Quantifying the Effects of Changing Deposition Rates and Hiatii on the Stratigraphic Distribution of Fossils

Full text

Friedrich-Alexander-Universität Erlangen-Nürnberg
Naturwissenschaftliche Fakultät
Bachelorarbeit
Vorgelegt von Niklas Hohmann
aus Erlangen
Betreuerin: Dr. Emilia Jarochowska
Bearbeitungszeit: 01.04.2018 - 07.06.2018
GeoZentrum
Nordbayern
Quantifying the Effects of Changing
Deposition Rates and Hiatii on the
Stratigraphic Distribution of Fossils

Contents
1 Introduction 2
2 The Basic Model 3
3 Expanding the Model 5
3.1 Incorporating Diastems, Hiatii and Erosion . . . . . . . . . . . . 5
3.2 First and Last Fossil occurrences in the Presence of Gaps . . . . 6
4 Evaluating the Effects of Different Deposition Models 7
5 Examples 9
5.1 The Effect of Different Deposition Models on the Recognition of
ExtinctionRates ........................... 9
5.1.1 The Relation of Extinction Rates and Rates of Last Ap-
pearanceDates........................ 9
5.1.2 Application: Analysis of the Mass Extinction at the K-Pg
Boundary on Seymour Island, Antarctica . . . . . . . . . 10
5.2 The Effects of Binning on the Distinguishability of Extinction
Hypotheses .............................. 12
6 Discussion 14
6.1 Limits of the Model . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.2 Obtaining Deposition Rates . . . . . . . . . . . . . . . . . . . . . 14
7 Summary 15
8 Acknowledgements 16
9 Appendix 16
10 Figures 17
11 Tables 35
1

1 Introduction
The stratigraphic distribution of fossils is an important source of information
on paleoecological processes such as origination or extinction rates [1], but it is
affected by a number of biases that can complicate the reconstruction of the un-
derlying process. This is most clearly demonstrated by the Signor-Lipps effect,
which shows that even a sudden extinction will lead to a ”backwards smearing”
of the last fossil occurrences of the taxa going extinct [2]. This makes the extinc-
tion appear more gradual than it originally was. Many other effects that alter
the relation between the stratigraphic distribution of fossils and the underlying
processes are known. They include destructive effects (diagenetic dissolution
due to shell chemnisty, destruction by bioturbation, any type of physical shell
destruction) and effects of mixing and unmixing (any type of time-averaging,
hydrodynamic sorting) and changes in deposition rates as well as the presence
of hiatii and diastems [3] [4] [5] [6]. Understanding and quantifying these effects
is crucial for any analysis that is based on the stratigraphic position of fossils,
since every fossil occurrence will to some extend be affected by them.
In this paper, I will focus on studying the effect of changing deposition rates,
hiatii, diastems, and erosion on the stratigraphic distribution of fossils. This
effect is based on the observation that different deposition rates place different
amounts of sediment between fossils occurrences that were placed in the sedi-
ment after fixed periods of time. This changes the distance between these fossils
occurrences and thereby alters the stratigraphic distribution of fossils [7] [8].
This effect has been known for a long time and has been studied in the field
as well as by means of simulations [7] [9] [10] [11]. Changing deposition rates
are also a big contribution to the effects of sequence stratigraphy on the strati-
graphic distribution of fossils. Simulations show that these effects can change
biostratigraphic accuracy up to 10000 % [9] and generate clusters of last fossil
occurrences [11]. These clusters can easily be mistaken for signs for elevated
extinction rates, especially since geochemical proxies of extinction causes can
also be affected by these processes to match the pattern of increased extinction
rates [11] [12]. This emphasises the importance of studying the effects of chang-
ing deposition rates and incorporating them into any statistical evaluation of
the stratigraphic distribution of fossils.
In this thesis, I present a model describing the effects of changing deposition
rates and hiatii on the stratigraphic distribution of fossils. This model allows
to exactely determine the effects of changing deposition rates on paleontological
2

hypotheses and can be naturally embedded into a propability theoretic context.
In this context, the effects of different deposition rates on paleontological hy-
pothesis can be approached with statistical methods. This allows, for example,
to correct data for effects of changing deposition rates, to assess the effect of dif-
ferent deposition models on the interpretation of extinction rates (section 5.1)
and to evaluate the distinguishability of hypothesis under different sampling
procedures (section 5.2).
2 The Basic Model
For the beginning, assume a nonzero deposition rate and no erosion. These
model assumptions will be generalized in section 3. Under these assumptions,
every point in a section can be uniquely identified by the age it has been de-
posited. Conversely every point in time corresponds to one and only one point
in the section that has been deposited at this time. This type of correspondence
is called a 1-1 correspondence, which in this case connects points in the section
with points in time. For a given sedimentation rate r, this 1-1 correspondence
is determined by the integrated sedimentation rate:
I(t) = Zt
r(x) dx. (1)
This function assigns to each moment in time tthe amount of sediment that has
been accumulated up to this moment. This is equal to the height in the section
that has been deposited at this moment (fig. 1). The inverse function I−1,
mapping each point in the section yto the time which it has been deposited is
given by the the integrated inverted deposition rate:
I−1(y) = Zy1
r(x)dx. (2)
This correspondence between the section and time can be used to transform
stratigraphic positions of fossils in a section back into time to reconstruct the
time of deposition of said fossils. It can also be used to transform this data fur-
ther into an imaginary section with constant deposition rate. In this imaginary
section, simple statistical methods based on the assumption of constant depo-
sition rates can be applied. This can for example be used to adapt simple first
generation methods [13] to situations with complex sedimentological settings.
3

The results of these methods can then be transformed back into the original
section. This procedure allows to adapt already existing statistical methods to
complex sedimentological settings and therefore ensures backward compatibility
of the presented method.
The correspondence between time and section can be expanded further. For
this, denote the number of fossils that can be found in the section by Fsand
the number of individuals that die, are deposited in the sediment, survive the
taphonomic process and form fossils by Ft. For convenience, i will call these
individuals designated fossils. For a discussion of this concept see section 6.1.
Next, note that the number of fossils found in an interval in the section is de-
termined by the number of designated fossil that are buried during the time in
which the sediment forming this interval of the section has been deposited. This
can be formalized as
Fs([x1, x2]) = Ft(I−1([x1, x2])),(3)
with [x1, x2] being an interval in the section. In a measure theoretic context, the
transformation in equation 3 is known as the pushforward measure or the image
measure ( [14], p. 129; [15], p. 42), here with the number of designated fossils
Ftbeing pushed forward by the integrated sedimentation rate Ito determine
the number of fossil occurrences Fsin the section.
In a more general context, the number of designated fossils Ftcan be replaced
by any function of time f.1For clarity, i will call these functions temporal
rates. In the context discussed here, these functions can have multiple meanings.
The most important temporal rates are the number of individuals of a taxon
(assuming this is proportional to the number of designated fossils generated) as
well as rates of first appearance dates (FAD) or rates of last appearance dates
(LAD) of taxa. These temporal rates can be pushed forward as a whole by the
analytical transformation ( [15], p. 42; [14], p. 132)
fr(y) = f(I−1(y))
r(I−1(y)) ,(4)
where yis a point in the section and fris the transformed temporal rate and
the result of the push forward. I will refer to these transformed temporal rates
or to any rate function defined in the section as observable rates. With this
1This is justified by taking the function fas a density of some measure µwith respect to
the Lebesgue measure λ.
4

transformation, any of the temporal rates mentioned above can be transformed
from time into the section. Here they represent the corresponding observable
rate, given that the sedimentation rate ris the case:
•The number of individuals of a taxon throughout time corresponds to the
rate of fossil occurrences
•The rate of last appearance dates corresponds to the rate of last fossil
occurrences
•The rate of first appearance dates corresponds to the rate of first fossil
occurrences
The transformation from time to the section is displayed in the figures 2, 3, and
4). It is implemented for R and can be found in the appendix under the name
timetostratbasic.R
Since the connection between time and the section is a 1-1 correspondence,
the same principle can be used to transform observable rates from the section
into their corresponding temporal rates in time. In this case, the function used
for the push forward is I−1, the inverse of the integrated sedimentation rate I.
This transformation is also implemented for R and can be found in the appendix
under the name. strattotimebasic.R.
3 Expanding the Model
3.1 Incorporating Diastems, Hiatii and Erosion
The model is based in the assumption of no erosion and nonzero deposition.
This assumption is unrealistic, since it is known that the rock record is not
complete and consists of long periods of nondeposition [3] [4]. Therefore di-
astems, hiatii and erosion must be incorporated in the model.
To do this, the strictly positive sedimentation rate ris expanded to a depo-
sition rate (also denoted as r) that can be positive, zero or negative. Here a
deposition rate of zero corresponds to nondeposition, a negative deposition rate
corresponds to the removal of sediment and a positive deposition rate corre-
sponds to actual sedimentation.
The implementation of the transformation from time to the section can be
adapted to this more general framework. For this, the integrated deposition
5

rate is calculated as before. Then sediment from intervals with erosion is re-
moved, which is done by replacing intervals that are eroded with intervals of
nondeposition. After that, the temporal rate of interest is transformed for each
interval with strictly positive deposition rate separately by applying the analyt-
ical transformation described above. In a last step, all transformed parts of the
temporal rate are joined together to form the observable rate. This procedure
is implemented in R and can be found in the appendix under the name time-
tostrat.R. For more details see the comments in the code and the documentation
of the R code in the appendix.
The transformation from the section into time can be adapted to the more gen-
eral framework in a similar manner. Here the implementation is designed in a
way that hiatii and diastems of any duration can be inserted at points in the
section, which leads to gaps in the resulting temporal rate. This implementation
is named strattotime.R and can again be found in the appendix.
It must be noted that both procedures create discontinuities in the temporal or
observable rate at the points where the hiatus is introduced or the transformed
temporal rates are merged. This is unavoidable, since equation (4), which is
the basis for the transformation, does not hold for deposition rates of zero. The
more general formulation on the basis of the pushforward measure is still valid
in this case, but has not been implemented.
3.2 First and Last Fossil occurrences in the Presence of
Gaps
It is important to note that the transformations of rates of first/last appearance
datums and fossil occurrences in the presence of diastems, hiatii and erosion
with the implementations described above will lead to erroneous results. This is
because rates of first and last occurrences are defined a posteriori in the section
based in the fossils that are present in the section and are therefore not affected
by erosion or nondeposition. This means that the designated first/last fossil
does not necessarily correspond to the first/last fossil occurrence, since it might
be destroyed by erosion.
To compensate for the lack of an analytical solution, an implementation on
the basis of a Monte Carlo approach is used to generate rates of LFO and
FFO in the presence of hiatii. It is attached in the appendix under the name
timetostratlastfirstocc.R and is based on the following scheme:
1. Simulate one last fossil occurrence based on the assumption that they
6

follow an inhomogeneous Poisson point process with the LFO rate as the
rate function. This is done using code from [16]
2. Simulate all appearances of designated fossils before the LFO. This is done
by assuming that they follow a homogeneous Poisson point process with
a given rate. This rate determines the number of appearances per time
unit. This is equivalent to the assumption that the number of individuals
in one taxon is constant throughout time and that all taxa have the same
number of individuals.
3. Transform all appearances into the section, removing the ones that appear
in periods of nondeposition or are deposited in sediment that is eroded
4. Determine the last (highest) fossil occurrence that can be found in the
section
5. Repeat this multiple times to approximate the distribution of LFOs
The same procedure (mutatis mutandis) is also used for first fossil occurrences.
4 Evaluating the Effects of Different Deposition
Models
The model created above can be used to evaluate the effects of different deposi-
tion models on the interpretation of fossil occurrence patterns. For this, the rate
functions introduced above are taken as representatives of probability distribu-
tions. This is done by interpreting them as density functions (with respect to
the Lebesgue measure) of an intensity measure EX that uniquely determines an
inhomogeneous Poisson point process (IPPP) ( [15], p. 530; [17], p. 77). This is
consistent with modeling fossil occurrences as rare and independent events that
occur with changing rates ( [18], p. 153-158).
Within this context, it is possible to define the relative entropy (RE) of two rate
functions, which is given by
E(f0;f1) = Zf1(x)−f0(x) + ln f0(x)
f1(x)f0(x) dx. (5)
[16]. It can be used as a measure of how good the two rate functions can be
distinguished by the samples they generate. This is directly supported by Stein’s
7

theorem ( [19], p. 425), which states that under increasing sample sizes, the rate
of convergence of the type two error is determined by the relative entropy. In
the context described here, the RE can be used in two ways.2
One is to look at two rate functions f0, f1and transform them with a fixed
deposition rate r. Then the relative entropy E(f0
r;f1
r) of the transformed rate
functions is a measure for the distinguishability of the rate functions under the
deposition rate r: A high relative entropy allows to tell the two transformed
rate functions more easily apart. This can for example be used to assess the
distinguishability of two hypotheses on extinction rates under a given deposition
model and to adapt the sampling procedure accordingly. A slightly different
application of this approach can be found in section 5.2.
The second possibility is to fix a rate function fand transform it under different
deposition rates ri, rj. For the resulting functions fri;frj, the relative entropy
E(fri;frj) can be determined. Here, a high relative entropy corresponds to a
higher chance of misinterpreting the underlying rate function f. This is since a
high relative entropy corresponds to friand frjbeing different. But since they
are both created by the same underlying rate function, but look very different
in the field, the outcome of the transformation is sensitive to changes in the
deposition rates and can therefore be mistaken easily. This can for example be
used to asses the sensitivity of the interpretation of an extinction rate under
different deposition models. This is demonstrated in section 5.1.
These two approaches allow the quantification of the effects of deposition rates
on the interpretation of fossil data. One example for this quantification is the
application of ordination methods such as NMDS using the symmetrized relative
entropy (SRE)
˜
E(f0;f1) = E(f0;f1) + E(f1;f0) (6)
as a generalized notion of distance. The SRE can be used to identify rate
functions that generate very similar or very different patterns in the section
under a given deposition model or to identify deposition models that can lead
to different interpretations of the same geological event. This is demonstrated
in the examples below.
2note that the only difference between these approaches is that one fixes a rate function,
while the other one fixes a deposition rate
8

5 Examples
5.1 The Effect of Different Deposition Models on the Recog-
nition of Extinction Rates
As a first example, the framework created above is used to study the effects
of different deposition models on the recognition of extinction rates.3. Since
until now, only rates of last appearance dates were discussed, it is necessary to
discuss the relation between extinction rates and rates of last appearance dates.
5.1.1 The Relation of Extinction Rates and Rates of Last Appear-
ance Dates
It should first be noted that all information about extinction rates is indirect,
since it can only be derived from the last appearance dates. These are however
dependent on the rate of designated fossils, which determines the time between
the actual extinction of a taxon and its last appearance date. If the rate of
designated fossils is high, the time between the actual extinction and the last
appearence date of the taxon, being the last designate fossil before the extinc-
tion, will be shorter. Therefore rates of last appearance dates become more and
more similar to the extinction rate if the rate of designated fossils is higher (fig.
6). So every rate of last appearance dates is a combination of an extinction rate
and a rate4of designated fossils, corresponding to an occurrence rate of some
taxa.
To be able to embed extinction rates into the framework built above, their cor-
responding rates of last appearance dates have to be determined. For this, the
following procedure is used:
1. Simulate an extinction based on the assumption that extinctions follow
an inhomogeneous Poisson point process with the extinction rate as the
rate function (fig. 5). This is done using code from [16]
2. Determine a random number following an exponential distribution with
parameter λ. This number corresponds to the distance between the last
appearance date and the extinction
3All code used for this example can be found in the appendix under the name example1.R.
It uses the R packages MASS [20] and numDeriv [21]
4In a more general setting, every taxon will have its own rate of designated fossils. This
case will not be considered here
9

3. Subtract the random number from the time of extinction to determine the
last appearance date.
4. Repeat this multiple times to approxinmate the distribution of last ap-
pearance dates
Here, the parameter λdetermines the rate of designated fossils. This approach is
based on the model assumptions that the designated fossils appearances follow
a homogeneous Poisson point process with rate λ. This translates into the
assumption that all taxa have the same number of individuals and go extinct
independently of each other.
5.1.2 Application: Analysis of the Mass Extinction at the K-Pg
Boundary on Seymour Island, Antarctica
As a real world example for the methods developed, the K-Pg boundary on
Seymour Island in Antarctica is chosen. On this island, deposits from the Late
Cretaceous to the early Paleogene crops out. It is rich in macrofossils such as
ammonites, bivalves, and gastropods and well sampled [22] [23] [24] [25]. There-
fore data from this locality has been frequently used to derive information about
the nature of the mass extinction at the K-Pg boundary [26] [27] [28] [29] [30].
Hypotheses that were discussed using data from this section include a gradual
extinction [25] as well as one or multiple extinction pulses [31] [26] [32]. Since this
locality is well sampled, data from it is also used as a benchmark for new statis-
tical methods that derive information about extinction events [26] [27] [33] [29].
This is based on the assumption that the fossils in the section on Seymour island
have uniform recovery potential [34] and has been deposited constantly [25].
As the extinction rate whose recognition under different deposition models will
be discussed, a single extinction pulse at the K-Pg boundary as was proposed
by [31] is chosen. This extinction rate is displayed in fig. 8, the rate of last
appearance datums that was derived with the procedure described in section
5.1.1 (with λ= 20 designated fossils per ma) is displayed in fig. 9.
The assumption of a constant deposition rate that is made by many authors is
contrasted by a sequence stratigraphic interpretation by Macellari (1988) [35]
with a maximum flooding surface and a stillstand at the K-Pg boundary(fig. 7).
This is is supported by the presence of hardgrounds [36], condensation horizons
( [24] section A, [35]) as well as lag deposits [37] and dinocyst data that suggests
a flooding [38]. The lateral homogeneity is also questionable ( [24] section A vs.
10

B, [39]), which makes correlation between sections problematic and correlated
datasets less reliable. Additionally there are cold-seep carbonates present in the
lower parts of the section and cold-seep faunas in the higher parts [40]. This
can alter the speed of cementation, which increases preservation [6].
These are all indicators that contrast the assumption of a uniform recovery po-
tential. The aim of this example is to asses the effects of deposition models that
do not assume a constant deposition rate on the recognition of a sudden spike
in the extinction rate at the K-Pg boundary. This is done by transforming the
rate of last appearance dates which was created above (fig. 9) by ten different
deposition models: one with a constant deposition rate and nine variations of
Macellaris sequence stratigraphic interpretation (fig. 10, subfigs. 1 to 10).
These nine deposition models were created by using the temporal framework
provided by Witts et al. (2016) [24] to generate boundary conditions with abso-
lute age. Within this framework, the maximum flooding surface was fixed at the
stratigraphic height where the data from section A from Witts et al. displays a
spike in overall fossil occurrences as well as last and first fossil occurrences. This
height roughly corresponds to the stratigraphic height of the maximum flooding
surface proposed by Macellari in section F from Macellari. This section is closest
to section A from Witts et al. among all sections from Macellari. Then three
deposition rates for the proposed stillstand at the K-Pg boundary (200, 50, and
5 m per ma deposition) and three different durations of the maximum flooding
surface (maximum flooding surface ending 0.199, 0.149, and 0.099 ma before
the K-Pg boundary) were chosen. Creating all possible combinations of these
parameters generates the nine different depostion models that are displayed in
fig. 10.
Three of the last fossil occurrence rates that were generated by pushing the rate
of last appearance dates forward are displayed in fig. 11. For each pair out
of the ten of the transformed rates, the symmetrized relative entropy (SRE) is
calculated and used as a measure of dissimilarity for a NMDS [41] [42]. The raw
values of the SRE can be found in table 1.
The result of the NMDS, which was performed using the R package MASS [20],
is shown in fig. 12. In this plot, the distance between two numbers corresponds
to how different the rates of last fossil occurrences appear in the field. Since
every rate of LFOs is generated by the same underlying extinction rate, but
transformed by a different deposition model, the distance between two numbers
also corresponds to how different the samples generated by the extinction rate
look like under different deposition models. The deposition models that corre-
11

spond to the numbers can be found in fig. 10.
Most relevant is the position of the constant deposition rate (denoted by the
number 1), since it is frequently used by different authors [33] [29] [34] [25] [43].
It is located close to the numbers 2 and 3, but has a marginal position in the
plot. This shows that deposition rates 2 and 3 yield similar results compared
to the constant deposition rate, but results derived under the assumption of a
constant deposition rate will in general differ from the results derived under the
other deposition models. The numbers 8,9 and 10 form a cluster at the right
side of the plot. This shows that results derived from these deposition models
will be very similar. So for a long duration of the maximum flooding surface,
the results are robust in terms of changes of deposition rates at the K-Pg bound-
ary. This does not hold for medium and short durations of maximum flooding
surfaces, since the numbers 2,3, and 4 as well as 5,6, and 7 do not form clusters.
5.2 The Effects of Binning on the Distinguishability of
Extinction Hypotheses
As a second example, the effects of binning on the distinguishability of hypothe-
ses on extinction rates are discussed.5For this, the dataset from Macellari
(1986) [22], consisting of ammonite occurrences collected on Seymour Island, is
used as an example. This dataset is frequently used to derive information about
the mass extinction at the K-Pg boundary (see first example) and to demon-
strate the performance of new statistical methods that derive information about
extinction rates [26] [27] [28] [29] [30] [44] [33] [45] [43] [46]. This data is usually
obtained by digitizing figure 5 from Macellari (1986) [26] [28], whose way of
representation suggests that the data was sampled continuously, as is seen in
figure 13. But this is not the case, since the data from Macellari (1986) is based
on Macellari (1984) [47], where the raw data from different sections is presented
in sampling bins with up to 39 meters length, and findings of ammonites are
recorded as present, abundant or very abundant within these bins. The data
presented by Macellari (1986) is also correlated across Seymour Island, which
might be problematic because of the lack lateral homogeneity that was noted in
section 5.1.
The goal of this example is to asses how the distinguishability of different pairs
of extinction rates changes due to binning. For this, nine extinction rates are
5All code used for this example can be found in the appendix under the name example2.R.
It uses the R packages MASS [20] and numDeriv [21]
12

defined. They consist of single extinction pulses of different duration (extinc-
tion rates 1-3), gradually increasing extinction rates (extinction rate 4) or two
extinction pulses that are differently weighted(extinction rates 5-9) (fig. 14).
For comparability, the extinction rates are chosen in a way that in average they
all generate the same number of extinctions. The first extinction interval in the
extinction rates 5 to 9 is created by taking the extinction interval proposed by
in fig. 5 from [26] and transforming it back into time under the assumption of a
constant deposition rate. The rates of last appearance dates derived from these
extinction rates are displayed in fig. 15.
For simplification, the deposition rate is assumed to be constant. Therefore the
pushforward by this deposition rate reduces to a rescaling of the rate functions
as they are displayed in fig. 14. Then for each pair of these rates of last fossil
occurrences, the symmetricized relative entropy (SRE) is calculated. The re-
sults are displayed in table 2 and taken as input for a NMDS6. The result of
this NMDS is displayed in figure 16. In this figure, the distance between two
numbers is proportional to the distinguishability of the corresponding extinc-
tion rates shown in figure 14. This shows that the single pulses with different
durations (numbers 1,2, and 3) are easy to distinguish, whereas the gradual ex-
tinction (4) and the two longer pulses are harder to distinguish. The extinctions
with two extinction pulses (5-9) are approximately equally hard to distinguish.
In a second step, each of the rate of last fossil occurrences is binned with the
bins taken from Macellari (1984), section A. One example for the continuous
as well as the binned version of a rate function can be found in fig. 17. Then
the SRE is calculated for each pair of the binned rate functions. The results
are shown in table 3, which are used for another NMDS. The results of this
NMDS are displayed in figure 18. It shows, up to a rotation, the same pattern
as the NMDS of the continuous functions. This demonstrates that the relation
of distingushability between different extinction rates does not change much due
to binning.
Last, for each pair of last fossil occurrence rate, the ratio of the SRE of the
continuous functions and the SRE of the binned functions is determined. These
ratios are presented in table 11. This ratio serves as a measure of how much
harder the different extinction hypotheses are to distinguish after binning. The
results show a range of approximately 1.1 to approximately 2.4. So in the best
case, extinctions just get roughly 10 % harder to distinguish after binning. This
6using the R package MASS [20]
13

is for example the case for any pair of extinction rates with two pulses (5-9). In
the worst case, distinguishability increases by 140 % because of binning. This
is the case for the extinction rate with one pulse of intermediate duration (2)
and the gradually increasing extinction rate (4).
6 Discussion
6.1 Limits of the Model
There are certain limits to the model presented here.
One implicit model assumption that was not discussed or mentioned above is
that shells have no volume, so they do not contribute to the deposition rate.
This is neglectable in periods with low input of shell volume and sufficiently
high sediment input. However in periods with high volume of shell input, com-
bined with intervals of nondeposition, this model assumption will lead to errors.
Another potential problem is the concept of the designated fossil. It was in-
troduced to ensure that the fossil occurrences in the section correspond to an
individual of some taxon in time. To make this correspondence meaningful, it
is necessary that the rate of designated fossils is proportional to the rate of the
actual individuals of the taxon. This basically translates to the model assump-
tion that all taphonomic processes are invariant. This model assumption can
be problematic, since there is a variety of processes like time-averaging, com-
paction, diagenetic dissolution and the effects of bioturbation that can disturb
this invariance. It is possible to incorporate these processes into the framework
provided in this thesis, but this requires a more detailed discussion than is pos-
sible here. Everything said above applies mutatis mutandis to designated first
fossils and designated last fossils.
6.2 Obtaining Deposition Rates
Deposition rates play a crucial role in the model presented here. It is however
hard to obtain reasonable estimates for these rates. This can be made easier by
the following considerations:
•Every age model is always also a deposition model. This is since both
connect point in time with point in the section and vice versa. So points in
the section with a fixed absolute age can be used as a temporal framework
14

and as boundary conditions for the deposition rates. This is for example
done in section 5.1.
•Deposition rates do not change arbitrarily, but within certain restrictions
that are set by the sequence stratigraphic framework. This can be used
to derive points with minimal or maximal deposition rates or to identify
intervals having lower or higher intervals than other intervals. This is for
example done in section 5.1.
These approaches can be used as a rough framework for deposition rates that
can then be filled with more detailed observations. These observations can for
example be obtained by using a sampling procedure that does not only aim
at the stratigraphic position of fossils, but also at the reconstruction of the
depositional environment.
And even in cases where no unique or precise deposition rate can be derived,
the model can still be used to assess the robustness of the results under the
available competing deposition rates.
7 Summary
The presented model embeds the effects of changing deposition rates and hiatii
on the stratigraphic distribution of fossils in a probability theoretical framework.
This allows, among others, to
•Correct data for effects of changing deposition rates (section 2)
•Analytically study the effects of changing deposition rates and hiatii on
paleontological patterns such as extinction rates and origination rates (sec-
tion 3)
•Analyse the sensitivity of results with regard to different deposition models
(sections 4 and 5.1)
•Assess the distinguishability of hypotheses that are based on the strati-
graphic distribution of fossils and adapt sampling procedures accordingly
(sections 4 and 5.2)
15

8 Acknowledgements
I would like to thank Emilia Jarochowska for being a very patient and encour-
aging supervisor, and Rafal Nawrot for taking the time to give feedback on the
project.
9 Appendix
In the digital version of this thesis, the following files are provided:
•A R file named example1.R, containing all code that was used for the
example in section 5.1.
•A R file named example2.R, containing all code that was used for the
example in section 5.2.
•The R files timetostrat,timetostratbasic,strattotime,strattotimebasic, and
timetostratlastfirstocc.R. They perform the transformations described in
the sections 2 and 3
•The file docrcode.pdf, containing descriptions of the transformations that
complement the comments in the code
•The R file functionexamples.R, containing simple examples of the trans-
formations described in the sections 2 and 3
16

10 Figures
0 2 4 6 8 10 12
0 5 10 15 20 25 30 35
Time (Arbitrary Unit)
Stratigraphic Height (in m)
Figure 1: An integrated sedimentation rate I. The thin dashed black lines
indicate how points in time are connected to heights in the section and vice versa
by I. The sedimentation rate rthat generated this integrated sedimentation rate
is qualitatively indicated by the red dashed line. Time intervals with a high
sedimentation rate lead to a steep integrated sedimentation rate, whereas time
intervals with a low sedimentation rate lead to a flat integrated sedimentation
rate.
17

0 2 4 6 8 10 12
02468
Time (ArbitraryUnit)
Rate
Figure 2: The sedimentation rate rfrom fig. 1(red dashed line, in sediment
deposited per time unit) and a temporal rate f(blue dashed line, in events
(fossils, extinctions etc.) per time unit).
18

0 2 4 6 8 10 12
0 5 10 15 20 25 30 35
Time (Arbitrary Unit)
Stratigraphic Height (in m)
Figure 3: The temporal rate ffrom fig. 2(blue dashed line) being pushed
forward by the integrated deposition rate I(red thick line) from fig. 1) to
form the corresponding observable rate fr(thick blue line). The thin black
dashed lines indicate the 1-1 correspondence between time and stratigraphic
height. Note that the maximum of the temporal rate has no corresponding
maximum in the observable rate, since it is accompanied by a high deposition
rate (compare with fig. 2), which dilutes the rate function. This is contrasted
by the maximum in the observable rate, which is created by low deposition rates
(compare fig. 2), although the temporal rate in the corresponding time interval
decreases.
19

0 5 10 15 20 25 30 35
0.0 0.5 1.0 1.5 2.0 2.5
Stratigraphic Height (in m)
Events per m Stratigraphic Height
Figure 4: The observable rate that was generated by the transformation dis-
played in fig. 3. The sediment between two thin dashed black lines has been
deposited in two time units. This emphasizes that the sediment containing the
peak in the observable rate has been deposited over a longer period of time than
all of the sediment forming the lowest 15 meters of the section.
20

0.0 0.5 1.0 1.5 2.0 2.5 3.0
0246810
Time (Arbitrary Unit)
Extinctions per Time Unit
Figure 5: An extinction rate (thick red line). On the six thin dotted lines,
different simulated positions of extinctions (black crosses) are displayed. Note
that in general, not only the position of the extinctions is random, but also the
number of extinctions generated by an extinction rate. In the case displayed
here, the number of observed extinctions ranges from five (second dotted line
from below) to thirteen (third dotted line from below). The maximum in the
extinction rate does also not necessarily generate a cluster of extinctions (fourth
dotted line from below), since the position of extinctions is nondeterministic.
The rates of last appearance dates derived from this extinction rate for different
parameters λare desplayed in fig. 6.
21

0.0 1.0 2.0 3.0
048
Lambda = 2
Time (Arbitrary Units)
Rate
0.0 1.0 2.0 3.0
048
Lambda = 4
Rate
Time (Arbitrary Units)
0.0 1.0 2.0 3.0
048
Lambda = 8
Rate
Time (Arbitrary Units)
0.0 1.0 2.0 3.0
048
Lambda = 16
Rate
Time (Arbitrary Units)
Figure 6: The rate of last appearence dates (black, in last appearance dates per
time unit), derived from the extinction rate in fig. 5 and assuming different rates
of designated fossils λ(see section 5.1). As λincreases, the shape of the rate of
last appearance dates is approximating the extinction rate (red, in extinctions
per time unit). This is since for increasing rates, the random numbers following
an exponential distribution are getting smaller, which corresponds to more des-
ignated fossils per time unit. This reduces the distance between an extinction
and the corresponding last appearance date. Transforming this into the section,
this can be seen as a depiction of the Signor-Lipps effect: as fossil occurrences
become rarer (corresponding to a lower rate λ), the last fossil occurrences show
more backwards smearing compared to the actual extinction rate. Note that
since the overall number of extinctions is constant, the overall number of last
fossil occurrences is also constant. Therefore the backwards smearing not only
spreads the the last fossil occurrences over a longer period of time, but also
makes the extinction appear less dramatic.
22

Figure 7: Figure 18 from Macellari (1988) [35]. The interval discussed in the
example starts slightly to the left of the writing maximum flooding and ends at
the transition of unit 9 to unit 10.
23

66.4 66.3 66.2 66.1 66.0
0 50 100 150 200
Age (ma)
Extinctions per ma
Figure 8: The extinction rate used for the first example from Seymour island
(section 5.1). It consists of one spike in the extinction rate right at the K-Pg
boundary and a low background extinction rate. The background extinction
rate generates in average one extinction per ma. The duration of the extinction
pulse is chosen to generate in average as many extinctions as the background
extinction would in one million years.
24

66.4 66.3 66.2 66.1 66.0
0 5 10 15 20 25
Age (ma)
Last Appearance Dates per ma
Figure 9: The rate of last appearance dates derived from the extinction rate
in fig. 8. The rate is λ= 20, which corresponds to an average of 20 fossil
appearances per taxon per one million years.
25

66.4 66.1
940 1000
Age
Stratigraphic Height
Time
accumulated sediment
Time
accumulated sediment
Time
accumulated sediment
Time
accumulated sediment
Time
accumulated sediment
Time
accumulated sediment
Time
accumulated sediment
accumulated sediment
accumulated sediment
940 1000
940 1000
940 1000
940 1000
940 1000
940 1000
940 1000
940 1000
940 1000
66.4 66.1 66.4 66.1 66.4 66.1
66.4 66.1 66.4 66.1 66.4 66.1 66.4 66.1
66.4 66.1 66.4 66.1
10
9
8765
4321
Figure 10: An overview of the deposition models from the example in section
5.1, displayed as accumulated sediment/integrated deposition rate. The x-axis
is age, the y-axis is stratigraphic height, both are taken from [24]. The depo-
sition rates for the proposed stillstand at the K-Pg boundary are 200 (high),
50 (medium), and 5 (low) m deposition per ma, the three different durations of
the maximum flooding surface end 0.199 (short duration mfs), 0.149 (medium
duration mfs), and 0.099 ma (long duration mfs) before the K-Pg boundary.
The numbers display the following deposition models: 1: constant deposition
rate, 2: short maximum flooding surface (mfs) duration and high deposition rate
(dr) at the K-Pg boundary, 3: short mfs duration and medium dr at the K-Pg
boundary, 4: short mfs duration and low dr at the K-Pg boundary, 5: medium
mfs duration and high dr at the K-Pg boundary, 6: medium mfs duration and
medium dr at the K-Pg boundary, 7: medium mfs duration and low dr at the
K-Pg boundary, 8: long mfs duration and high dr at the K-Pg boundary, 9:
long mfs duration and medium dr at the K-Pg boundary, 10: long mfs duration
and low dr at the K-Pg boundary.
26

940 960 980 1000
0.00 0.05 0.10 0.15 0.20
Stratigraphic Height (in m)
Last Fossil Occurrences per Meter
Figure 11: Three transformations of the rate of last appearance dates as dis-
played in fig. 9. The black line corresponds to the rate of last fossil occurrences
under a constant deposition rate (deposition model 1 in fig. 10), the red line to
the rate of last fossil occurrences under a short mfs and high deposition rates at
the K-Pg boundary (deposition model 2 in fig 10) and the red line to the rate
of last fossil occurrences under a long mfs and low deposition rates at the K-Pg
boundary (deposition model 10 in fig 10). The spike in the green rate of last
fossil occurrences reaches values of over two. Note that the overall number of
last fossil occurrences is constant, so more condensation at the mfs with more
last fossil occurrences automatically leads to less last fossil occurrences at the
K-Pg boundary.
27

1
2
3
4
5
6
7
8
9
10
−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0
−1.0 −0.5 0.0 0.5
NMDS Axis 1
NMDS Axis 2
Figure 12: The results of the NMDS from the example in section 5.1. The
numbers correspond to the deposition models as described in fig. 10. The
distance between two numbers corresponds to how different the signal of the
extinction described in the figures 8 and 9 looks if the different deposition models
from figure 10 are used for the transformation from time into the section, see
section 4, approach 2. Therefore the distance between two numbers is a measure
of how strongly the interpretation of the extinction by an sudden extinction
pulse at the K-Pg boundary is altered by the choice of the deposition model.
28

Figure 13: Figure 5 from Macellari (1986) [22]. The mode of representation
suggests that every point corresponds to one ammonite occurrence.
29

0 50 150
Time
Ext. per ma
5 10 20
Time
Ext. per ma
2 4 6 8
Time
Ext. per ma
5 10 20
Time
Ext. per ma
0 50 150
Time
Ext. per ma
0 50 100
Time
Ext. per ma
0 40 80 140
Ext. per ma
0 40 80 120
Ext. per ma
0 40 80
Ext. per ma
123
456
9
87
66.4 66.25 66.1 66.4 66.25 66.1 66.4 66.25 66.1
66.4 66.25 66.1 66.4 66.25 66.1 66.4 66.25 66.1
66.4 66.25 66.1 66.4 66.25 66.1 66.4 66.25 66.1
Figure 14: The nine different extinction rates used for the example in section
5.2. The x-axis is age, the y-axis is extinctions per ma. Note the different scales
on the y-axis. All extinction rates are adjusted to generate in average the same
number of extinctions.
30

66.4 66.25 66.1
0 5 10 20
time
Rate of LAD
0 5 10 20
time
Rate of LAD
0 5 10 20
time
Rate of LAD
0 5 10 20
time
Rate of LAD
0 5 10 20
time
Rate of LAD
0 5 10 20
time
Rate of LAD
0 5 10 20
Rate of LAD
0 5 10 20
Rate of LAD
0 5 10 20
Rate of LAD
66.4 66.25 66.1 66.4 66.25 66.1
66.4 66.25 66.1 66.4 66.25 66.1 66.4 66.25 66.1
66.4 66.25 66.1 66.4 66.25 66.1 66.4 66.25 66.1
123
456
9
87
Figure 15: The nine continuous rates of last appearance dates derived from the
extinction rates shown in figure 14, using the procedure described in section
5.1.1 and a rate of λ= 20 designated fossils per ma. The x-axis is age, the
y-axis is last appearance dates per ma.
31

1
2
3
4
5
6
7
8
9
−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6
−0.6 −0.4 −0.2 0.0 0.2 0.4
NMDS Axis 1
NMDS Axis 2
Figure 16: The result of the NMDS of the continuous rates of last fossil occur-
rences from section 5.2. The distance between the numbers corresponds to how
well the corresponding extinction rates in fig. 10 can be distinguished via their
derived continuous rates of last fossil occurences.
32

66.40 66.30 66.20 66.10
0 5 10 15
Age (ma)
Last Appearance Dates per ma
Figure 17: The continuous rate of last appearance dates (red) and its binned
counterpart (blue), derived from the extinction rate number six in fig. 14. The
bins from Macellari (1984) [47], section A were transformed into time assuming
a constant deposition rate. The thin dashed lines indicate the borders of these
bins. For each bin, the area of the bin and the area under the continuous rate
of last appearance dates are equal. Note that the two extinction pulses are well
visible in the continuous rate of last appearance dates, whereas they are not
visible at all in the binned rate of last appearance dates.
33

1
2
3
4
5
6
7
8
9
−0.5 0.0 0.5
−0.6 −0.2 0.2 0.4 0.6
NMDS Axis 1
NMDS Axis 2
Figure 18: The result of the NMDS of the binned rates of last fossil occurrences
from section 5.2. The distance between the numbers corresponds to how well the
corresponding extinction rates in fig. 10 can be distinguished via their derived
binned rates of last fossil occurrences.
34

11 Tables
model 1 2 3 4 5 6 7 8 9 10
1 0 0.2460 0.8402 1.4690 2.5539 2.8857 3.1526 3.4248 3.5420 3.6339
2 0.2460 0 0.5469 1.2009 1.4084 1.6809 1.9647 1.7088 1.7914 1.8796
3 0.8402 0.5469 0 0.2680 1.8965 1.4445 1.5143 2.1448 1.8278 1.8100
4 1.4690 1.2009 0.2680 0 2.4968 1.7201 1.4810 2.6480 2.1168 1.9599
5 2.5539 1.4084 1.8965 2.4968 0 0.2460 0.5014 0.7538 0.8297 0.9089
6 2.8857 1.6809 1.4445 1.7201 0.2460 0 0.0816 0.9845 0.7866 0.7838
7 3.1526 1.9647 1.5143 1.4810 0.5014 0.0816 0 1.2004 0.8840 0.8096
8 3.4248 1.7088 2.1448 2.6480 0.7538 0.9845 1.2004 0 0.0755 0.1419
9 3.5420 1.7914 1.8278 2.1168 0.8297 0.7866 0.8840 0.0755 0 0.0151
10 3.6339 1.8796 1.8100 1.9599 0.9089 0.7838 0.8096 0.1419 0.0151 0
Table 1: The raw values of the symmetrized relative entropy (SRE) from the
example in section 5.1. The SRE of the transformation of the rate of last
appearance dates (fig. 9) by the i-th and by the j-th deposition model (fig. 10)
can be found in the i-th column and the j-th row. Note that the table, taken as
a matrix, is symmetrical, since the SRE is symmetrical.
model 1 2 3 4 5 6 7 8 9
1 0 0.4253 0.8857 0.5004 0.0460 0.1582 0.3273 0.5513 0.8254
2 0.4253 0 0.2432 0.0363 0.4116 0.4749 0.5938 0.7657 0.9787
3 0.8857 0.2432 0 0.0997 0.7923 0.7816 0.8235 0.9217 1.0541
4 0.5004 0.0363 0.0997 0 0.4643 0.5065 0.6036 0.7542 0.9442
5 0.0460 0.4116 0.7923 0.4643 0 0.0342 0.1279 0.2775 0.4795
6 0.1582 0.4749 0.7816 0.5065 0.0342 0 0.0305 0.1174 0.2582
7 0.3273 0.5938 0.8235 0.6036 0.1279 0.0305 0 0.0292 0.1127
8 0.5513 0.7657 0.9217 0.7542 0.2775 0.1174 0.0292 0 0.0282
9 0.8254 0.9787 1.0541 0.9442 0.4795 0.2582 0.1127 0.0282 0
Table 2: The values of the SRE of the continuous rates of last fossil occurences
from the example in section 5.2. The SRE of the i-th and the j-th continuous
rate of last fossil occurrences can be found in the i-th column and the j-th
row. Note that the table, taken as a matrix, is symmetrical, since the SRE is
symmetrical.
35

model 1 2 3 4 5 6 7 8 9
1 0 0.2020 0.6527 0.3242 0.0410 0.1431 0.2970 0.5023 0.7524
2 0.2020 0 0.1325 0.0150 0.1993 0.2619 0.3744 0.5398 0.7418
3 0.6527 0.1325 0 0.0582 0.5764 0.5733 0.6204 0.7222 0.8549
4 0.3242 0.0150 0.0582 0 0.2987 0.3408 0.4328 0.5784 0.7585
5 0.0410 0.1993 0.5764 0.2987 0 0.0307 0.1163 0.2539 0.4384
6 0.1431 0.2619 0.5733 0.3408 0.0307 0 0.0273 0.1074 0.2361
7 0.2970 0.3744 0.6204 0.4328 0.1163 0.0273 0 0.0264 0.1028
8 0.5023 0.5398 0.7222 0.5784 0.2539 0.1074 0.0264 0 0.0252
9 0.7524 0.7418 0.8549 0.7585 0.4384 0.2361 0.1028 0.0252 0
Table 3: The values of the SRE of the binned rates of last fossil occurences from
the example in section 5.2. The SRE of the i-th and the j-th binned rate of last
fossil occurrences can be found in the i-th column and the j-th row. Note that
the table, taken as a matrix, is symmetrical, since the SRE is symmetrical.
model 1 2 3 4 5 6 7 8 9
1 - 2.1058 1.3570 1.5434 1.1198 1.1054 1.1018 1.0976 1.0970
2 2.1058 - 1.8354 2.4148 2.0647 1.8132 1.5861 1.4185 1.3192
3 1.3570 1.8354 - 1.7141 1.3745 1.3633 1.3274 1.2762 1.2330
4 1.5434 2.4148 1.7141 - 1.5546 1.4862 1.3945 1.3040 1.2447
5 1.1198 2.0647 1.3745 1.5546 - 1.1111 1.0996 1.0932 1.0938
6 1.1054 1.8132 1.3633 1.4862 1.1111 - 1.1152 1.0931 1.0939
7 1.1018 1.5861 1.3274 1.3945 1.0996 1.1152 - 1.1078 1.0957
8 1.0976 1.4185 1.2762 1.3040 1.0932 1.0931 1.1078 - 1.1198
9 1.0970 1.3192 1.2330 1.2447 1.0938 1.0939 1.0957 1.1198 -
Table 4: The values of the ratio of the SREs of the continuous rate of last fossil
occurrences and the binned rate of last fossil occurrences from the example
in section 5.2. The ratio of the i-th and the j-th continuous and binned rate
function can be found in the i-th column and the j-th row. Note that the table,
taken as a matrix, is symmetrical, since the SRE is symmetrical.
36

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40

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41

Friedrich-Alexander-Universität Erlangen-Nürnberg
GeoZentrum Nordbayern
Name: ………………………………………………….. Vorname: ……………………………………………….
Matrikelnummer:………………………………….
Selbständigkeitserklärung zur Bachelorarbeit
Ich erkläre ausdrücklich, dass es sich bei der von mir eingereichten schrilichen Arbeit
mit dem Titel
............................................................................................................................
............................................................................................................................
um eine von mir selbstständig und ohne fremde Hilfe verfasste Arbeit handelt.
Ich erkläre, dass ich sämtliche in der oben genannten Arbeit verwendeten fremden Quellen,
auch aus dem Internet (einschließlich Tabellen, Grafiken u. Ä.) als solche kenntlich gemacht
habe. Insbesondere bestäge ich, dass ich ausnahmslos sowohl bei wörtlich übernommenen
Aussagen bzw. unverändert übernommenen Tabellen, Grafiken u. Ä. (Zitaten) als auch bei in
eigenen Worten wiedergegebenen Aussagen bzw. von mir abgewandelten Tabellen,
Grafiken u. Ä. anderer Autorinnen und Autoren (Paraphrasen) die Quelle angegeben habe.
Ich erkläre hiermit weiterhin, dass die vorgelegte Arbeit zuvor weder von mir noch – soweit
mir bekannt ist – von einer anderen Person an dieser oder einer anderen Hochschule
eingereicht wurde.
Mir ist bewusst, dass Verstöße gegen die Grundsätze der Selbstständigkeit als Täuschung
betrachtet und dass die Unrichgkeit dieser Erklärung eine Benotung der Arbeit mit der Note
nicht ausreichend zur Folge hat und dass Verletzungen des Urheberrechts strafrechtlich
verfolgt werden können.
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(Datum) (Unterschri)