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251
12.1 Introduction and Background
A common goal for studies using isoscapes is to determine geographic areas in which
organic material was developed, particularly in studies of wildlife or human forensics.
For example, studying animal migration by direct observation is difficult and expen-
sive for any species, and is impossible for many small species and for questions
involving archaic systems. The use of stable isotope tracers, however, has proven use-
ful for inferring geographic origins and spatial connections for a range of species or
materials and a variety of temporal extents. Examples include estimating single-
season movement patterns of songbirds (Norris et al. 2004), reconstructing geo-
graphic histories for modern butterflies (Wassenaar and Hobson 1998), prehistoric
humans (Muller et al. 2003; Sharp et al. 2003) and historic human trade routes
(Giuliani et al 2000), as well as forensic work to determine recent human move-
ment (Fraser et al. 2006), or the origin of illicit substances such as drugs or explo-
sives (Ehleringer et al. 2000; Benson et al. 2006).
The availability of isoscapes has generated much excitement about potential
GIS-based analyses to determine geographic origins in studies of wildlife or human
forensics. However, there are some misconceptions in the literature about the nature
of isoscapes and isotope-based geographic assignments. Most significantly, many
studies fail to recognize that GIS-based information in isoscapes is not really data
at all, but rather is a set of expectations from the isoscape model. That is, values in
isoscape grids are predicted values from a model; they describe a general pattern,
and for the most part disregard the degree to which observed data are scattered
about that pattern. A secondary and less pervasive misconception is that some
isoscapes are universal in application. An isoscape that can be effectively applied
M.B. Wunder ()
Department of Biology, University of Colorado Denver, Campus Box 171, Denver, CO,
80217-3364
e-mail: michael.wunder@ucdenver.edu
Chapter 12
Using Isoscapes to Model Probability Surfaces
for Determining Geographic Origins
Michael B. Wunder
J.B. West et al. (eds.), Isoscapes: Understanding Movement, Pattern, and Process
on Earth Through Isotope Mapping, DOI 10.1007/978-90-481-3354-3_12,
© Springer Science + Business Media B.V. 2010
252 M.B. Wunder
to one study may be less useful for another; in nature no single geographic location
is expected to produce exactly the same isotope value for every organism using
resources there, nor is any location value expected to stay the same from year to
year. Isoscapes must be custom-tailored to individual forensic studies not only for
best results, but also in order to effectively explore mechanistic hypotheses for
explaining differences in efficacy among studies.
Perhaps for the sake of convenience, deviations from mean isoscape values
are rarely formally addressed in models, but they are occasionally documented.
Because of this, many sources of within-location variance in isoscapes are
well known and the magnitudes of these sources of variance can be easily
estimated. This can be extremely useful for not only characterizing certainty
in geographic assignments (Wunder and Norris 2008a), but also for provid-
ing insight into the most fruitful directions for future research (Bowen and
Revenaugh 2003).
Along with work presented elsewhere in this volume, this chapter concerns
the application of, rather than the generation of, isoscapes. In contrast, however,
the ideas presented here are centered primarily on the consideration of variances
rather than of means. Because of this, I de-emphasize prediction in favor of
probabilistic assignment, which marks this work as fundamentally different
from the many examples using GIS-based isoscapes for wildlife forensics
reported in the literature (e.g. Wassenaar and Hobson 1998; Hobson et al. 1999,
2006, 2007; Meehan et al. 2001; Cryan et al 2004; Norris et al. 2004; Bearhop
et al. 2005; Bowen et al. 2005; DeLong et al. 2005; Lott and Smith 2006; Paxton
et al. 2007). I describe here the use of variance estimates directly to generate
probability surfaces and to identify gaps in knowledge. I discuss approaches for
both discrete and continuous frameworks to characterize the probability of
specific geographic origins and I discuss the differences between such charac-
terizations for individuals and for populations.
This chapter is intended as a proof-of-concept work, and will not prescribe when
or where to apply or avoid various isoscapes. It is intended to provide generalized
guidance for using isoscapes in any research that features geographic forensics as
a goal. In an effort to carry along the chapter and the ideas that shape the approach,
I refer to a generalized example problem of assigning summering locations to birds
sampled during winter via a precipitation-based hydrogen isoscape, a widely-
reported application of isoscapes. I constrain the presentation in this way to high-
light the relevant features of the approach without distraction from the unavoidable
list of caveats that riddle any particular case-study and without the added complex-
ity of addressing issues of covariance and added dimensionality that arise from
considering multiple isoscapes and other information sources simultaneously.
However, I point out that the efficacy of the approach is considerably improved by
using multiple isoscapes and higher-dimension covariates in case-specific applica-
tions; the exploration and reporting of such extended applications are strongly
encouraged.
25312 Using Isoscapes to Model Probability Surfaces for Determining Geographic Origins
12.2 Example Problem
The example here concerns the use of stable isotope compositions of bird feathers
to infer cross-seasonal geographic connectivity in North American bird migration
systems, the basic examples of which were first presented by Chamberlain et al.
(1997) and Hobson and Wassenaar (1997). The premise is that we can use an
isoscape to find the geographic location where bird feather or claw material (kera-
tin) was synthesized. Bird feathers and claws are useful in this regard because they
are metabolically inert once formed, and so in theory the stable isotope values of
the keratin will stay the same as the bird migrates to a new location. The idea is to
link two different points in time and space as part of the migratory history of an
individual bird using isotopes in keratin.
Migratory birds typically molt feathers once or twice per year, often depending
on the type of feather in question. For example, many North American migratory
birds molt flight feathers (wings and tail) on the summer grounds late in the breed-
ing cycle and rely on these feathers to get them to the wintering grounds and back
(Pyle 1997). Thus, by sampling flight feathers during winter, one can determine
locations within the breeding region where the feather was likely synthesized dur-
ing the previous summer. It is worth noting that many of the same migratory spe-
cies molt their contour feathers (the body feathers) twice per year; once into basic
(winter) plumage near the end of the breeding cycle, and again into alternate
(breeding) plumage just prior to spring migration to the breeding grounds (Pyle
1997), suggesting that it might be possible to also determine winter locales for the
same species of birds sampled during summer. The point in either case is that
feathers can be used to infer geographic linkages between summer and winter
regions by sampling in the alternate season from when the feather was grown. The
goal of such migratory bird studies is to learn how geographic dependencies
within species are shaped through the course of a full year in seasonal systems
(Webster and Marra 2005).
12.2.1 Study Design and Model Development
For the migratory bird example, I calibrate an available precipitation-based
isoscape using freshly grown feathers sampled from young and adult birds across
the target range (the range of isoscape values spanning the spatial extent of the
breeding range). I then sample feathers during winter that are known to have been
synthesized the summer before. The calibrated isoscape is used to estimate the
otherwise unknown breeding origins of birds sampled during winter. The list of
assumptions and expectations necessary for this plan include (1) the assumption
that the precipitation-based hydrogen isoscape provides a reasonable process-
model for the expected spatial pattern in hydrogen isotopes found at the base of
avian food webs, (2) the assumption that values in that precipitation isoscape are
254 M.B. Wunder
fixed and known (e.g. model-based variance around the mean spatial pattern is
negligible), (3) the expectation that “nice” transfer functions modeling water
isotope values through the food web to bird keratin isotopes can be estimated
empirically, (4) the assumption that carbon-bound (non-exchangeable) hydrogen
isotope values of organic keratin do not change with time, (5) the expectation that
lab practices reliably calibrate hydrogen measurements in organic keratin to Vienna
Standard Mean Ocean Water (VSMOW) using multiple organic keratin standards,
(6) the assumption that keratin is synthesized by birds during summer prior to
migration, (7) the expectation that age of bird influences isotope values of keratin,
and (8) the expectation that all other factors influencing keratin isotope values are
either unknown or un-measurable for birds sampled during winter. This list of
assumptions and design plans is nearly universal in these types of studies. Study
design and model construction follow directly: In order to understand the extent to
which the precipitation-based isoscape model is reasonable (assumption 1), points
2–7 must be supported either directly or from the literature, and data models must
incorporate points 3, 7 and 8.
12.3 Model Aim and Overview
Isoscapes describe average spatial patterns in isotope values; that is, an isoscape
provides an expected isotope value, given a geographic coordinate. This is useful
for studying and understanding processes that lead to changes in isotope values
over space. The distinguishing problem in forensics is to determine the extent to
which isotope patterns can differentiate among locations when the isoscape model
is inverted. That is, we want probabilistic statements about the geographic coordi-
nate, given an isotope value. This implies that the goal in most forensic applications
of isoscapes is to determine geographic origins for individual organisms or sam-
ples; rarely are we interested in making inferences about population means.
Therefore it is useful to understand the extent to which distributions of isotope
values from different locations are the same, which is different from understanding
the extent to which mean values differ among locations. For example, if there are
statistically significant differences between the means from different locations, but
the observed ranges of isotope values of feathers from those different locations
overlap substantially, it is difficult to reliably assign individual birds to one of these
discrete locations (Wunder and Norris 2008a). Often, these statistically significant
but pragmatically insignificant differences among mean values for locations result
from large sample sizes (in cases where means are proximal in isotope space).
Because GIS-based isoscape models define only one isotope value for each
geographic location, we need more than the isoscape to determine the probability
that a feather was grown at a given location. The typical work-around solutions
have been to either (1) arbitrarily describe large geographic ranges a priori and
summarize isoscape values within each range (e.g. Royle and Rubenstein 2004), or
(2) to arbitrarily determine a minimum magnitude that is on the order of the
25512 Using Isoscapes to Model Probability Surfaces for Determining Geographic Origins
measurement (analytical) error for measuring d
2
H. The former is appealing in terms
of simplicity but may be unnecessarily conservative, whereas the latter ignores
heterogeneity among individuals at a given location and is therefore not conserva-
tive enough. The modeling approach described here provides another alternative by
combining a stochastic component based on known, estimated, or hypothesized
residual variance with a calibrated isoscape.
The calibration function can be based on theory (i.e. represent mechanistic
understanding of the system) or it can be empirically based (i.e. use a generalized
linear model to relate feather values to isoscape values). As the simplest example,
consider early exploratory studies that suggested d
2
H of feathers is related to that
of precipitation by constant linear shift of −25‰ (Rubenstein and Hobson 2004).
If we assume this to be true, then we would calibrate a precipitation-based hydrogen
isoscape by subtracting 25‰ from the value of each modeled value in the isoscape
for use as the calibrated isoscape. Calibrations will rarely be this simple; many
applied studies will allow for the construction of far more complex calibrations and
I point out that the calibration can be (and should be) as complex or as simple as
the state of the science allows.
The stochastic component models unknown or un-measurable random processes
for every point in the isoscape; it represents all sources of uncertainty associated
with the state of the science. As with the calibration, the stochastic model structure
can be based on theory (i.e. constructed hierarchically), or it can be empirically
estimated from the residuals of the calibration step. This coupled model forms the
basis for generating geographically-explicit probability densities (probability sur-
faces) for isotope values from any sample of unknown origin.
12.3.1 Model Assembly, Deterministic Calibration
Because isoscapes are average representations of spatial processes, they operate as
the deterministic component of the model framework here. Ideally, isoscapes com-
bine data with results from experiments and theory that together identify relevant
factors governing the isotopic variance among individual samples and over space.
Failing any underlying mechanistic theory, however, the spatial model can be
entirely data-based and fit using ordinary kriging, inverse distance weighting, etc.,
but the universality of the resultant model is compromised by comparison. Once fit
to the data, the isoscape is a deterministic model that depends on the sample at
hand. In many cases the sample material of interest is available from across the full
extent of the geographic range of interest; in such cases, the isoscape can be mod-
eled directly and no specific calibration is needed. The alternative for cases where
saturated sampling of the material of interest is not possible, an existing isoscape
developed for other applications can be calibrated from a smaller sample of known-
origin material relevant to the study at hand. The basic idea for practitioners is that
the deterministic isoscape should reflect as much as is known about baseline pat-
terns (spatial and otherwise) in the system of interest.
256 M.B. Wunder
The example migratory problem relies on a calibrating an isoscape for hydrogen
in precipitation. The general form of this calibration is flexible and can be written
in simplest form as y
ij
= f(x
j
) + e
ij
where y
ij
is the observed isotope value for feather
i that was known to have been synthesized at location j, f(x
j
) is the function that
relates the expected value for the isoscape (x = d
2
H) at location j to d
2
H of the
feather ij, and e
ij
is the distance between the expectation of that function for location
j and the measured d
2
H for feather ij. For example, we might calibrate the precipita-
tion isoscape via a simple linear regression of measured summer-grown feather
values on predicted water values. The calibration model is y
ij
= b
0
+ b
1
x
j
+ e
ij
where
x
j
is the expected d
2
H for precipitation at location j and where y
ij
and e
ij
are as above.
The regression coefficients are estimated from the feather data and the isoscape
model output. Notice that the calibration f(x
j
) need not be so simple; it can be multi-
dimensional and non-linear. It almost always should include structure not only from
the modeled isoscape (the universal geographic influences), but also from covari-
ates that are relevant to the study organism or material (e.g. based on natural his-
tory, experimental results and/or theory). Our migration problem assumed that
young developing birds use resources for synthesizing keratin differently than do
adults (c.f. Meehan et al. 2003). Adding an indicator function for the adult age
group, we have:
(
)
[ ]
(
)
( )
[ ]
(
)
( )
01 01
1
ij a a j c c j ij
adult adult
y xI x Ibb j bb je=+ ++ − +
(12.1)
where I(j) = 1 if the feather y
ij
is from an adult bird and I(j) = 0 if it is from a
chick; b
0
and b
1
are additionally indexed with a and c to differentiate regression
coefficients for adults and chicks respectively. Figure 12.1 shows this model for a
simulated dataset of adult and young birds sampled across a landscape with water
isoscape d
2
H values from −125‰ to −45‰. For the simulated data, b
a0
= −22, b
c0
= −24, b
a1
= 0.85, b
c1
= 0.92, and e
ij
~ N(0,s
2
) where s
2
= 196, values selected to
fall within the range of published values for studies of migratory birds (reviewed
by Wunder 2007).
So far, most studies coupling water d
2
H isoscapes with bird feather data to deter-
mine geographic origins of migratory birds have used simple linear regressions like
this to calibrate precipitation isoscapes (see Wunder and Norris 2008b for a review).
In this literature, there has been much discussion about so-called discrimination
factors (the intercept in these simple linear regressions; e.g. Hobson and Wassenaar
1997; Meehan et al. 2001; Rubenstein and Hobson 2004; Lott and Smith 2006).
However, the more important factor to consider for linear calibrations is the scaling
(the slope parameter estimate). This is especially true for hydrogen models because
few measurements actually occur near the origin; most are 50 or more units away.
For example, a calibration function with b
0
= −20‰ and b
1
= 1.0 gives −120‰ for
isoscape values of −100‰ whereas a calibration with the same −20‰ intercept but
with scaling of 0.9 gives −110‰ for the same isoscape value, a difference of 10‰,
which is five times greater than the typical magnitude of analytical uncertainty.
This simple rule of scaling has been generally underappreciated thus far, perhaps
25712 Using Isoscapes to Model Probability Surfaces for Determining Geographic Origins
because it implies that we still really need to calibrate isoscapes using tissue of
known origin for every case study application. For this reason, I encourage every
effort to calibrate isoscapes using known-origin samples from as much of the target
range as possible, and discourage the use of fixed offsets to adjust isoscapes.
12.3.2 Model Assembly, Stochastic Component
Obviously, we would like e
ij
to be small for all samples at all locations in the
isoscape. Small e would indicate that we have modeled the relevant mechanistic
processes that govern change in isotope values across geographic and covariate
space. Such understanding remains a distant goal for most organic systems, and
almost certainly will never be fully realized for any system. In the meantime, study-
ing the shape and magnitude of the distribution on e for sample populations helps
to quantify exactly how far away from that understanding we are.
Because even the most carefully calibrated isoscapes are (thus far) static mod-
els, they provide only a single value for each geographic location. This geographi-
cally indexed set of values is the best guess for long-term, large-population
−150
−100
−50
−130 −120 −110 −100 −90 −80 −70 −60 −50 −40
−150
−100
−50
Water isoscape value
Feather value
Adult Birds:
y = (0.85)x−22
Young Birds:
y = (0.92)x− 24
Fig. 12.1 Simulated d
2
H data for feathers of known origin from birds of two different age classes.
Lines are regressions of d
2
H in feathers on predicted values for d
2
H at the known origins from a
water-based isoscape (Bowen et al. 2005). Simulated data were generated from the equations
shown along with error structure given by the Normal distribution N(m,s
2
) with m = 0 and s
2
= 196
for both age classes. The linear models shown were used to calibrate the water-based isoscape
from Bowen et al. (2005) for use in determining spatially explicit probability densities for feathers
of unknown origin. See text for details
258 M.B. Wunder
average values, given the model structure used to generate the isoscape surface.
That is, the isoscape is a population-level inference; given a population of observed
isotope values for a single pixel, the isoscape value for that pixel should be proxi-
mal to the mean of the values for that population. Because of this, few individual
isotope data observed at a single geographic location are expected to align exactly
with the isoscape value for that locale. So long as the distribution of feather values
at a location in the isoscape has a single mode, the value of the rescaled isoscape
for any locale will put us generally in the correct region of the isotope space for
feathers from that location. However, it will not tell us how far from that point we
can wander before we have enough reason to believe that the feather was grown
elsewhere.
In problems of forensics, it is far more common to seek inference about the
geographic origin of a single sample than it is to be concerned with the origin asso-
ciated with the mean of a population of samples. Even in cases where we have a
sample population, such as with the wintering bird example, we are most often
interested in understanding the structure of geographic origins within the sample,
as opposed to the geographic location associated with the sample mean, a point I
discuss in more detail later. This is why it is useful to carefully consider the how
the stochastic term relates the isoscape to observed samples.
Often, the stochastic term is called error or noise. Such terminology invokes the
sense that there are problems, abnormalities, or mistakes in our data or model, but
these terms are really just used to describe the difference between observed nature
and models for nature. They represent random processes and are considered as
random variables. Sometimes these processes (variables) have known structure. For
example, the Normal random variable circumscribes the familiar bell-shaped curve.
Given values for the mean and variance for the Normal random process, the prob-
ability density for any value of y is
(
)
2
2
2
()
2
y
e
fy
m
s
sp
−
−
=
(12.2)
where y is the observed value and m is the population mean scaled by s. This
model is often written more succinctly as Y ~ N(m , s
2
). This simple Normal
structure can be used as the stochastic component for the calibration model given
by Eq. 12.1; we just need estimates for m and s
2
. Following from the calibration
model in Eq. 12.1, the mean value for d
2
H of feathers at location x
j
is an estimate
for m:
( )
[ ]
( )
( )
[ ]
(
)
( )
01 01
1
a aj c cj
adult adult
xI x Imbb j bb j=+ ++ −
(12.3)
Substituting into Eq. 12.2, we have:
25912 Using Isoscapes to Model Probability Surfaces for Determining Geographic Origins
(
)
[ ]
(
)
( )
[ ]
(
)
( )
{ }
(
)
2
0101 0 1 0 1
, , , , ,~ 1 ,
j j a a c c a aj c cj
adult adult
Yx N x I x Ibbbb b b j b b j sβ + ++ −
(12.4)
leaving s
2
to be structured by one of many different methods. Once a structure
for s
2
is determined, the model given by Eq. 12.4 describes a weighted distribu-
tion of feather isotope values for each location x
j
in the calibrated isoscape,
thus overcoming the limitations of using a single value for any given location.
Although f(x) can vary in scaling of expected feather values over individual loca-
tions in an isoscape (constant variance is not a requirement), in practice it is often
better to try to adjust f(x) for homogeneity of variance in the residuals so that they
have the same overall form across geographic space. For example, the residuals
from the model for the simulated feather data are homoscedastic across the
sampled range (Fig. 12.1); maximum likelihood unsurprisingly returns s
2
= 196
for the residuals in both adult and young bird feathers, which was the value used
to simulate the data.
It is important to recognize that the distribution of errors need not be normal; the
structure can be informed from the observed distribution of the calibration residu-
als. Error structures can take any shape, ranging from a simple uniform distribution
with support over some narrow pre-determined range, to an unknown complex
hierarchical structure that cannot be written in closed form with support over an
infinite range. Most isoscape applications in the literature thus far have defaulted to
a uniform distribution. For example, Lott and Smith (2006) used U(m − 8, m + 8)
where m is the feather value; this model evenly distributes the nonzero probability
density over all isoscape locations that have values within 8‰ of the observed
feather value and assigns a probability density of 0 to all outside this range.
Limiting the support to a range of 16‰ was reasoned by comparing model output
with the model-generating data. They could just as easily have used 4‰ as an esti-
mate for s in a Gaussian kernel to similarly constrain approximately 95% of the
density to a range of 16‰ centered on the observed feather value. In either case
however, as well as in the example problem discussed here so far, the estimated
error structure is relevant only to the study at hand. More importantly, because this
structure is modeled as a single all-inclusive process we gain little to no insight
about mechanisms that generate differences from the expectation of the isoscape
model. A more informed approach is to partition the variance in ways that reflect
the state of understanding for the study system. I discuss this in more detail in
Section 12.3.4.
12.3.3 Model Application, Bayes Rule Inversion
In our example so far, we have a probability distribution function for feather values,
given any location. But ultimately we are interested in probability distribution func-
tions for locations, not feathers. Let J be a random variable defining the probability
260 M.B. Wunder
distribution for all locations j, given a feather value. It usually makes sense to
constrain the potential region of origin to some a priori suspected range; for our
case, we would clearly want to restrict the range of possible geographic locations
to within the known breeding range for the species in question. To accomplish this,
let the prior distribution on J be defined by an indicator function identifying the
breeding range: f
J
(j) = I
[range]
(j). This prior on location assigns a multiplier of zero
to all locations outside the known summer range and a multiplier of one to all loca-
tions within the range. Note that again the general form of this prior is flexible and
it can be written to reflect any and all non-isoscape information about the probabil-
ity of any location as an origin. For example, Royle and Rubenstein (2004) advo-
cate using relative abundance of various locations when available; this would
extend the indicator function above such that the prior probability is not evenly
distributed across the breeding range.
Bayes’ rule defines conditional probability inversions. Following from above,
(
)
(
)
(
)
(
)
(
)
== =
== ==
== =
∫
,
,
Y ij j J
X
J ij j
YX
Y ij J
X
f Y yX x f J j
f J jY y X x
f Y yX x f J d
z
zz
(12.5)
where f
Y|X
is the function associated with the calibration for feathers of known ori-
gin (Eq. 12.4, the conditional distribution on Y
j
from the previous section) for an
isoscape over x locations. This equation describes the posterior probability density
function for location j as the true origin given the measured d
2
H value for a feather
and the calibrated isoscape. The denominator integrates out to a constant, so that
the probability density is proportional to the numerator. Using this relation and
substituting detail from Eqs. 12.2 and 12.4, we have:
( )
( )
[ ]
(
)
( )
[ ]
(
)
(
)
{ }
( )
2
− + ++ −
−
=
∝
2
01 01
2
1
()
2
[]
2
,,, ,
1
2
a aj c cj
adult adult
j
y xI x I
j
range
P J jy x
eI
bb j bb j
s
bj s
ps
(12.6)
where j is a geographic location, y is d
2
H for a feather of unknown origin (from the
individual bird of interest),
β
is the vector of regression coefficients from the cali-
bration function, f is the age class for the bird from which the feather came, x
j
is
the expected value for d
2
H at location j from the original (precipitation) isoscape,
and s
2
is variance observed in the residuals from the calibration.
For large geographic areas or for isoscapes with fine-scale resolution, the resul-
tant probability density surface can be comprised of very small values for the map
locations. Because of this and for ease of interpretation, it is often useful to project
these values onto the unit scale on a per-location basis using either a logistic trans-
formation or by rescaling all values relative to the largest observed density value.
26112 Using Isoscapes to Model Probability Surfaces for Determining Geographic Origins
Figure 12.2 shows the geographically explicit probability surface for a feather with
d
2
H = −75‰ from the model just described fitted with the simulated adult bird data
and the isoscape for water d
2
H from Bowen et al. (2005). The posterior density
surface in Fig. 12.2 has been rescaled on a per-location basis relative to the largest
observed density value among all locations.
12.3.4 Partitioning the Variance
Although it is convenient to estimate the variance structure of the model directly
from the calibration residuals as just described in Sections 12.3.2 and 12.3.3, it can
be more informative to structure the model from first principles. For example, we
might consider a two-tiered hierarchy that models two independent sources of error,
one from lab measurement (analytical) errors and another from individual hetero-
geneity in physiology, diet, and behavior (within population variance). The assump-
tion of independence in variances implies that measurement error does not change
proportional to the magnitude of within population variance. This independence
assumption also implies that the variances are additive as long as they are similarly
Fig. 12.2 Probability density surface for a hypothetical adult bird feather of unknown origin with
d
2
H = −75‰ modeling bulk variance from the calibration shown in Fig. 12.1. The geography of
possible origins for the feather is restricted to the hypothetical breeding range chosen to represent
a generic migratory bird that breeds in the eastern deciduous forests of North America. For pre-
sentation purposes, the density value for each pixel in the probability surface has been rescaled by
the largest observed density value. Fig. 12.2, see Appendix 1, Color Section
262 M.B. Wunder
distributed, giving s
2
= s
2
a
+ s
2
p
, where the subscripts a and p indicate analytical
and within-population variance, respectively.
Because continuous-flow isotope ratio mass spectrometry (CF-IRMS) depends
on linear calibrations using multiple lab standards to estimate d
2
H for samples, one
can use these calibration curves to estimate the value of the standards and quantify
the difference between those estimated values and the accepted values for the stan-
dards. Similarly, an estimate of s
2
can be made from observed isotope values for
known-origin feathers collected at single sites, from which s
2
a
can be subtracted to
find s
2
p
(since the observed data include analytical error).
I will again assume Gaussian kernels for illustrating the isoscape calibration
function, this time incorporating the hierarchical structure from the two indepen-
dent components of variance. Let y
ij
be the observed d
2
H for a feather i at location
j which includes measurement error. Let z
ij
be the true value of feather i at location
j. We can then adjust the model for Y
j
to reflect the new structure using the follow-
ing hierarchy:
( )
[ ]
(
)
( )
[ ]
(
)
( )
{ }
( )
22
0 1 01
,,, ~ 1 ,
j j p a aj c cj p
adult adult
Zx N x I x Ij s b b j b b j sβ + ++ −
(12.7)
( )
22
,~ ,
jja ja
Yz Nzss
(12.8)
In this formulation, Z
j
is the expected distribution of feather values for location j,
including among-individual variance and centered on the value predicted for pre-
cipitation from the original isoscape; this is the distribution we would observe if
there were no additional variance from the process of obtaining lab measurements.
The next level in the hierarchy, Y
j
, incorporates uncertainty from CF-IRMS mea-
surements. The hierarchy can be represented in the probability model for J
(Eq. 12.6) as:
(
)
( )
[ ]
(
)
( )
[ ]
(
)
( )
[ ]
=∝
++
+−
22
01
22
01
,, ,, ,
, , ()
1
j pa
a aj
adult
pa
range
c cj
adult
P J jy x
xI
NN I j
xI
bj s s
bb j
ss
bb j
(12.9)
One can continue in this way to add additional components of variance that are
either known from the literature or estimated from data. In this simplified exam-
ple, only variances associated with feather values are considered. However, it
should be clear the calibration function coefficients (b
a0
, b
a1
, b
c0
, b
c1
) are estimated
with error, the model used to generate the isoscape value x
j
also involves param-
26312 Using Isoscapes to Model Probability Surfaces for Determining Geographic Origins
eters that are estimated with error, and so too might the determination of age class
(f) be done with some estimable degree of error. All of these sources of variance
can be incorporated hierarchically (Clark 2007).
This hierarchical arrangement provides a key advantage over the use of a single
estimate for bulk variance: it provides the flexibility to ask questions about relative
effects on uncertainty in geographic assignments from measurement error and from
among-individual differences. For example, the standard deviation associated with
the population of errors for d
2
H measurements on typical instrumentation (s
a
) is
about 2‰ (Wassenaar 2008). Using this value for s
a
we can compute s
p
for the
migration example as being about 12‰ (using the value of s
2
= 196 from the simu-
lation data). Plugging these values into the hierarchical model above produces the
same probability density as was produced by the non-hierarchical model with s
2
=
196 (e.g. Fig. 12.2). Figure 12.3a shows the same density as in Fig. 12.2 but derived
from this hierarchical model and left on the natural scale (the pixel values sum to
unity). The hierarchically structured model provides the flexibility to let s
2
a
= 0 to
ask the question “How would my geographic inference change if I were to com-
pletely remove all measurement error?” Figure 12.3b shows the probability surface
for this case. Alternately, we can let s
2
p
= 0 and leave s
2
a
= 4 to ask what things
would look like if measurement error were the only source of variance (e.g. for the
unlikely event that our calibrated isoscape captured all of the natural processes
influencing variation in isotope values among individuals across space). The prob-
ability density surface for that case is given in Fig. 12.3c. In this example, it is clear
that efforts to minimize measurement error will result in virtually no improvement
in geographic precision of assignment, and that future research would be better
directed at understanding mechanisms of individual heterogeneity. To the extent
possible, I discourage the use of case-study-specific estimates for bulk variance
terms; most progress will come from first principles-based hierarchical models that
partition the estimable variance in thoughtful ways that help prioritize future
research needs.
12.4 Discrete and Continuous Frameworks
Technically, because isoscapes are often distributed as GIS grids at fixed geographic
resolution, all applications of these isoscapes are discrete; the GIS grids give values
for a discrete set of points. Therefore, the distinction I make here is one of resolution
and applies to whether the interest is in estimating kernels for every possible grid
point (continuous framework) or in estimating kernels from a collection of points
that fall within some geographic boundaries that are imposed a priori (discrete
framework). Discussion thus far has involved continuous frameworks, cases where
interest was in kernels for individual geographic coordinates. Discrete frameworks
are coarser and address the question of whether sampled material derived from juris-
diction A or jurisdiction B. For discrete frameworks, the collection of points within
each jurisdiction is treated as a population of distributions and Monte Carlo methods
264
M.B. Wunder
Fig. 12.3 (a) Probability density surface for an adult bird feather of unknown origin with mea-
sured d
2
H = −75‰ using a hierarchical model for two sources of variance; one from measurement
errors in the lab and the other from among individual differences at a single location. The geog-
raphy of possible origins for the feather is additionally restricted as described in Fig. 12.2. (b)
Probability density surface for the same adult bird feather (d
2
H = −75‰) and hierarchical model,
26512 Using Isoscapes to Model Probability Surfaces for Determining Geographic Origins
are used simulate the posterior distribution from the population of distributions.
Wunder and Norris (2008a) illustrate an extension for just such a case (Norris
et al. 2006).
The limitations of a discrete framework are generally that the boundaries defin-
ing the regions can be somewhat arbitrary and may not nicely divide the isotope
space (e.g. Fig. 12.4). If the isotope space is not nicely divided, the distributions of
values for the defined regions are likely to overlap widely, or result in some wide
geographic regions getting mapped to narrow regions in isotope space, thereby
reducing the efficacy of discerning among potential regions of origin. For example,
regions B and D in Fig. 12.4 are geographically vast, but isotopically narrow. Thus,
these regions are not expected to end up as assigned origins for many samples
despite the broad geographic coverage. Also notice that adjacency in geographic
space does not necessarily translate to adjacency in isotope space: isotopic neigh-
A
B
D
E
C
−140 −120
Feather d2H
−100 −80 −60 −40
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Density
A BC DE
Fig. 12.4 Map showing broadly-circumscribed geographic target ranges for assigning as breed-
ing origins to American redstarts (Setophaga ruticilla) sampled during winter as described in
Norris et al. (2006) and Wunder and Norris (2008a). Inset shows empirical probability density
(smoothed histogram) for each labeled geographic region in hydrogen isotope space, using values
from the isoscape in Bowen et al. (2005). Vertical lines occur at assignment cutoff thresholds for
distinguishing between neighboring regions
Fig. 12.3
(continued) but setting the parameter for analytical error to zero. This illustrates the
gains in geographic precision from improving lab practices for measuring d
2
H to the point of
perfect repeatability among samples. (c) Probability density surface for the same adult bird feather
(d
2
H = −75‰) and hierarchical model, but setting the parameter for within-location variance to zero,
and restoring the original parameter value for analytical error (see text). This illustrates the gains
in geographic precision from improving mechanistic understanding of the variance-generating
processes to the point of perfect prediction, but still allowing for measurement error in the lab. All prob-
ability densities appear on the same scale as defined by the color bar
266 M.B. Wunder
bors are not necessarily geographic neighbors. Another potential limitation of the
discrete approach is that each jurisdiction of potential origin must be fully
characterized, ideally with sample material of known origin. Otherwise any
uncharacterized region is considered with probability zero as the origin.
The advantages of a discrete framework are likewise obvious: in many cases, we
simply do not have enough training data to adequately calibrate the full extent of
the isoscape of interest; a conservative workaround is to use the discrete framework
to define regions at broad resolutions that capture the extent of the limited training
sample. In other cases, there may be applied management or otherwise politically-
motivated situations where the real interest is in fact to determine the weight of
evidence for one region as the origin relative to another. In such cases, it makes
sense to adopt a discrete framework. However, for many, if not most cases, it is
generally more desirable to apply a continuous framework.
12.5 Origin of Populations versus That for Individuals
It is tempting to think that we can improve our models (gain better geographic
precision) by increasing sample sizes. However, because isoscape calibrations are
not always 1:1 mappings, the geographic location of the sample mean is not the
same as the mean of the geographic locations, a point to which I alluded earlier.
Furthermore, in cases where the true sample structure is multi-modal, mapping
the location for the arithmetic mean will describe a geographic region from which
none of the samples actually derived. In other words, for research seeking to find
spatial structure, it is universally more advantageous to first generate the
probability density surfaces for each individual in the sample and then summa-
rize over the set of spatially explicit surfaces. It is generally uninformative to
generate a single spatially explicit probability density surface from population-level
summary statistics.
To illustrate this effect, I simulated a bimodal distribution to represent a sample
population of unknown-origin birds. The simulated population was a mixture of
two normal random variables, one centered on −50‰ and the other centered on
−115‰. Figure 12.5a shows the bimodal density associated with this mixture and
also the density assuming a normal structure, which has a mean of −92‰ for the
mixture. Already, it is clear that by falsely assuming all samples are from the same
random process (e.g. are normally distributed) the mean is nowhere near the sample
values.
Figure 12.5b and c show the geographically indexed posterior probability densi-
ties for the two different approaches. In Fig. 12.5b, the density was generated by
first modeling a probability density for each individual sample simulated from the
mixture, then computing the mean field from these densities. The density in
Fig. 12.5c was generated as a single surface for the mean value of the simulated
samples (−92‰). The model presented in Section 12.3.3 was used in both cases.
26712 Using Isoscapes to Model Probability Surfaces for Determining Geographic Origins
That is, the kernel parameters depended on the isoscape calibration using known-
origin samples. The simulated data in the mixture shown in Fig. 12.5 represent
samples of unknown origin that are assumed to individually follow the random
process models used in the calibration.
The geographic structure in Fig. 12.5b reflects the true bimodal structure of the
mixture. The gradients in Fig. 12.5b faithfully translate variance shape in isotope
space to variance shape in geographic space. By contrast, Fig. 12.5c shows the
probability density associated with the mean of the simulated samples. Notice that
not only does the density for the mean value put most of the probability density in
a geographic location from where no samples actually derived, but it also appears
more peaked, giving a false sense of heightened precision. The gradients in
Fig. 12.5c do not reflect the variance structure in isotope space among the simu-
Fig. 12.5 (a) Simulated feather d
2
H data from a mixture of two normal random variables. Solid
black line shows the probability density for d
2
H based on the true bimodal distribution for the
simulated feather data. Dashed red line indicates the probability density for d
2
H when assuming
a normal distribution with a single mode. (b) Probability density surface constructed from densi-
ties modeled separately for each individual sample. Posterior density shown is average of all
individual densities normalized by the sum of all individual densities, clearly showing bimodal
structure in geographic space corresponding to bimodal structure in isotope space. Black arrows
link modes in isotope space with those in geographic space. (c) Probability density surface using
mean and standard deviation of simulated samples, showing single geographic mode corresponding
to single mode in isotope space, with most of the mass where no actual data exist. Red arrow links
center of mass in isotope space with that in geographic space. Fig. 12.5, see Appendix 1, Color
Section
268 M.B. Wunder
lated samples. Clearly, the probability surface model for the mean is not the same
as the mean of the probability surface models. It is also clear that the more informa-
tive approach is to find the geographic structure using individual level probability
surfaces, rather than assuming the population structure from the start.
12.6 Concluding Remarks
Wonderfully predictable geographic patterns emerge in many isoscapes. In order to
best see these geographic patterns, it is important to focus acutely on the mean value
from the isoscape model. Considering variances around the mean isotope pattern can
easily obscure pattern elucidation and theory formulation and therefore is of less
value for theoretical work. However, for applied problems, patterns from isoscape
models are only as useful as their inverses. The most informative inversions will
include direct consideration of uncertainty in the original model construct and will
provide for an unbiased two-way translation of information. Inverting models with-
out considering variance can lead to very biased results, especially when variances
are large and causes of those variances are poorly understood.
I suggest the following points for consideration in the early stages of designing a
study using isoscapes to determine origins. Whenever possible, build the isoscape
with material that has a known history in space and time; that is, use material of
known geographic and temporal origins. Think about how best to characterize vari-
ance among sample material synthesized at the same place and time. Is there theory
to suggest ways to partition the variance or does the structure of the calibration residu-
als suggest some particular process or mechanism? Partition the variance whenever
possible as this is the shortest way to develop strong feedback between experimental
results and applied observational work; reserve the use of single estimates of bulk
variance to case-specific studies that feature some pressing applied need. It rarely
makes sense to find the probability density for the mean of a population of samples
of unknown origin. Because there is not a 1:1 mapping from isotope space to geo-
graphic space, it is universally more advantageous to find probability densities for
individual samples prior to aggregating over the population of interest.
Isoscapes are powerful constructs for applied studies with geographic forensics
as a goal because of their potential to translate isotopic measurements into geo-
graphic locations. The goal in this chapter was to determine specific geographic
locations from which sampled tissue may have derived, given isotope measure-
ments for the tissue. The isoscape in this case was the translator; it was used to
translate the tissue isotope value for a single feather sample into a geographic value.
The trick from an applied standpoint is in understanding how much is lost in such
a translation and how best to recover the lost information. It is my hope that the
methods described in this chapter provide a simple first step in that direction, and
that the framework presented here can be readily extended to cases where much
more information is at hand.
26912 Using Isoscapes to Model Probability Surfaces for Determining Geographic Origins
Acknowledgements I thank the editors for the opportunity to write this chapter. Comments from
Gabe Bowen, Jason West, and two anonymous reviewers substantially improved earlier versions
of this manuscript. I thank Mirgate and Basin for funding my travel to the Isoscapes meeting in
Santa Barbara, CA where I presented the methods described here. In addition, I am grateful to
Fritz Knopf, Colleen Webb, Cyndi Kester, Craig Stricker, Craig Johnson, Len Wassenaar, Keith
Hobson, Ryan Norris, Pete Marra, Jeff Kelly, and Carlos Martinez del Rio for discussions that
shaped both the direction and presentation of this work.
References
Bearhop S, Fiedler W, Furness RW, Votier SC, Waldron S, Newton J, Bowen GJ, Berthold P,
Farnsworth K (2005) Assortative mating as a mechanism for rapid evolution of a migratory
divide. Science 310:502–504
Benson S, Lennard C, Maynard P, Roux C (2006) Forensic applications of isotope ratio mass
spectrometry – a review. Forensic Sci Int 157:1–22
Bowen GJ, Revenaugh J (2003) Interpolating the isotopic composition of modern meteoric pre-
cipitation. Water Resour Res 39:1–13
Bowen GJ, Wassenaar LI, Hobson KA (2005) Global application of stable hydrogen and oxygen
isotopes to wildlife forensics. Oecologia 143:337–348
Chamberlain CP, Blum JD, Holmes RT, Feng X, Sherry TW, Graves GR (1997) The use of isotope
tracers for identifying populations of migratory birds. Oecologia 109:132–141
Clark JS (2007) Models for ecological data: an introduction. Princeton University Press,
Princeton, NJ
Cryan PM, Bogan MA, Rye RO, Landis GP, Kester CL (2004) Stable hydrogen isotope analysis
of bat hair as evidence for seasonal molt and long-distance migration. J Mammal
85:995–1001
DeLong JP, Meehan TD, Smith RB (2005) Investigating fall movements of hatch-year flammu-
lated owls (Otus flammeolus) in central New Mexico using stable hydrogen isotopes. J Raptor
Res 39:19–25
Ehleringer JR, Casale JF, Lott MJ, Ford VL (2000) Tracing the geographical origin of cocaine.
Nature 408:311–312
Fraser I, Meier-Augenstein W, Kalin RM (2006) The role of stable isotopes in human identifica-
tion: a longitudinal study into the variability of isotopic signals in human hair and nails. Rapid
Commun Mass Spectrom 20:1109–1116
Giuliani G, Chaussidon M, Schubnel HJ, Piat DH, Rollion-Bard C, France-Lanord C, Giard D,
deNarvaez D, Rondeau B (2000) Oxygen isotopes and emerald trade routes since antiquity.
Science 287:631–633
Hobson KA, Wassenaar LI (1997) Linking breeding and wintering grounds of neotropical migrant
songbirds using stable hydrogen isotopic analysis of feathers. Oecologia 109:142–148
Hobson KA, Wassenaar LI, Taylor OR (1999) Stable isotopes (dD and d13C) are geographic
indicators of natal origins of monarch butterflies in eastern North America. Oecologia
120:397–404
Hobson KA, Wilgenburg SV, Wassenaar LI, Hands H, Johnson WP, O’Meilia M, Taylor P (2006)
Using stable hydrogen isotope analysis of feathers to delineate origins of harvested sandhill
cranes in the central flyway of North America. Waterbirds 29:137–147
Hobson KA, Wilgenburg SV, Wassenaar LI, Moore F, Farrington J (2007) Estimating origins of
three species of neotropical migrant songbirds at a gulf coast stopover site: combining stable
isotope and GIS tools. Condor 109:256–267
Lott CA, Smith JP (2006) A geographic-information-system approach to estimating the origin of
migratory raptors in North America using stable hydrogen isotope ratios in feathers. Auk
123:822–835
270 M.B. Wunder
Meehan TD, Lott CA, Sharp ZD, Smith RB, Rosenfield RN, Stewart AC, Murphy RK (2001)
Using hydrogen isotope geochemistry to estimate the natal latitudes of immature Cooper’s
hawks migrating through the Florida Keys. Condor 103:11–20
Meehan TD, Rosenfield RN, Atudorei VN, Bielefeldt J, Rosenfield LJ, Stewart AC, Stout WE,
Bozek MA (2003) Variation in hydrogen stable-isotope ratios between adult and nestling
Cooper’s hawks. Condor 105:567–572
Muller W, Fricke H, Halliday AN, McColluoch MT, Wartho JA (2003) Origin and migration of
the Alpine Iceman. Science 302:862–866
Norris DR, Marra PP, Montgomerie R, Kyser TK, Ratcliffe LM (2004) Reproductive effort, molting
latitude, and feather color in a migratory songbird. Science 306:2249–2250
Norris DR, Marra PP, Bowen GJ, Ratcliffe LM, Royle JA, Kyser TK (2006) Migratory connectiv-
ity of a widely distributed songbird, the American redstart (Setophaga ruticilla). Ornithol
Monogr 61:14–28
Paxton KL, vanRiper IIIC, Theimer TC, Paxton EH (2007) Spatial and temporal migration pat-
terns in Wilson’s warbler (Wilsonia pusilla) in the southwest as revealed by stable isotopes.
Auk 124:162–175
Pyle P (1997) Identification guide to North American birds. Slate Creek Press, Bolinas, CA
Royle JA, Rubenstein DR (2004) The role of species abundance in determining breeding origins
of migratory birds with stable isotopes. Ecol Appl 14:1780–1788
Rubenstein DR, Hobson KA (2004) From birds to butterflies: animal movement patterns and
stable isotopes. Trends Ecol Evol 19:256–263
Sharp ZD, Atudorei V, Panarello HO, Fernandez J, Douthitt C (2003) Hydrogen isotope systemat-
ics of hair: archeological and forensic applications. J Archaeol Sci 30:1709–1716
Wassenaar LI (2008) An introduction to light stable isotopes for use in terrestrial animal migration
studies. In: Hobson KA, Wassenaar LI (eds) Tracking animal migration with stable isotopes.
Academic, San Diego, CA
Wassenaar LI, Hobson KA (1998) Natal origins of migratory monarch butterflies at wintering
colonies in Mexico: new isotopic evidence. Proc Natl Acad Sci U S A 95:15436–15439
Webster MS, Marra PP (2005) Importance of understanding migratory connectivity and seasonal
interactions. In: Greenberg R, Marra PP (eds) Birds of two worlds: the ecology and evolution
of migratory birds. Johns Hopkins University Press, Baltimore, MD
Wunder MB (2007) Geographic structure and dynamics in mountain plover. Ph.D. Dissertation,
Colorado State University
Wunder MB, Norris DR (2008a) Improved estimates of certainty in stable isotope-based methods
for tracking migratory animals. Ecol Appl 18:549–559
Wunder MB, Norris DR (2008b) Analysis and design for isotope-based studies of migratory ani-
mals. In: Hobson KA, Wassenaar LI (eds) Tracking animal migration with stable isotopes.
Academic, San Diego, CA
