate time scale an interpolated age may be replaced by a more honest age range. For example, consider a 900 m section of platform limestones between two horizons dated at 10 and 25 m.y. respectively. If the interval is divided proportionally into 5 m.y. intervals, a level 300 m below the 10 m.y. determination would be assigned an age of 15 m.y. (and should be given an average standard error). However, using the data here we may estimate the probable completeness of the limestone section at a time scale of 5 m.y. units: expected completeness = (t,/t)-m = (5/15)0o. 0.68. Now only about 2 in 3 of the 5 m.y. time intervals should be expected to be represented by sediment. Let us suppose that the 5 m.y. increments are uniform, but examine the extreme cases in which the total lacuna in the section is concentrated at the top or the bottom. The horizon 300 m from the top may now be calculated to be 13.4 or 18.2 m.y. At this scale of 5 m.y. then, its age should be stated as 10-20 m.y. Similarly, if the interval is divided down to units of 1 m.y., the appropriate completeness estimate falls to 0.39 (i.e., only 5-6 of the 1 m.y. units should be expected to be represented by sediment). The horizon which would have been dated at 11 m.y. by direct interpolation can now be given the possible age range 10-19 m.y. Reversals in the earth's magnetic field seem to be rapid, global events, ideally suited to time-correlation. However, the degree to which the boundary between normal and reversed remanent magnetization in a sedimentary section can be expected to identify the field reversal itself, rather than the two sides of a gap in the record depends upon the completeness of the record. Magnetostratigraphic correlation techniques include comparison of patterns of reversed and normal intervals in sedimentary sections, but the uniformity of these patterns is controlled by the uniformity of accumulation rates and section completeness. We are now in a position to assess the likelihood of preservation of a given pattern element in a particular section. It is simply the expected completeness of that section when t, is given by the duration of the interval in question. In the hypothetical section above, there is a 68% probability that some sediment records a magnetic interval of 5 m.y. duration. For a 1 m.y. interval the probability is 39%. Calculating such probabilities, interval by interval, will allow the magnetostratigrapher to justify misfits in correlative patterns. Where magnetostratigraphic patterns are strictly comparable, even though all sections cannot be expected to be very complete, this reflects closely comparable accumulation histories. Indeed, similarity of magnetostratigraphic pattern would seem to be a powerful argument for similarity of diastrophism, and thus for delineating and comparing tectonic provinces, rather than simply attempting correlation. This step has already been made in reading the patterns of magnetic stripes on oceanic crustal plates, but has clear potential for continental structural provinces. Geologic time is available for subdivision only when it is represented by the rock rec582 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). SEDIMENT ACCUMULATION RATES ord, and so the completeness of that record cannot be eliminated from a discussion of time-stratigraphic precision. In their treatment of the fundamental character of biological evolution, Eldredge and Gould (1972) are concerned with the completeness of the record. Unfortunately they tackle only the question of the abundance of fossils. It is equally important to determine whether the completeness of the sediment accumulation is adequate for the recognition of "punctuations" in the fossil record that are phyletic in origin rather than indicators of sedimentary lucanas. We have attempted to quantify the sedimentary aspect; these numbers need to be matched by estimates of the durations implied by "punctuations." Shaw (1964) has pointed out that the allure of radiometric time scales lies in the anthropocentricity of the year in geology. He has established an elegant method by which an absolute, non-annual time scale of superior precision can be erected by statistically ordering and compositing unique taxonomic events. The method does not assume steady accumulation rates, but rather that the first differential of rate, with respect to time, is uniform. The units of the scale are based on sediment increments at a standard section, and Shaw sets limits on the precision of the method according to the abundance of fossil taxa. By compositing many sections the gaps in any one section may be "filled" by information from another, but the completeness of the standard section is still important since precision is measured in standard section thickness intervals. Thus, where the standard section is relatively incomplete it may accommodate more information from other sections, and the limits to significant precision will be relatively fine. It is a simple matter to use compilations of rates of sedimentation to quantify the expected time spans of thin stratigraphic samples, as Schindel (1980) has shown. Schindel's use of a "constancy of sedimentation" factor, however, is very subjective and becomes invalid wherever there have been intervals of erosion. His "calibration" of stratigraphic samples should be combined with estimates of completeness, as outlined here, to assess properly the resolution of a sequence of samples. CONCLUSIONS The data presented in this paper are real measured rates of accumulation from extant and ancient sedimentary systems characterized by net sediment accumulation. Conclusions may be drawn concerning real sedimentary stratigraphic sections, however thin, but may not be extended to describe natural systems without net accumulation. Neither may they be taken as generalizations about.environments in which long-term accumulation is unusual. The shortest time spans considered in this complication may seem irrelevant to many geological problems; often they are. But these are the time spans of a geologist's experience, and are at the basis of his intuition. For example, the relatively very incomplete nature expected for fluvial sequences is a feature of rather short time spans only. Indeed, should a fluvial system build up a section spanning more than 1,000 yr, its a priori expected completeness must be the same as a shallow marine section (fig. 12). We may summarize the conclusions as follows: 1. Sediment accumulation rates are extremely variable, but are inversely related to the time span for which they are determined. 2. The slope of the regression of accumulation rate on time span varies with environment of deposition, and is an indicator of the completeness of stratigraphic sections. 3. This relationship between the net rate and the duration of accumulation may be explained primarily as a direct consequence of unsteady accumulation. 4. Whether the fundamental pattern of accumulation is cyclic or random, the resulting unsteadiness is characterized by fluctuations, the magnitudes of which increase with recurrence interval. 5. Sediment accumulation at or above wave base seems to experience similar regimes of unsteadiness beyond time spans of 583 This content downloaded from 154.059.124.102 on October 23, 2016 20:36:09 PM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). PETER M. SADLER 1,000 yr, regardless of environment. This 7. Ratios of long- to short-term accumulamay reflect the controlling influence of crustal tion rates provide a measure of the completesubsidence. ness of stratigraphic sections, at the time 6. 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