Geophys. J . R. astr. SOC. (1987)88,81-95
Extremal bounds on geotherms in eroding mountain belts
from metamorphic pressure-temperature conditions
Marcia McNutt and Leigh Royden Department of Earrh, Atmospheric
& Planetary Sciences, Massachusetts Institute of Technology, Cambridge. Ma 02139, USA
Accepted 1986 May 28. Received 1986 May 27; in original form 1985 November 22
Summary. We present a technique based on linear programming for inverting
metamorphic pressure-temperature (PT) information from one or more
stratigraphic horizons in an eroded mountain belt to determine maximum and
minimum bounds on geotherms as a function of time in the orogen. These
extremal bounds characterize the set of all possible geotherms at a particular
time in the history of the mountain belt consistent with the metamorphic
data. Using synthetic PT data generated from several assumed initial thermal
profiles, we demonstrate the sensitivity of the extremal bounds, determined
by this inverse technique, t o the number, distribution, and accuracy of the PT
measurements. We conclude that 20 Myr after orogeny, the set of all possible
geotherms is reasonably well-constrained. Reconstruction of tight bounds on
possible geotherms as early as 10 Myr after the orogeny requires accurate PT
measurements from more than one structural horizon. Finally, we provide an
example of extremal bounds consistent with real metamorphic PT data from
the Lower Schieferhulle in the Tauern Window of the Austrian Alps. Combined with estimates of the mechanical strength of the continental lithosphere supporting the Alps derived from gravity observations, our results
suggest that the temperature at the elastic/ductile transition in continental
lithosphere exceeds 540°C.
Key words: geotherm, metamorphism, orogeny, inverse theory, linear
programming
Introduction
Many minerals or mineral assemblages contained within metamorphic rocks preserve a record
of pressure and temperature conditions encountered during metamorphism. Using a simple
forward model, England & Richardson (1 977) calculated temperature structures that might
be created during deformation in orogenic belts and subsequently modified by thermal
relaxation and surface erosion. They demonstrated that such theoretical thermal histories are
consistent with peak metamorphic conditions recorded in many of the most common metamorphic mineral facies currently exposed on the earth’s surface.
82
M. McNutt and L . Royden
More recently, a better understanding of reaction kinetics in metamorphic systems and
improved calibrations of the equilibrium coefficients governing those reactions have made
possible more precise reconstructions of metamorphic pressure and temperature conditions.
Additionally, several different methods have been developed t o analyse the changing
pressure-temperature conditions present during prograde and retrograde metamorphism (for
example, garnet zonation, fluid inclusions and exchange reactions). Application of these and
other techniques to metamorphic rocks from many orogenic belts has begun to produce a
wealth of quantitative pressure-temperature-time (PTt) data for a variety of metamorphic
rocks in different orogenic settings.
With the rapidly increasing availability of such data, it is now n o longer expedient to use
forward models t o test initial geotherms and erosion histories for consistency with observed
PTt paths for individual rock parcels. Indeed, for any set of PTt data, an infinite number of
geotherms exist that are consistent with the PTt data. Therefore, there is an obvious need for
a proper inverse technique to construct the entire suite of possible time-dependent geotherms that are consistent with the available PTt data. In this paper, we apply linear programming techniques t o a general conductive cooling model described by Royden & Hodges
(1984) and show how this method yields upper and lower bounds on the temperatures
present throughout the entire lithosphere during uplift and erosion, subject t o PTt constraints. The purpose of this study is t o examine the precision with which true geotherms
can be recovered from synthetic PT data, so that the results of similar inversions or real PT
data can be evaluated realistically.
The forward problem
Before we attempt the inverse problem of predicting temperature from a known PT path, we
must be able t o d o the forward problem of calculating the PT path for a stratigraphic
horizon initially at depth zo, with a specified temperature structure, T(z, t ) ,as a function of
depth and time in the mountain belt. We treat the mountain belt as a conductively cooling
slab of infinite horizontal dimensions which is uniformly eroded at the surface at a constant
rate. This 1 -Dforward model grossly oversimplifies erosional and thermal processes in real
orogenic belts, and completely neglects late-stage tectonic movements. However, unless one
can demonstrate that the inversion technique yields accurate and useful constraints on
temperatures generated synthetically with a simple, idealized forward model, it is fruitless
t o consider more realistic and more complex situations.
?
h
k
I
I
I
I=/-
’
h
Figure 1. Simple model assumed for the lithosphere. Plate is infinite in the horizontal dimensions and is
uniformly uplifting and eroding at the rate u . Temperature is maintained at 0°C at z = O and at
temperature T , at depth /. ‘A’ is the layer of heat generation with thickness z - uf.
Bounds on geotherms in eroding mountain belts
83
Following Royden & Hodges (1984), we fix the base of the lithosphere at a depth/
below the surface (Fig. 1). For the purposes of this problem, /is the depth below which
convection, as opposed t o conduction, is the dominant mode of heat transport. The slab is
uplifted at the velocity u and eroded a t the surface z = 0. At z = 0 and z = 8,temperature is
fixed at 0°C and T,, respectively. The equation describing the temperature in this eroding
slab as a function of depth z and time t is (Royden & Hodges, 1984),
T(z. t ) = T,(z, t ) + T ,
1 - exp (- 2Rz/l')
-)+T,
I - exp ( - 2 R )
exp [-Rz/8] sin (nnz//)
-
00
cnexp[-(n2nZ+RZ)~]
n= 1
in which
R = the Peclet number (equivalent t o a dimensionless uplift rate) = rik/2a
7=
the thermal decay constant = at/P
cr = thermal diffusivity
T R = contribution to T from radiogenic heating
The coefficients c, are similar to Fourier coefficients and reflect the initial temperature
conditions T(z,0):
2
Jb'
c, = o m
(T(z, 0 ) - T R ( z ,0 ) - T ,
[ I - e x p _~
(- 2RP)I
exp ( R z / t )sin (nnzl.4)
[l -exp(-2R)]
The simple solution we adopt for T K is that of a uniform layer of heat productivity A .
extending from z = 0 t o z = a - u t , where u is the thickness of the radioactive layer before
erosion. The expression for TK corresponding t o this model is given in Royden & Hodges
( 1 984).
The second term in (1) is the time invariant or steady-state component of the geotherm.
Note that this term yields a geothermal gradient that decays exponentially with increasing
depth and that the exponent is directly proportional t o the uplift rate. The steady-state
geotherm therefore reflects the uplift rate, with steeper gradients and higher near-surface
temperatures for faster uplift rates (see, for example, Albarede 1976). The third term in ( I )
represents the departure of the geotherm from the steady-state, and this term decays with
time. If the third term is small relative t o the steady-state geotherm, then the complete geotherm will still reflect the uplift rate, as described above. If, however, the third term is large
relative t o the steadystate, there will be n o clear relationship between the geotherm and the
uplift rate. This is generally the case with complex initial geotherms, such as those used in
this paper.
From expression ( 1 ) for the temperature, Royden & Hodges (1984) calculate the PT
path, S(z '), for any given stratigraphic horizon a t depth zo at time to :
S ( Z ' ) = T , [ Z ~ , ( Z-O- z ' ) / u t t o ] + T ,
a
1 - exp (- 2 Rz'/f )
1 - exp (- 2 R )
a,, exp [(n2n2/2R- R / 2 ) z ' / t ]sin nnz
+ T,
n= I
in which
z = 20
-
u(t
-
to)
a,, = c, exp [ -(n2n2/2R + R / 2 ) ( z o / /
+ ~RT,)].
yt
(3)
84
M. McNutt and L . Royden
The individual terms in (3) have the same significance as the corresponding terms in (1).
The second term is the contribution t o the PT path from the steady-state part of the geotherm, and similarly scales with uplift rate. The third term is the contribution t o the PT path
from the non-steadystate portion of the geotherm, and decays with time (with decreasing
z’). If the third term is small, the PT path scales with uplift rate, while if it is large there is
no correlation. Because the third term decays with time, the late portions of the uplift path,
where z ’ is small, tend t o reflect the uplift rate directly, while the early portions d o not.
Thus given an arbitrary temperature profile T(z, 0) in the lithosphere at the onset of the
uplift and erosion phase of orogeny, we can calculate from ( 2 ) the coefficients c,, which in
turn yield both the thermal history of the mountain belt according t o (1) and the PT path of
a particular suite of rocks as described by (3). In addition t o the initial geotherm, values for
heat production A o , thickness of the radiogenic layer a , lithospheric thickness ,! thermal
diffusivity 0 , and uplift rate u must be specified.
The inverse problem
The problem which confronts earth scientists is t o infer something about the temperature
structure T(z, t ) in the lithosphere given a few noisy, discrete samples of the PT path from
one or more stratigraphic horizons. This inverse problem is difficult for several reasons:
(1) Given perfect knowledge of the PT path (e.g. the infinite number of u,’s), one can in
theory recover exactly the c, series. However, there will always be more unknowns, c,, than
PT measurements, S(z ’). Therefore, an infinite number of solutions are consistent with the
data. This fact precludes a simple solution for T(z, r ) by least squares procedures.
( 2 ) Not all sets (c,} correspond t o physically possible initial models T(z, 0) because
many produce geotherms with negative or unrealistically high temperatures. This fact
renders Monte Carlo inversion procedures inefficient.
(3) The effect of an individual harmonic coefficient c, decays as exp (-n2?r2T) with time.
Thus estimation of the higher order c,, which are important in the early thermal history of
the orogen, from PT measurements corresponding t o mostly later times is unstable, with
errors growing as -exp(n2). The instability is analogous to the problem encountered in
downward continuing potential fields.
(4) There are many constraints on the problem which are physically realistic but cannot
be expressed as linear constraint equations. An example is the estimation of the maximum
thermal metamorphic grade for a particular PT path.
Given the difficulties described above, this inversion problem is well-suited t o the
technique of linear programming (Gass 1975). Using the linear programming algorithm we
can solve for any unknowns that enter (1) linearly ( A o , T,,,, c l , c 2 . . .) subject t o constraints
imposed by the PT data S(z ‘), subject to any other linear inequalities (e.g. requiring
0 < T(z, t ) < T,,,), and such that a specified quantity (penalty function) is minimized. There
are numerous ways t o formulate the linear programming inversion, depending on the choice
of penalty function. Here we discuss in detail only one particular choice for the inversion,
namely the calculation of extremal bounds on true geotherms at any point in time. Certainly
an understanding of how tightly geotherms are constrained by metamorphic data is a key
question for studies of the thermal history of orogens.
We set up the linear programming problem in the following manner. Using the revised
simplex algorithm, we find the solution vector
x = { A o , T m ,c1,
c 2 , c3
subject t o constraints:
. .. Crnax}
Bounds on geotherms in eroding mountain belts
85
(1) the resulting PT path S fits the N data points S(z/. i = 1 ,N)t o within a prescribed
error 2 6s;
(2) temperatures along the entire PT path S(z ’) are always less than or equal t o S(zk,,)
where z A a x corresponds t o the depth at which the maximum thermal grade of metamorphism is observed;
(3) T(z, 0)is greater than or equal t o zero for all 0 < z </;
(4) T(z, 0) is less than or equal t o T,, the constant temperature below the lithosphere for
0 < z < /. (Constraints (3) and (4) are necessary and sufficient to ensure 0 < T(z) < T, for all
t.1
(5) A,, (radiogenic heat production) and T, must lie within prescribed bounds:
Aornin < A o < A o r n a x
Tmmin
< T, < T,,,,.
For a particular time t b , we ask for a solution vector x compatible with the above
constraints such that the temperature is either minimized or maximized at a specified depth
z b . Successive inversions trace out the entire envelope of minimum and maximum bounds on
T(z,t b ) . The extremal bounds themselves are generally not possible geotherms, nor are all
geotherms within the bounds feasible solutions. However, any geotherm which is compatible
with the above constraints must lie within the bounds. Furthermore, the bounds are optimal
in that there is indeed a thermal model that achieves the extremal temperature a t t = tb.
In performing this inversion, we have made no assumptions about smoothness of the PT
path or of the initial geotherm, except in so far as the heat-conduction equation implicitly
guarantees an increasing smoothness in both the geotherm and the PT path through time.
The only constraint on the initial geothermaat time t = 0 is that O < T < T, at all depths.
Clearly this introduces a large number of geologically unreasonable solutions. Although it is
easy t o impose more stringent constraints on the t = 0 geotherm, it would be difficult t o
obtain a consensus from earth scientists as t o what those constraints should be. Therefore,
we have chosen t o include all solutions that satisfy the criteria O < T < T, at t = 0, so as t o
produce the broadest possible bounds on the geothermal structure as determined with the
inversion method.
The linear programming method of inversion requires that the solution vector x be a
function of a finite number of coefficients, c,, despite the fact that full representation of
the temperature T in ( I ) involves an infinite expansion of c,’s. Fortunately, at any time
t b > 0, the contribution from the higher order terms will be proportional t o
exp(-n2nzatb/kZ). Thus we find in practice that for a given tb, the extremal bound does
not change when terms above some n = n m a xare included in the solution vector. For the
examples used here, we chose n m a x= 12, although for bounds on geotherms at t b = 20 Myr,
fewer terms usually, but not always, would suffice. In all cases, due t o the conductive
smoothing of large temperature gradients, we can be assured of finding the true extremal
bound even with a finite number of c, as long as tb > 0.
The above argument breaks down at t b = 0. We cannot use the linear programming
technique t o produce extremal bounds on the initial thermal profile in the lithosphere
because we cannot deal with an infinite vector x. A little thought, however, shows that this
exercise would be pointless anyway. First of all, our physical model of instantaneous
emplacement of thrust sheets at time zero must break down. Our assumptions of uniform
uplift and purely conductive cooling at best begin to approximate the actual conditions
several million years after the orogeny. Even if the physical model did hold, it is easy to
visualize what the extremal bounds will look like for the infinite-dimensioned x without
M. McNutt and L. Rojden
86
performing any inversion. Clearly models can be found, fitting any PT constraints. that at
any given depth z b have thermal spikes bounded above by T,,,and below by zero at time
Lero except at the one depth where we might have at YT measurements at time zero. Such
spikes will instantly decay away and not influence later parts of the PT path.
The fact that bounds on T(z,0) are not useful does not mean that PT measurements d o
not constraint initial thermal conditions in the orogen. While it may not be possible to
conclude much about temperature at specific depths. it is possible t o constrain certain
integrals of temperature in the lithosphere, such as the total heat content at time zero. We
have investigated, using simulated tyT measurements, the bounds one can place on integrated
temperature in the lithosphere. The exact value for the bound, of course, depends on the
choice for n=n,,,.
However, Comparison of the initial geotherms T(z,0) as tz,,,
is
increased shows a regular pattern which makes i t easy to extrapolate to the expected shape
of T(z.0 ) for II,,,
= such that the integrated temperature is extremal.
Inversion of synthetic data
The linear programming method of bounding geotherms was tested on synthetic PT data
corresponding to three different initial thermal profiles (Fig. 2a-c). In the results presented
below, we investigate the form of the bounds as a function of the error attributed to the PT
MODEL A
DEPTH
.
TEMPERATURE (degrees C)
c
200 T-4$0
, 6!0-7
lq00
DEPTH
~
\i
60
b
1
~
TEMPERATURE (degrees C )
4vo , 6fo , avo
iooo
zoo
,
d
-
TEMPERATURE (degrees C )
200
, wo-,
KO
, avo
I
io,oo
..-
TAUERN YlNDOY
4
4
+
3
1
i
1-.1
*-.
L
I
1-
I
.I
L A -
.
:
Figure 2. lsamples of pressure-temperature paths. (a--c) Dashed lines show the PT path that would be
experienced by a stratigraphic horizon initially at 5 0 k m depth uniformly uplifting at 0.6 km Myr-'
assuming the initial geotherm shown by the solid line. 'The lithosphere is assumed to be 120 km thick with
a basal temperature of 1333°C and thermal diffusivity of 6.4 X 10 ' m2 sec '. The radioactive layer is 50
km thick with heat production of 0.84 U W IT-^. Large dots show discrete samples of PT paths used for
inversions. (d) Actual PT paths for rocks originating near 40 km depth in the Tauern Window of western
Austria (data from Selverstone e t a / . 1984; Selverstone 1985).
Rounds on geothernis in eroding mountain belts
87
measurements, errors in assumptions concerning the lithospheric thickness and thermal
diffusivity, the number and location of PT data, and the shape of the initial profile. Again
we stress that the extrenial bounds produced cannot begin to quantify the uncertainty in
geotherms caused by breakdown of the physical model governing ( I ) and (3). Rather, we
demonstrate the latitude allowed in the solution to an inverse problem due to limitations in
number and accuracy of the data.
SYNTHETIC CASE A
T h e solid line in Fig. ?(a) shows the initial geotherm for Model A , which corresponds t o a
60 km thick thrust sheet with a thermal gradient of 10°C km-' instantaneously emplaced on
another 60 km thick plate with a 15°C km-' geotherm. The model might, for example,
correspond to underthrusting of the Himalaya by the Indian Shield (see general discussion in
Oxburgh & Turcotte 1974). This example provides a good test of the inversion procedure
because the anomalously low temperatures at mid-lithospheric depths persist for many
million of years after emplacement of the thrust sheets.
The sinuous curve with long dashes in Fig. 2(a) traces the PT path experienced by a rock
parcel initially a t 50 km depth. The PT path was calculated from the starting geotherm in
Fig. ?(a) using (3) with the following values for variables:
u = 0.6 km Myr-l
k'= 120 km
a = 6.4 x
A.
= 0.84 pW
mz sec-'
m-3
a=50km
T , = 1333°C
The horizon cools early in its uplift trajectory as heat is lost t o the underlying thrust sheet.
Between 40 and 20 krn, the stratigraphic level warms slightly due to radioactivity and heat
input from the base of the lithosphere. The rocks experience rapid cooling once again as
they near the surface. The large dots show five discrete samples of the PT path which
provide the data base for inversions. In some of the following examples, we assume that the
temperatures along the PT path are accurate to 6s = ?lO°C. about the width of the dashed
line. This error estimate is overly optimistic, but might represent what we could expect with
improved understanding of reaction kinetics, more statistical samples, and better analytical
techniques. In the more realistic examples, we increase 6s to *50"C, which is about twice
the width of the sample dots in Fig. 2 ( a x ) . The best metamorphic data currently available
are probably more accurate than *50°C, but because we have ignored the uncertainty
( 2 a few kilometres) in the depth estimate, this uncertainty is not overly pessimistic.
Fig. 3(a) shows the actual geotherm for Model A 20 Myr after the orogeny as calculated
from the forward model (solid line) and the extremal bounds (shaded area) calculated by
inversion of the discrete samples of the synthetic PT path assuming uncertainties of *lO"C
in the data. In calculating the extremal bounds on all possible geotherms consistent with the
five discrete PT samples in Fig. 2(a), we assume that the values of u, A o , /, a,and T,,, are
known perfectly. Thus the shaded bounds represent the uncertainty in the geotherm
introduced almost solely by the limited number o f samples of the PT path, because the error
assigned to those observations is minimal. With an infinite number of PT samples, the
shaded bounds would converge onto the true geotherm. With only five samples of the PT
M.McNutt and L . Royden
88
a
TEMPERATURE (degrees Cl
7oo
400
600
8oo
7
7
77
-
looo
-
7
7
MOOEL A
20 million y e a r s
-
ideal
7
oounds
DEP
(k n
b
TEMPERATURE (degrees Cl
T--8?0
10:O
200 _50$,6",0-
d
,
20 m i l l i o n years
TEMPERATURE (degrees
400
600
zoo
c)
000
-7
7
year
I
r
-'?deal"
bounds
,w/errors
i n data
ana unLt.7 Ld I I1 t y
DEF 1
( k n1)
Figure 3. Geotherms and extremal bounds on geotherms from PT data from Model A. (a) 20 Myr geotherm (solid line) and extremal bounds on 20 Myr geotherm (shaded area) assuming errors of +1O"Con
the five PT samples in Fig. 2(a) and perfect knowledge of A , , T, t', u, a , and a. Dashed bounds show
increase in extremal bounds when error in data is increased to +50"C. Dotted lines are bounds for errors
of i5O"C and allowing 0.63 < A , < 1.05 ALWI I - ~ ,1300°C < T , < 1350°C. (b) 20 Myr geotherm for
Model A (solid line) and extremal bounds (shaded region) corresponding to dotted lines from Fig. 3(a).
Dashed lines give extremal bounds when a is overestimated at 8 X lo-' mz sec-'. Dotted lines correspond
to extremal bounds when a is underestimated a t 5 X lo-' m' sec-I. (c) Same as (b) except that dashed
lines are extremal bounds when lithospheric thickness is overestimated at 160 km. (d) Geotherm for
Model A at 10 Myr (solid line) and extremal bounds on geotherms (shaded area) assuming 6s = t10"C and
perfect knowledge of physical parameters. Dashed lines give more realistic bounds for SS = r50"C,
0.63 < A , < 1.05 p W m-3, 1300 < T , G 1350°C. Dotted lines show modification to dashed bounds for
samples of the PT path at 4 4 , 4 0 , 3 0 and 20 km.
path, however, it appears that 20 Myr after the orogeny temperature at any given depth can
be determined to better than 270°C at all depths, with the narrowest constraints near the
surface and near 38 km depth, the location of the sampled horizon at 20 Myr. If we had
sampled the PT path at 38 km depth, the extremal bounds would taper in to *lo" at that
point.
Assuming a larger, more realistic error estimate of +-5O"Cseriously degrades resolution of
the 20 Myr geotherm. The average width of the extremal bounds shown as the dashed lines
in Fig. 3(a) is greater than +-loo", and locally exceeds +-2OO"C. Note, however, that
increasing the uncertainty in the observations by a factor of five by no means increases the
uncertainty in the temperature by the same amount.
The dotted lines in Fig. 3(a) define the bounds on the 20 Myr geotherm for an example
which is one step closer to characterizing our true level of ignorance of the governing
Bounds on geotherms in eroding mountain belts
89
physical parameters. For this run, we again set 6s at 5 0 ° C and, in addition, allowed the
and the
radiogenic heat production t o assume any value between 0.63 and 1.05
mantle temperature to lie anywhere between 1300 and 1350°C. The bounds become even
broader, with most of the increase in uncertainty attributable to the latitude allowed in
radioactive heat production. For the assumed ranges in A , and T,,, , however, it is clear that
uncertainty in these physical parameters is less important than the size of the errors in the
data in contributing to the uncertainty in the geotherm.
In Figs 3(b) and (c) we investigate the effect on the extremal bounds of using the wrong
input for parameters which. unlike A , and T,,,, d o not enter the problem linearly. The
variables to which we must assign exact values are u, a,/, and a . Potentially an incorrect
choice for the uplift rate u would have the greatest influence on the geotherms. However.
with the increased availability of K-Ar dates t o complement the PT values it should be
possible to assign a value t o use with some degree of confidence. (Note, however, that
cooling rates cannot be translated directly into uplift rates without a thermal model or wellconstrained PT path.) The thickness of the radiogenic layer, a, must also be specified, but
the size of a t o a large extent trades off with A , , so that it is possible t o enlarge the bounds
on A , sufficiently t o absorb uncertainty in a. Similarly, the assumed form of the radioactivity is probably not critical. Tests using finite difference solutions to the forward
problem show that the geotherms and PT paths for slabs with exponentially decreasing
radiogenic heat production are almost indistinguishable from the uniform layer case with a
proper choice of A. and u (Hubbard 1985).
The bounds in Fig. 3(b) give some indication of the effect of incorrectly assigning a,
assuming again that 6s = +5O”C, 0.63 G A , G 1.05 pW m-3, 1300 G T,,, G 1350°C. Overestimating Q at 8 x
m2 sec-’ (dashed line) very slightly underestimates the true bounds
for the correct ar (shaded area), while underestimating Q at 5 x
m2 sec-’ (dotted line)
gives overly pessimistic bounds. It is doubtful that errors in Q would ever result in the true
geotherm lying outside the extremal bounds.
The dashed lines in Fig. 3(c) show the extremal bounds when we overestimate F a t 160
km. Compared t o the extremal bounds with the correct choice of /‘(shaded area), the upper
bound is quite similar but the lower bound allows significantly cooler temperatures in the
lithosphere below 70 km depth. The initial thermal profiles for models that actually achieve
those lower bounds below 70 km depth at 20 Myr as shown in Fig. 3(c) look nothing like
the starting model in Fig. 2(a) that generated the PT data. Fig. 4 plots the geotherrns at 0
and 20 Myr for one such model corresponding t o the minimum temperature permissible at
20 Myr at 100 km depth. Although the t = 0 geotherni represented here is improperly
resolved with an n , a x = 12 model, we could prove that the n m a x= m expansion of the time
zero geotherm would be a finite number of thermal spikes. In this case, the spikes are
located at 50, 80 and 160 km. After 20 Myr, the geotherms from the n m a x= = and
n , a x = 12 models are. however, indistinguishable. The small thermal spike at 50 km depth
at time zero is necessary to fit the first PT observation. The upward advection and
conduction of the larger thermal spike at 80 km depth along with the maximum allowed
radioactive heat generation is sufficient to match the later PT measurements without needing
a mantle flux component from a thinner 120 km thick plate. In effect, the low temperatures
below 100 km depth act as a heat sink t o achieve the lower bound on temperature at 100
km depth by 20 Myr, while the thermal spike at 80 kni depth acts as an alternate heat
source. Clearly such a contrived model that attempts to overcome the imposed constraint of
a thicker plate by placing a ‘mock asthenosphere’ a t 80 km depth could be discarded as
being unrealistic, but undoubtedly errors in 8 will affect our bounds on the deep thermal
structure of the lithosphere. This example underscores the futility of attempting to recover
90
M.McNutt and L . Royden
TEMPERATURE
(degrees C )
'
I
'
'
'
'
'
I
I
l ' * l - * o . l ,
1
7
ooy--
Figure 4. Geotherms at 0 Myr (solid line) and 20 Myr (dotted line) for estremal model that actually
achieves lower bound of T(z = 100 km, f = 20 Myr) = 365°C in Fig. 3 ( c ) assuming / = 160 km.
true geotherms at time zero. The solid lines in Figs 2(a) and 4 both represent initial geotherms which produce the same set of PT data t o within *5OoC, yet one model involves
anomalously cold temperatures between 60 and 100 km, while the other displays exactly the
opposite.
We also attempted to invert the PT data while underestimating the lithospheric thickness
at 100 km, but found no solutions consistent with the constraints. As seen in Fig. 3(c), the
observations required temperatures below about 1200°C a t 100 km depth. It is simply not
possible to set 7,,1 between 1300 and 1350°C at that depth.
Fig. 3 ( d ) shows the actual geotherni and the extremal bounds on temperature at t = 10
Myr for the same PT data from Model A. Even for the 'ideal' case with perfect knowledge of
T,. A o . and setting 6.7 = *lO°C, the uncertainty in temperature locally exceeds +-2OO"Cand
averages more than +lOO"C (shaded bounds). Clearly with the larger error estimate and
allowing uncertainty in A,, and T,,, (dashed lines), we can conclude only a little about
temperatures in the lithosphere at or before 10 Myr after the orogeny. Temperatures cannot
exceed 600°C in the upper plate, and can remain extremely cool in the lower lithosphere.
Again. as in the 20 Myr geotherm, the bounds converge on the true temperature at the
location of the reference horizon at this point in time. An obvious tactic for improving
resolution of the geotherm is t o incorporate PT measurements from other structural
horizons to increase the number of taper points at any epoch.
Bounds on geotherms in eroding mountain belts
91
We also investigated the effect on the extremal bounds at 10 Myr of having fewer samples
of the PT path. Eliminating the first, deepest and last, shallowest PT points has virtually no
effect on the dashed bounds in Fig. 3(d). This result is consistent with our argument that a
PT measurement from the horizon at time zero only constrains the temperature at one
single depth on the initial geotherm and the observation that any PT path passing from the
value at 20 km depth to 0°C at the surface must pass within +50°C of the PT point at 10
km depth regardless of the thermal model. Eliminating the point at 40 km depth has a rather
damaging effect on the 10 Myr geotherms, because this level was reached by the horizon
only 6 Myr later. Of the five PT points sampled, the values at 30 and 40 km are most critical
for bounding the temperatures in the orogen.
Based on the observation that the PT measurements from 50 km and 10 km are of little
value, in Fig. 3(d) we compare the extremal bounds substituting a PT point at 44 km for the
ones at 50 km and 10 km. The extremal bounds (dotted lines) are everywhere identical to or
tighter than the dashed bounds. In particular, the bounds converge to +50°C of the actual
geotherm at 4 4 km depth where we have an actual PT measurement. Note that by replacing
the useless points at 50 and 10 km with a useful point at 44 km, the extremal bounds are
little worse than the shaded bounds at 40 to 50 km even though the assumed error is five
times greater. It is possible to partially compensate for noisy data by simply having
information from the right depths, and vice versa.
A general feature of all the geotherms in Fig. 3 (and in Figs 5 and 6) is that they are well
constrained near the horizon from which the PT data were sampled (44 km in Fig. 3(d) and
38 km in Fig. 3(a-c)), and more poorly constrained at some distance from this horizon. This
TEMPERATURE (degrees
400
600
800
200
TEMPERATURE (degrees C )
200
4O
:
-:06
: :0
1
C)
1000
TEMPERATURE (degrees C)
TEMPERATURE (degrees C)
200
400
600
800
1000
MODEL B
10 m i l l i o n years
i
I
1
I
1
I
Figure 5. Geotherms (solid line) and extremal bounds for models 8 and C in Fig. 2. Shaded bounds
assumed 6S = ?lO°C and perfect knowledge of A , , T,, L, u , a,a n d a . Dashed bounds allow 6 S = t50°C,
0 . 6 3 < A 0 c1.05 ~ . r W m -1300<Tm~135OoC.
~,
M.McNutt and L . Royden
92
TEMPERATURE (degrees C )
200
400
600
800
1000
100
I
I
I
TEMPERATURE (degrees C )
200
400
800
800
1000
I
TEMPERATURE (degrees C )
200
400
600
BOO
1000
20 million years
TEMPERATURE (degrees C )
200
400
600
800
1000
mi
ion
years
DEPTH
60
(km)
80
Figure 6. Extremal bounds on geotherms consistent with the metamorphic data from the Tauern Window
at 10, 20,30 and 40 Myr. We assume u = 1 km Myr-', 2 < A , < 5 fiWm-3, 1300<T,< 1350"C,a = 50
mz sec-'.
km, i = 120 km, and LY = 6.4 X
occurs because the temperature of a given particle is only influenced by the temperature
structure in adjacent regions that lie closer than about Az = lM,where t is in millions of
years and Az is in kilometres. Thus the early part of a PT path provides little information
about the temperature structure in distant parts of the lithosphere, while the late part of a
FT' path provides much more. This explains conceptually both the pinching in of the reconstructed geotherms near the horizon from which the PT data were sampled, and the tighter
constraints on the geotherms at later times (compare Figs 3a, d).
SYNTHETIC CASE B
Fig. 5 shows how the widths of the bounds at 20 and 10 Myr depend upon the form of the
initial geotherm. Model B (Fig. 2b) has the same thermal gradients as in Model A, but the
thickness of the upper thrust sheet is only 30 km. The PT path for Model B, also shown in
Fig. 2(b), is very different and simpler than the path for Model A for the same 50 km
reference horizon moving upward at 0.6 km Myr-'. Comparing the extremal bounds on geotherms at 20 Myr in Fig. 5(a) with those in Fig. 3(a) reveals much tighter constraints on
possible geotherms for the thinner thrust sheet. Even for rather large uncertainties in the
data of 250°C (dashed lines) the temperatures are fairly well determined. At 10 Myr (Fig.
5b) the picture is similar. Temperatures in the thinner thrust sheet are better constrained.
The reason for this behaviour can be understood by considering the differences in the
Bounds on geothenns in eroding mountain belts
93
coefficients c, for models A and B. For Model A , the size of the c,'s for n = 5 t o 15 are of
the order 0.1 t o 1. For Model B, the size of those higher order coefficients is 0.01 t o 0.1.
Therefore, the PT path for Model B is simpler, with less contribution from higher order
harmonic functions. The thermal models consistent with the PT path have correspondingly
fewer degrees of freedom, so can be more tightly bounded for Model B.
SYNTHETIC CASE C
The third synthetic example shown in Fig. 2(c) suggests a very different sort of initial
condition t o the orogenic belt. In contrast t o the two overthrust cases, the geotherm for
Model C might correspond to a situation in which the upper lithosphere has been heated
from below by intrusions and other volcanic activity. The extremal bounds on geotherms
at 10 and 20 Myr for this model are displayed in Figs 5(c) and (d) assuming both the ideal
and more realistic bounds on 6S, A,,, and T,,,. In spite of the uncertainty in the true geotherms, the bounds clearly reflect the extremely high thermal gradients in the upper plate
and more nearly isothermal temperatures below 50 km depth.
Inversion of PT data from the Tauern Window
Finally, we present the bounds on geotherms as a function of time in the Alpine region as
constrained by metamorphic data from the Lower Schieferhulle series in the Tauern Window
of western Austria. In this region, the Hercynian basement rocks of the lower thrust sheet
have been exposed by denudation o f the overlying Austroalpine nappes. The PT path as
shown in Fig. 2(d) is derived from standard geothermometry/barometry, pseudomorph
relations, garnet zoning, and fluid inclusion data (Selverstone et al. 1984; Selverstone 1985).
The PT path from the Tauern Window has a similar shape t o that of Model B in Fig. 2(b).
This is not surprising, because other workers have shown that the metamorphic facies
present in the Eastern Alps are consistent with an initial geotherm similar t o that shown in
Fig. 2(b) (Oxburgh & Turcotte 1974; Oxburgh & England 1980). Note that in both cases the
structural horizon sampled for PT data is positioned in the lower plate near its contact with
an overlying thrust sheet.
In inverting the PT data for bounds on temperature, we set the uplift rate u = 1 km Myr-'
as indicated by K-Ar ages of the metamorphic rocks (Selverstone 1985). Radiogenic heat
production was allowed t o take on any value between 2 and 5 p W m-3, and the asthenospheric temperature was constrained t o lie between 1300 and 1350°C. Values for 8,a , and
Q are set at 120 km, 50 km, and 6.4 x lo-' m2 sec-', respectively. The bounds on geotherms
as a function of time are plotted in Fig. 6(a-d) beginning 10 Myr after the onset of uplift
and erosion and ending at the present. The bounds on geotherms are broad, as might be
expected given the generous errors of +50"C assigned t o the data and large uncertainty in the
radioactive heat production. The bounds on temperature in the lower lithosphere are
particularly wide compared t o those derived from the synthetic samples. The combination of
a faster uplift rate and a shallower initial depth for the horizon sampled in the Tauern
Window means that its PT data contain relatively less information constraining temperatures
in the lower lithosphere compared t o the situation in any of the synthetic examples.
Nevertheless, we can extract some useful information from the bounds in Fig. 6. The shift
in time of the geotherm envelope strongly suggests that at least on average, temperatures
have been heating u p over the past 40 Myr through a combination of rapid advection of hot
rocks t o the surface and radioactive heat generation. In fact, if the uplift rate is reduced
much below I km Myr-', the linear programming inversion yields no models consistent with
94
M. McNutt and L . Royden
the higher temperatures implied by the low-pressure PT data within the allowed constraints
of radioactive heat production.
If we focus only on the relatively well constrained temperatures in the uppermost plate,
the extremal bounds can be used t o help characterize the thermo-mechanical properties of
continental lithosphere in the Alpine region. Karner & Watts (1983) have fit elastic plate
models to a number of gravity profiles across the Alps. The value they obtain for elastic
plate thickness near the Tauern Window is about 40 km. This value of elastic plate thickness
should reflect the weakest (hottest) point in the evolution of the mountain belt since any
subsequent cooling will merely ‘freeze in’ the flexure signal of the weakest plate. The plots
in Fig. 6 suggest that the temperature a t the base of the elastic plate exceeds 540°C due to
the relatively high temperatures required by the PT data at 20 Myr between 20 and 30 km
depth. Of course, the model we use t o invert the metamorphic data for temperature is overly
simplistic, and we have ignored the possibility of local thermal anomalies in the vicinity of
the Tauern Window.
Conclusions
Even under ideal conditions, when all the physical parameters governing the thermal history
of an orogen are known, and when the orogenic belt follows the simplest possible uplift and
cooling history, the fundamental nonuniqueness of this inversion problem, with infinite
degrees of freedom constrained by a finite number of metamorphic FT measurements,
prevents us from recovering any details of the initial geotherm in the lithosphere. By 10 t o
20 Myr after the onset of uplift and conductive cooling, we are able to characterize the set
of permissible geotherms with resolution depending on the errors in the metamorphic data,
uncertainty in physical parameters, number and location of PT measurements, and the form
of the PT path itself (i.e. initial geotherm). The linear programming algorithm is an efficient
technique for probing the thermal history implied by metamorphic measurements because
of the ease with which one can incorporate other physical, geological, or geophysical constraints and produce extremal models which characterize properties common t o all feasible
solutions.
The bounds generated from synthetic PT data such as we have shown here d o not treat
the full range in uncertainty which can also arise from breakdown of the physical model
used t o invert the data. Nevertheless, such examples can serve as a guide t o field programs, in
helping t o answer such questions as
1 . From what depth range are PT observations critical?
2. How many discrete samples are enough before we reach the point of diminishing
return?
3 . How much effort should be put into heat flow studies t o characterize the productivity
and thickness of the radioactive layer?
4. At what point is it unimportant t o further reduce the errors in the data?
Problems we have not attempted t o deal with in this report include the effects of finite
emplacement time of the thrust sheets, nonuniform uplift rates, variations in radioactive
heat generation and thermal conductivity with depth, lateral temperature gradients,
uncertainty in the proper boundary condition a t the base of the lithosphere, etc. Clearly,
forward models are of great importance in assessing the contributions of these various
factors (see, e.g. England & Thompson, 1984). One topic for future research will be t o use
such forward models t o generate synthetic geotherms and PT paths under more realistic
conditions using finite difference and finite element solutions to the heat conduction
equation. The resulting data can then be inverted using average values of the nonuniform
Bounds o n geotherms in eroding mountain belts
95
properties t o investigate a t what point our simple approximations break down. Theoretically, there is no impediment (beyond computing expense) t o using linear programming to
generate extremal thermal models with nonuniform radioactive heat production assuming
known but variable uplift rate and thermal conductivity.
Acknowledgments
We wish to thank Norman Sleep for helpful comments that improved the manuscript. This
work was funded by NSF 85-07816EAR.
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