The Influence of Hiatuses on Sediment Accumulation Rates

Hiatuses pervade the stratigraphic record at all scales from grain boundaries to inter-regional unconformities. Every attempt to measure a rate of accumulation must average together sediment increments and surfaces of hiatus. As the time span of measurement lengthens, longer hiatuses tend to be incorporated into the estimated rate. Consequently, short term rates are systematically faster than longer term rates. As would be expected for a fractal time series, the empirical relationship between accumulation rate and the time span of measurement is a negative power law. Just as a measured length for a coastline is meaningless without a statement of the map scale, a measured rate of sediment accumulation requires a statement of the time scale of measurement. Fractal scaling laws offer a means to estimate and interpret the changes in average accumulation rate from one time scale to another. The steepness of the negative power law that relates rate to time span increases with the incompleteness of the stratigraphic record. For steady accumulation and complete sections the slope would be zero. For the most incomplete and unsteady accumulation, the slope approaches minus one. A purely random accumulation process produces a power law with a slope of minus one half; less steep negative slopes indicate some degree of persistence or positive correlation between successive sediment increments; steeper negative slopes prove a negative correlation. Regular periodic fluctuations in accumulation are one form of negative correlation; they produce steep negative slopes at time scales close to the period of the fluctuations. In shallow marine carbonate environments the negative power laws steepen markedly at time spans of tens of thousands to hundreds of thousands of years; the time scale suggests that the steepening records the preponderance of hiatuses that have been generated by Milankovitch-scale changes of sea-level. A plot of the age and level of sediment increments preserved in a stratigraphic section has a staircase-like shape, in which the treads represent surfaces of hiatus. Sequence boundaries of known age fix the coordinates of points on these treads. Rocks dated by radiometric techniques fix the coordinates of points on the risers. Horizons across which the fauna, isotopic composition or magnetic polarity are found to change may correspond to points on either the treads or the risers. The staircase plot forms the lower bound of the true accumulation history which would be determined by continuous monitoring of the elevation of the sediment surface during accumulation. Thus, the relationship of accumulation rate to time span also depends to some extent upon the method used to date the sediment. On the Determination of Sediment Accumulation Rates edited by P. Bruns and H. C. Haas 16 On the Determination of Sediment Accumulation Rates Rates and Unsteady Processes Accumulation rate has a deceptively simple formulation -measure the thickness of the sedimentary deposit and divide by the time that elapsed between the start and finish of deposition. Thus, changes in thickness of a deposit or changes in the elevation of the sedimentary surface are, in effect, standardized for differences in the time span of observation. Nevertheless, a measured accumulation rate has no general power to convert deposit thickness to duration of deposition, or stratigraphic elevation to age. If sediment accumulation were steady, then a rate determined from any portion of the accumulation history (regardless of age and duration) would indeed be the correct conversion factor for every other portion. We could interpolate age linearly between dated horizons and one measured rate would support extrapolation beyond dated intervals. But sedimentation is, of course, inherently unsteady and discontinuous. Surfaces of hiatuses record the discontinuities. This paper will show that the presence of hiatuses causes the expected accumulation rate to become a strong negative function of the time span of measurement. Empirical data demonstrate that rates measured at one time span cannot be extended with impunity to other time spans anywhere in the range from minutes and hours to hundreds of millions of years. The underlying principle becomes almost intuitively obvious when we consider the deposition of sediment as a succession of instantaneous arrivals of discrete particles with finite diameters. Each particle diameter is added to the net thickness at an instant in time. Consequently, instantaneous accumulation rates can be infinitely high. Of course, nobody would consider applying such instantaneous accumulation rates to longer time intervals of geologic interest. But rates measured on the relatively short time spans of human observation are routinely applied to much longer stratigraphic time scales. The conceptual error is the same as extrapolating from the instantaneous rates. Rates from different time scales are not directly comparable and hiatuses are the cause, whether the hiatuses are recorded at the scale of grain-to-grain boundaries or inter-regional unconformities. Fortunately, fractal scaling laws offer a means to estimate and interpret the changes in average accumulation rate from one time scale to another. After establishing how accumulation rates scale with time span for a variety of depositional environments, the paper reviews some standard results from fractal mathematics that help to relate the scaling laws to properties of the underlying geologic processes. Then, two examples serve to demonstrate how the failure to appreciate the time-scale dependence of rates does lead to false geologic interpretations. Next, we examine how the scaling laws can vary with the method of determination of age. With these insights, the paper concludes by analyzing a single stratigraphic section that has been dated by both radiometric and paleomagnetic methods. Before any of this, however, it is useful to review the nature of hiatuses and the options for displaying stratigraphic data. The Pervasive and Composite Nature of Hiatuses Hiatuses pervade the stratigraphic record at all spatial scales from grain boundaries to bedding planes to inter-regional unconformities [1,2]. Surfaces of hiatus record time intervals without net deposition. The duration of these intervals ranges from the fractions of a second between successive grain arrivals at the sediment surface to the millions of years that may elapse between marine flooding events on continental interiors. Tiny hiatuses must be pervasive because sediment consists of discrete particles. The areal extent of intergranular hiatuses must be extremely limited. Bedding planes and laminae record innumerable discontinuities in accumulation at larger spatial scales and longer times scales. And practitioners of sequence stratigraphy have advanced the case that interregional hiatuses are sufficiently commonplace and areally extensive to provide a basis for global time correlation. During the time interval in which a surface of hiatus is generated, deposition may simply have ceased or erosion may have removed some of the previously deposited sediment. Consequently, three distinctly different components of the accumulation history may all be condensed into a single GeoResearch Forum Vol. 5 17 surface of hiatus: episodes of non-deposition; episodes of erosion; and time intervals occupied by the accumulation of sediment that is subsequently eroded [3]. The closing sections of this paper explore the fact that some methods used to determine the age of sedimentary deposits can produce dates that fall within these intervals of non-deposition, erosion, and temporary deposition that make up hiatuses. Figure 1. The distinction between the accumulation history (a), staircase plot (b), columnar section (c), and time lines (d, e) for a hypothetical sedimentary deposit. Time line d distinguishes intervals of deposition from intervals of erosion. Time line e distinguishes time intervals recorded by preserved sediment increments from intervals of hiatus. The magnifying glasses caution that smaller hiatuses exist below the resolution of this diagram. In the limit of extremely fine resolution, the solid bars in time lines d and e become sets of discrete points and the risers in curves a and b become vertical. At any given level of resolution, the sedimentary stratigraphic record may be regarded as a stack of sediment increments separated by surfaces of hiatus. But closer inspection should always reveal that the sediment increments include smaller hiatuses, down to the resolution of individual grains. Systems like this in which a common motif is nested within itself at different scales are often usefully viewed from the perspective of fractal geometry. Sediment accumulation is no exception. 18 On the Determination of Sediment Accumulation Rates Every attempt to measure a rate of accumulation averages across sediment increments and the intervening surfaces of hiatus. As the time span of measurement increases, larger hiatuses may be incorporated into the estimated rate. As the time span of measurement decreases, the revelation of smaller hiatuses requires that increasing numbers of discrete sediment increments are revealed too. The net time allotted to the increments must be reduced because more time is seen to be apportioned to the hiatuses as the resolution increases; but the net thickness does not change; so the accumulation rate of the depositional increments must be found to increase as their individual thicknesses decrease. Thus, the mean (or expected) rate of sediment accumulation becomes inversely related to time span [2,4]. The empirical relationship between accumulation rate and time scale will be shown to be a “negative power law” this term conveniently describes an inverse linear relationship between the logarithms of two variables. Such relationships characterize fractals [5]. Table 1: Properties of Different Representations of Sediment Accumulation ACCUMULATION HISTORY (FIG. 1a) STAIRCASE PLOT (FIG. 1b) TIME LINES (FIGS. 1d ,1e) Appearance Time series with positive and negative slopes Time series with only zero and positive slopes Broken lines Represents Complete history of the deposit thickness during accumulation process Age and elevation of horizons in preserved sedimentary section Portions of elapsed time recorded by deposition (1d) or by preserved sediment (1e) Metric space Thickness Age Elevation Age Age Shortcoming Practical, but not explicit, lower limit of age resolution Range of fractal dimension 1.0 2.0 1.0 [19] 0.0 1.0 Possible fractal analogs Fractional Brownian walks Devils Staircase Cantor Bar, Levy Dust Physical significance of fractal dimension Degree and nature of feed-back during accumulation A borderline fractal! [19] Completeness of stratigraphic record (Fig. 1e) or continuity of deposition (Fig. 1d) Graphical Representation of Sediment Accumulation The most common representation of sedimentary deposits is a columnar section, drawn against a thickness scale (Fig. 1c). To appreciate the fractal nature of sediment accumulation and the role of hiatuses, however, we need a graphic that includes a time scale. Six options are considered below. In the first four, time is represented by a scale of age (Table 1). Two of these options (accumulation history and staircase plot) retain the thickness scale, so that the slope of the graph shows accumulation rate. The two time-line options do not; they record only the presence or absence of sediment for a given time interval (Fig. 1d, 1e). Notice (Fig. 1) that none of these options can be drafted in practice without an implicit and artificial lower limit of time-resolution. The fifth and sixth options make time-resolution explicit; they plot the durations of time intervals (i.e. resolution), rather than their ages. Although this ploy loses all information about the sequence of the intervals, it allows empirical data from many different deposits to be combined seamlessly into one graph. The Accumulation History: If all the successive elevations of the sediment surface are plotted in historic sequence, the resultant time series represents a complete accumulation history (Fig. 1a). It includes details of all the depositional and erosional episodes during time spans that are ultimately recorded by surfaces of hiatus. Notice that those portions of the accumulation history that represent hiatuses begin and end at the same elevation. If erosion has occurred, these portions assume an asymmetric form: they begin with positive slope in a depositional interval, arch over a maximum, GeoResearch Forum Vol. 5 19 and end with zero slope at minimum. These portions of the plot would not be recoverable from real stratigraphic sections even if every preserved horizon were dated. The Staircase Plot: A complete accounting of the elevation/age coordinates for all preserved horizons in a sedimentary deposit would assume the form of an irregular staircase (Fig. 1b) in which hiatuses appear as horizontal treads. The staircase plot omits those segments of the accumulation history that have negative slope (erosion episodes) or are higher than subsequent minima (erosional losses of sediment). Reconstruction of the staircase plot from real sections is limited only by the number of datable horizons. Notice that the accumulation history lies on or above the staircase plot, never below it. The recoverable staircase plot provides a lower bound for the true accumulation history; thus, the staircase plot is a systematically biased estimate of the accumulation history. Two Time-Line Plots: Sediment accumulation may also be represented as one-dimensional time lines (Fig. 1d, 1e) that ignore the elevation of the sediment surface. The solid portions of the time line report that the corresponding time interval is characterized by some deposition, regardless of thickness. Two versions of the time line are possible. One reports only the intervals of deposition that survive subsequent erosion (Fig. 1e) and thus represents the completeness of a stratigraphic section. The breaks in this line mark the duration of hiatuses. Alternatively, all depositional intervals may be included (Fig. 1d), whether later lost to erosion or not. In this case, the time line records the continuity of deposition, i.e. what an observer would have noted at the site of deposition. Line 1e is a one-dimensional projection of the risers in the staircase plot; line 1d is the corresponding one-dimensional projection of the accumulation history. Plots of Rate or Thickness against Time Span: The previous four diagrams are useful images for single sedimentary sections. But they are difficult to reconstruct in detail because real deposits typically include too few dated horizons. Neither do they lend themselves readily to combining data from more than one stratigraphic section onto a single line -the demands upon time correlation are typically too hard to meet. If thickness or rate are plotted against the duration of the dated interval rather than its age, however, measurements from many different deposits or stratigraphic sections can be combined into one graph. Logarithmic plots of average accumulation rate against time-span summarize nearly all of the empirical data that support later sections of this paper (Figs. 4-6). The logarithmic scales effectively display the wide range of measured values and reveal the negative power law relationships. But the graphs may initially trouble some readers, especially those unfamiliar with fractal geometry, because they plot a fraction (thickness/time) against its denominator (time). This practice deliberately highlights the scale-dependence of rates. In attempting to prevent application of rates to the wrong time spans, however, it invites an improper interpretation of correlation coefficients. It is crucial to realize that contours of constant thickness can always be drawn on a plot of rate against time. On a logarithmic graph, these contours are straight lines with a slope of -1. On a plot of accumulated thickness (y-axis) against time span (x-axis), a regression with zero slope would imply that accumulated thickness is independent of time span; the correlation coefficient would be zero. Plotted as accumulation rate against time span, however, the same data would generate a -1 slope and a non-zero correlation coefficient. Thus, some readers may worry that regressions of rate against time span include a spurious component of correlation. The degree of correlation is spurious if attributed to thickness and time. Some negative correlation of accumulation rate and time span is indeed a mathematical inevitability whenever the deposit includes hiatuses -the highest rates can be sustained only for relatively short time intervals. The peculiarity of the plot and the trouble with accumulation rates are one and the same. Unfortunately, more geologists tend to worry about the plots than to worry about using rates without regard to time scale. Concern about improper use of correlation coefficients should translate directly to concern about improper use of accumulation rates. Notice also that if thickness were truly independent of time span (power law gradient of -1 on these plots) this would be a highly significant observation for 20 On the Determination of Sediment Accumulation Rates stratigraphy. It would imply that the thickness of deposits generally provides no guide whatever to the amount of time that they span. Such steep negative slopes are, in fact, quite rare in the empirical data and persist for limited ranges of time span. We shall see that they can be explained in terms of periodic components in the accumulation history. For steady accumulation without hiatuses, rate would be a constant and independent of time scale; the plot of rate against time span would assume a zero slope if accumulation were steady. Positive slopes should never result from an accurate and complete set of measurements. Negative slopes indicate the presence of hiatuses and an incomplete stratigraphic record [2]. Steeper negative slopes reflect increasing incompleteness. The limiting steepness is -1 -constant expected thickness regardless of time span. Only an unrealistically stylized cyclic accumulation history could reach this limit. It would have to be imagined to operate as follows: the depositional process must generate rapid periodic alternations of depositional and erosional phases in which all depositional increments have the same thickness and are precisely eroded away before the next increment is added. Regardless of time span, all measurements of accumulation rate would then capture no more than one sediment increment. There is no net accumulation in this model at time spans longer than the alternating phases; it is the limiting extreme case of stratigraphic incompleteness. The significance of slopes between 0 and -1 will be interpreted in a later section from some basic equations of fractal mathematics. Our crucial lesson from fractal mathematics will be that single measurements of a property that is dependent upon time scale may have no validity beyond the time span of measurement. The lesson is generally better appreciated in the context of length scales and coastlines. And so, for readers who are not familiar with fractals, the next section reviews the role of spatial scale and the hierarchy of embayments in the problem of coastline length. A firm foundation in purely spatial patterns eases the conceptual leap to the fractal view of time series, which is at the heart of understanding the influence of time scale and hiatuses on accumulation rate. Of course, the time scale introduces significant differences between coastlines and accumulation histories which should be understood as well as the analogies between coastline length and accumulation rate (Table 2). The analogy is intended to provide a more familiar example of a property that is in part a function of the scale of measurement. Figure 2. Coastlines drawn at different scales. Details of the coast of Lewis (a: 10km scale bar), which is an island on the coastline of northern Great Britain (b: 100 km scale bar), which is an island on the coastline of northwest Europe (c: 1000km scale bar). All three maps have comparable complexity; nothing inherent in the pattern of the coastline would reveal the scale of the maps to a stranger. The Fractal View of Coastlines It is simply not reasonable to seek a single value to describe the length of a coastline. Unless the coast is uncommonly smooth, it has no unique length [6,7]; the mapped length of a coastline varies GeoResearch Forum Vol. 5 21 with map scale (Fig. 2). Introductions to fractal geometry routinely explain this coastline paradox. For reasons that are closely related, but only rarely explained [8,9], no single value may reasonably be expected to describe the rate of accumulation of a sedimentary deposit. Mean measured accumulation rates vary with the degree of resolution achieved within the hierarchy of sediment increments and hiatuses. Given a map, measurement of the length of a coastline presents no great difficulty. Yet the value obtained has very little meaning: if the map scale had been different, the measured length of the coastline would probably have been different too. A larger scale generally reveals more detail in the hierarchy of embayments (Fig. 2), leading to a greater measured length. The length of a coastline is evidently a property of both the coastline and the scale of measurement. The scaling relationship is a negative power law (Fig. 3). Figure 3. Two examples of negative power laws that describe coastlines. The circles represent measurements from a more intricately complex coastline with a steeper negative regression and a larger fractal dimension than the coastline reported by the rectangles; its length changes more dramatically with map scale than the simpler coastline represented by the rectangles. The slope of the power law describes the complexity of the coastline. As the shape of the coastline becomes more intricate, the negative slope steepens. A uniform slope means that the coastal complexity appears the same across all scales. The bays and headlands are self-similar at all scales and only somebody familiar with the region can recognize the map scale from the form of the coastline alone. If the slope of the power law changes at some spatial scale, then the complexity of the coastline changes as a function of scale; in other words, bays are not equally well developed at all scales. More intricately complex coastlines are said to have a larger fractal dimension. Fractal dimension is a scale-independent measure of how densely the spatial fractal occupies the space within which it lies. For example, a set of points arrayed to form a straight but broken line is a pattern intermediate between a single point (dimension = 0) and a complete line (dimension = 1). The term fractal describes this concept of “fractional” dimensions. A coastline occupies part of a two dimensional map space; so its fractal dimension may range between 1 and 2. The map of a perfectly smooth straight coast would be a Euclidean line. Its fractal dimension would be 1.0. The decimal form is used deliberately to admit the possibility of dimensions with fractional values. More intricate coastlines effectively occupy more of the map. In the limiting case, which is unattainable for a real coastline, a whole area of the map would be filled by a line that keeps doubling back upon itself. Table 2: Comparison of Fractal Properties 22 On the Determination of Sediment Accumulation Rates PARAMETER: LENGTH OF COASTLINE RATE OF SEDIMENT ACCUMULATION Coordinate system Latitude and longitude (two length scales) Elevation or thickness and time (a length scale and a time scale) Simple formulation Number of measuring rods, laid end-to end, that mimic coastline (or number of map cells which coastline enters) Net thickness of deposited sediment divided by the time elapsed between onset and cessation of deposition Pitfalls Single values should not be extrapolated or compared without attention to length scale Single values should not be extrapolated or compared without attention to time scale Source of fractal complexity Nested hierarchy of bays, headlands and islands Nested hierarchy of sediment increments and surfaces of hiatuses Critical scale Length of measuring rod (or size of counting box) Time span over which rate is averaged Diagnostic regression Log.-measured-length as a function of log.-size of the measuring device Log.-rate as a function of log.-time-span of measurement Range of possible gradients 0.00 to -1.00 A smooth straight coast would generate 0.00 0.00 to -1.00 Steady accumulation generates 0.00 A random walk generates -0.50 Corresponding fractal dimension 1.00 to 2.00 0.00 to 1.00 for the completeness of the sedimentary section (a broken line) Between 1.0 and 2.0 for the accumulation history (a time series) Meaning of steeper negative gradient More intricate and complex pattern of embayments Less complete stratigraphic record; i.e. more time recorded by hiatuses Lower limit of determination Surface texture of cliff materials or beach particles (length becomes impractically large) Instantaneous arrival of individual grains at sediment/fluid interface (rate becomes infinitely large as time span approaches zero) Upper limit of determination: Size of the Continent Overall duration of the sedimentary deposit or stratigraphic section Significant differences 1. Coastline may intersect same longitude at different values of latitude and vice versa. 2. The nested bay-headland motif may be self-similar. 3. Coastlines are a sets of continuous loops (continents and islands). 1. Sediment surface may have the same elevation at different times, but only one elevation for each moment in time. 2. The nested hiatus-increment motif may be self-affine. 3. If single grain arrivals are resolved, then accumulation histories are seen to have discontinuities in elevation. Sediment Accumulation as a Fractal Just as the measured length of a coastline requires a statement of the distance scale before it has real meaning, so too the measured rate of sediment accumulation or an estimate of the completeness of a stratigraphic section should be accompanied by a statement of the time scale. In other words, the rate of accumulation is a property of both the depositional system and the time scale. The diagrams in Figure 1 all have an arbitrary limit on time resolution. If redrawn with finer resolution, the time lines (Fig. 1d, 1e) include more breaks and the staircase plot includes more steps. Notice that the inclusion of more treads (in the magnified images) forces the steps to become shorter and steeper because the long-term slope is fixed. The incremental accumulation rates inevitably increase when examined at finer resolution. The broken time lines that represent the completeness of a stratigraphic section may obviously be considered as fractals with a dimension between zero and one. Readers familiar with fractal GeoResearch Forum Vol. 5 23 geometry may be reminded of the “Cantor Set” fractal [20]. A Cantor Set is a string of dashes (Cantor Bar) or points (Cantor Dust) generated by erasing part of a straight line, and then repeating the erasure pattern on the shorter line segments that remain. We might use the Cantor Bar as an analog for stratigraphic time lines, but the rules for preparing the bar hardly compare with real depositional processes. Furthermore, the Cantor Bar generates a wide range of hiatus lengths, whose ages always reveal a nested regularity. When a random distribution of hiatuses is preferred, then the fractal mathematics of the Levy Dust are appropriate [21]. The Cantor Set is intimately related to another fractal, the Devil’s Staircase [19], which could be used to model the stratigraphic staircase plots. The Devil’s Staircase has repeating and nested patterns in its sequence of steps; stratigraphic staircase plots should be free to exhibit random step distributions. Figure 4. Mean accumulation rates for terrigenous sediments on passive continental margins. a-a’: deltas (diamonds; 2,988 empirical rate determinations); b-b’: shelf seas (filled circles; 22,636); c-c’: continental slopes (crosses; 6,421); d-d’: continental rises and abyssal plains (squares: 10,821); e-e’: abyssal red clays (open circles; 2,215). Rates are averaged for logarithmically scaled windows of time span; there are five, nonoverlapping windows for each order of magnitude. Although the Cantor Set and Devil’s Staircase produce patterns that could simulate some graphical representations of sediment accumulation, it is important to realize that the algorithms used to generate these fractal images are not necessarily analogous to any sedimentary stratigraphic process. These fractals are two dimensional spatial images and some simple operations in space make no sense in the time dimension because time has a real irreversible sense of direction. Of course, mere comparison of images is the least sophisticated application of fractals. Fortunately, a large body of fractal mathematics concerns time series, rather than spatial patterns, and is these fractal relationships that are directly applicable to understanding accumulation rates. 24 On the Determination of Sediment Accumulation Rates The accumulation history of a sedimentary deposit is a time series that must incorporate the fluctuations in the level of the sediment surface which represent a nested hierarchy of hiatuses. By analogy with coastlines, we might expect the complexity of the accumulation history to be described by a fractional dimension between 1.0 and 2.0. But the coordinates of each point are now elevation (or thickness) and time. Technically, alteration of one axis from a length-scale to a time-scale has moved us from self-similar fractals to the realm of self-affine fractals. The term self-affine acknowledges that the elevation and time coordinates cannot be measured in the same units. There is no intrinsically correct proportion for the vertical and horizontal scales of an accumulation history. Experts differ on the propriety of assigning a fractal dimension to self-affine fractals. Instead, there are named fractal coefficients that assume values in a fixed range to describe the relative complexity of time series. These coefficients are analogous to the fractal dimension of self-similar fractals. Both self-similar and self-affine fractal scaling are described by power laws and the slopes of the power laws relate directly to the fractal dimensions and coefficients respectively. The best proof that sediment accumulation has fractal properties is a power-law dependence of accumulation rate upon time span. Figure 5. Mean accumulation rates for marine carbonate sediments. a-a’: peritidal platform and reef deposits (filled circles; 27,029 empirically determined rates); b: periplatform apron deposits (open squares; 7,693); c-c’: calcareous oozes remote from carbonate platforms (open circles: 68,315). Power Law Dependence of Accumulation Rate upon Time Span The empirical data for this paper are large compilations of accumulation rates and their dependence upon the time span of measurement. Plots of rate against time span have been prepared for a wide range of depositional environments (Figs, 4-6) [2,4,10]. For the length of a particular coastline, a few measurements at different scales suffice to reveal that the scaling follows a power law. To characterize the expected rates of accumulation for a whole depositional environment considerably larger samples are needed. A small number of rate determinations may produce a quite spurious slope in this exercise, because the range of measurable rates at any one time scale is quite large. The empirical relationships in figures 4 and 5 stabilized after the compilation of thousands of rate determinations from many different sites. GeoResearch Forum Vol. 5 25 Although each time span is characterized by a large variance in the measured rates, this does not mask the overall negative regression for large samples. Notice that the negative gradients characterize a very wide range of time scales; we must reckon that the sedimentary stratigraphic record incorporates hiatuses with this same wide range of durations. The steepness of the negative slope and, thus, the expected incompleteness of stratigraphic sections varies with environment of deposition, especially at short time spans. But differences that may be attributable to environment of deposition decrease with increasing time span [2]. This probably means that depositional processes are not the primary influence on the distribution of hiatuses at time scales longer than about ten thousand years. At long time scales the most influential factor must be less dependent upon depositional environment and operate on a larger spatial scale to explain the convergence of the power laws. Lithospheric subsidence likely becomes the most significant determinant of the longterm accommodation of sediment. It can explain the long-term convergence of plots from different shallow-marine environments. Accumulation on abyssal ocean floors would not be limited by subsidence; and so, the difference between abyssal and shallow-marine accumulation rates can be sustained to very long time spans. The Problem of “Representative” Accumulation Rates It is evident from figures 4-6 that far more of the total variability in accumulation rate is explained by the time span over which the rate is determined than by either the depositional environment or the age of the sediments. The characteristic accumulation rates for one depositional environment may be represented by a single power law that describes how rate scales with time span, but not by a single rate. Although it is reasonable to seek a single accumulation rate that represents the average for a depositional environment at a given time scale, even this exercise causes trouble. Figure 6. Mean rates of accretion of marine manganese nodules and crusts from shelf seas (squares a-a’, 85 determinations) and abyssal depths (circles b-b’, 1,054). A random, short term observation of most modern flood plains and shelf seas, for example, would most likely be unable to measure any sediment accumulation. Much of the time, sediment does not accumulate. Only a very long series of short-term observations would ensure that the mean rate included a representative number of zero values. And in practice it would be impossible to know when the series was long enough. Empirical logarithmic plots like figures 4-6 seem, at first glance, to compound the problem because they cannot show negative or zero values anyway. They show the average accumulation rate only for intervals in which there is some measurable net deposition. The proper proportion of intervals of zero net-deposition, which is so difficult to measure directly, is now 26 On the Determination of Sediment Accumulation Rates recoverable, however; it is revealed by the reduction in mean rate at longer time spans. If the average rate for ten thousand-year intervals is four times that for hundred thousand-year intervals, for example, then only one in four of the ten thousand year intervals need to be represented by sediment increments, in the longer term, and the other three fourths must be hiatuses. In this way the steepness and length of the negative slope of any segment of the plot describes the incompleteness of the sedimentary record at the time scales of its end points [2,11]. The ratio of the long-term to the short-term accumulation rate is the proportion of elapsed time represented, on average, by sediment when a stratigraphic section at the long time scale is resolved into sediment increments at the scale of the short term end point. The ratio decreases as the length and steepness of the segment increases. This quantifies our expectation that the stratigraphic record must reveal more and more hiatuses when viewed at finer and finer resolution. Because the empirical scaling laws in figures 4-6 are based upon rates from many different locations, purely local differences in the frequency distribution of hiatuses are likely to cancel one another out and have no impact on the general slope. The resulting negative power laws capture only the globally persistent features of the hierarchy of hiatuses. They may reflect the characteristic time scales of processes that accumulate and accommodate sediment. In order of increasing time scale, these processes would be the depositional mechanisms, global sea-level and climate change, and lithospheric subsidence. Tides and storms may dominate shallow marine accumulation at time scales of a few years down to a few hours. At time scales of tens and hundreds of thousands of years, astronomically forced climate change is likely to become a more powerful influence of sediment accommodation. Cooling and subsidence of the oceanic lithosphere accommodates stacks of eustatic cyclothems at time scales of tens of millions of years. In shallow marine environments (Figs. 4ab, 5a) the negative slopes tend to steepen markedly at time scales of tens of thousands to hundreds of thousands of years. The simplest explanation [11] is that hiatuses with this periodicity are more prevalent than hiatuses at longer or shorter intervals. Thus, the negative power laws seem to capture the impact of Milankovitch scale rhythms in eustasy and sediment accommodation. Because these cycles are driven by global climate change, it is reasonable to suppose that their influence upon accumulation rates would survive the process of compositing measurements from hundreds of different localities. For carbonate platforms (Fig. 5a), average accumulation rates fall to 10-20% of their shorter term values over the Milankovitch range of time scales. This implies that the surfaces of hiatus, which many stratigraphers use to delimit meter-scale “cyclothems,” record 80-90% of the time in each accommodation cycle [11]. Thus, interpretation of the slope of the power law has answered the difficult question of the proper apportionment of intervals with no net deposition. As the time span tends to zero, the negative scaling laws in figures 4-6 imply that accumulation rate could increase without bound. As we have already seen, this is an entirely realistic expectation because sediment is deposited as a succession of discrete particles. At the moment of arrival of a single sediment grain, accumulation rate is infinitely rapid; the entire finite grain thickness instantaneously becomes part of the deposit. Interpreting the Slope of the Power Law Numerical models can reveal many aspects of the significance of the slope of the power laws. The models allow the accumulation histories to be exactly prescribed; they facilitate much more intense sampling of age and elevation coordinates than is possible in real stratigraphic sections; and they eliminate the measurement error associated with empirical data. For some fractals that can mimic key properties of stratigraphic accumulation histories, staircase plots and time lines, there are standard equations from which the form of the negative power law may be determined directly. For other numerical models of sediment accumulation, however, plots of rate against time span can more easily be prepared from the outcomes of numerous computer runs. GeoResearch Forum Vol. 5 27 Both approaches will admittedly investigate much simpler time series than the accumulation histories of real sedimentary deposits; they are models rather than simulations. The goal of the exercise is to produce time series that isolate different aspects sediment accumulation histories with regard to the distribution of hiatuses and to look for corresponding differences in the power laws. By manipulating the various components of time series it is hoped to recognize the contribution of each to the power law. These components are the steady trend or ‘drift’ (which might model long term subsidence), periodic components (climate and sea-level cycles, perhaps), and the stochastic component (some aspects of the depositional process). We shall start with a simple periodic model and then examine the Cantor Set analogy. The stochastic element will be modeled as a Brownian Walk, which is a fractal time series with standard equations that can be adopted for geologic situations. Figure 7. Mean accumulation rates for a sine-wave model of periodic accumulation. The model adds a single sine wave (period and wave-height shown) to a secular trend (constant positive slope). The wave-height plots as a line with slope -1 because it is a line of constant thickness. Mean rates are illustrated as determined from: a-a’: dated rocks in cycles that have completed their erosional phases; b: dated changes; and c: dated hiatuses; d-d’: dated rocks from the youngest cycle, prior to any erosional losses. Regular Periodic Models of Accumulation: To isolate the influence of a regular periodicity in the distribution of hiatuses, eliminate all stochastic components from the model and leave just one sine wave and the trend component [2]. The model generates hiatuses with fixed size and spacing. The resulting regressions of rate upon time span (Fig. 7) have three segments with different slope. In the longestand shortest-term segments, the slope approaches zero. Measurements at time spans much longer than the interval between hiatuses are insensitive to the shorter and regularly spaced hiatuses; they see only the steady trend. At time spans much shorter than the duration of the hiatuses, measured rates capture only the rates of accumulation within the sediment increments. Between these two time scales, the slope is steeper, approaching -1.0. The steep portion approximates a line of constant thickness equal to the thickness of the individual sedimentary cyclothems that remain after completion of the erosional phase of the cycle. The break in slope at the shorter-term end of the steep segment approximates the duration and thickness of the sediment increments between hiatuses. The height of the steep segment increases 28 On the Determination of Sediment Accumulation Rates with the fraction of the cycle that is recorded by hiatus. Thus, the pronounced step in the empirical scaling law for accumulation rates in shallow carbonate environments (Fig. 5a) was explained in terms of a pervasive periodic control upon sediment accumulation -Milankovitch-scale eustatic cycles [11]. For shallow-marine terrigenous sediments, the corresponding step is less distinct (Fig. 4b); perhaps the difference is attributable to fluvial processes that continue to distribute sediment on sandy shelves during low-stands of sea level. If more sine waves with different period and amplitude are added to the periodic model, the scaling law must incorporate more steps [2,11]; the steps tend to merge and interfere with one another, giving the impression of a less steep overall slope. Almost any accumulation history could be simulated by including a large enough number of different periodic components with carefully selected period and amplitude; this principle underlies Fourier analysis. Eventually, however, plots with multiple periodic components become indistinguishable from those generated by models which are dominated by the stochastic component. Although the numerical model has revealed the influence of a dominant periodic component on the negative power laws, the exercise should not be compounded in an attempt to reach detailed simulation of the empirical plots unless it is realized that the solutions will not necessarily represent the real depositional dynamics. The Cantor Bar Analogy: It is instructive to examine briefly a standard result for the Cantor Bar fractal because it reveals how simply the slope of the negative power law can be related to fractal dimension or fractal coefficient. Like a stratigraphic time-line, the Cantor Bar does not model thickness. We must recast its descriptive equations into the language of sedimentary stratigraphy by including a constant, K, that depends upon the overall rate of accumulation for the whole section [9,21]. The expected rate of deposition, r, predicted by the Cantor model for a time interval of length t is then a decreasing power law function of that time interval: