Re: Allometric equations for predicting body mass in dinosaurs

[Mike Taylor <mike@indexdata.com>]

2009/8/20 Zach Armstrong <zach.armstrong64@yahoo.com>:
> I have a question about this paper: Allometric equations for predicting body 
> mass of dinosaurs. Journal of Zoology, June 21, 2009 DOI: 
> 10.1111/j.1469-7998.2009.00594.x.
>
> I do not have the paper, but was wondering if an estimate of 18 tonnes for 
> Brachiosaurus is reasonable (estimate reported here: 
> http://www.livescience.com/animals/090621-dinosaur-size.html)? That seems way 
> too light for a 70 ft long animal.

Yes, it's too light -- quite a bit too light (though not as much too
light as you might think).  The Packard et al. paper is about getting
better regression equations for estimating mass from limb-bone
measurements, but that method is always going to give misleading
results for brachiosaurs, however good the equations, because they
have crazy-weird super-gracile humeri (and their femora are
surprisingly narrow anteroposteriorly, though they are transversely
broad).

> The reason I bring this up was because I was trying to come up with a better 
> estimate of the weight of Puertasaurus (here: 
> http://ztwarmstrong.deviantart.com/). In the end, I get an estimate of 58 
> tonnes for Puertasaurus using Argentinosaurus as a model, and using the 
> drastically reduced weight estimates of the above paper. Any thoughts would 
> be appreciated.

(Nothing to say on this, sorry -- I've not really looked at
Puertasaurus.  Execept of course to say that any mass estimate based
on one cervical and one dorsal is going to be, shall we say, subject
to uncertainty.)

> Do these equations affect how those, like Greg Paul, get there weight 
> estimates?

No.  Greg Paul -- like Don Henderson, Alexander McNeill and (going
further back) Colbert and Gregory -- got his estimate by measuring a
model, scaling up the volume and multiplying by density.  There are
potential sources of error in this approach, for sure, but there's no
allometric equation involved, and each estimate is based on its own
specimen.

By the way, reputable published estimates of the mass of A SINGLE
SPECIMEN (the "Brachiosaurus" brancai lectotype HMN SII) have varied
by a factor of 5.75 (yes!) as shown by the following table from an
in-press book chapter:

Table 2. Changing mass estimates for Brachiosaurus brancai.

Author and date        Method                      Volume   Density    Mass
                                                   (l)      (kg/l)     (kg)
-------------------------------------------------------------------------------
Janensch 1938          Not specified               --       --         "40 t"
Colbert 1962           displacement of sand        86953    0.9        78258
Russell et al. 1980    limb-bone allometry         --       --         13618(1)
Anderson et al. 1985   limb-bone allometry         --       --         29000
Paul 1988a             displacement of water       36585    0.861(2)   31500
Alexander 1989(3)      weighing in air and water   46600    1.0        46600
Gunga et al. 1995      computer model              74420    1.0        74420
Christiansen 1997      weighing in air and water   41556    0.9        37400
Henderson 2004         computer model              32398    0.796      25789
Henderson 2006         computer model              --       --         25922
Gunga et al. 2008      computer model              47600    0.8        38000
Taylor in press        graphic double integration  29171    0.8        23337
-------------------------------------------------------------------------------

NOTES

1. Russell et al. give the mass as "14.9t", which has usually been
interpreted as representing metric tonnes, e.g. 14900 kg. However,
they cite "the generally accepted figure of 85 tons" (p. 170), which
can only be a reference to Colbert (1962). Colbert stated a mass of
85.63 U.S. tons as well as the metric version, so we must assume that
Russell et al. were using U.S. tons throughout.

2. Paul used a density of 0.9 kg/l for most of the model, and 0.6 kg/l
for the neck, which was measured separately and found to constitute
13% of the total volume, yielding an aggregate density of (0.9 × 87%)
+ (0.6 × 13%) = 0.861 kg/l.

3. Alexander did not state which Brachiosaurus species his estimate
was for, only that it was based on the BMNH model. This model is
simply stamped "Brachiosaurus".

> BTW, if anyone has a pdf of the paper I would be much obliged to receive a 
> copy...

I'll send it offlist.

> P.S.: How many cervicals and dorsals do titanosaurs have respectively?

It's impossible to say -- these numbers vary wildly throughout other
major sauropod clades and will likely do the same in Titanosauria
(which after all includes more than a third of a sauropod genera).  We
have very, very few complete titanosaurs to model on.  Rapetosaurus
has 16 cervicals and 11 dorsals (Curry Rogers and Forster 2001),
Malawisaurus may have 13 cervicals but even that is an estimate based
on others' estimates (Gomani 2005).

> How would you estimate the length of the cervical series and the dorsal 
> series (and caudal series) respectively?

You basically can't do it based on published measurements.  Hopefully
that will start to change with the long-awaited monographic
description of Rapetosaurus, which is either in review or in press at
JVP.