Tutorial 11: Graphic Double Integration, or, Weighing dinosaurs on the cheap

January 20, 2011

Why we do mass estimates

Mass estimates are a big deal in paleobiology. If you want to know how much an animal needed in terms of food, water, and oxygen, or how fast it could move, or how many offspring it could produce in a season, or something about its heat balance, or its population density, or the size of its brain relative to its body, then at some point you are going to need a mass estimate.

All that is true, but it’s also a bit bogus. The fact is, people like to know how big things are, and paleontologists are not immune to this desire. We have loads of ways to rationalize our basic curiosity about the bigness of extinct critters. And the figuring out part is both very cool and strangely satisfying. So let’s get on with it.

Two roads diverged

There are two basic modes for determining the mass of an extinct animal: allometric, and volumetric. Allometric methods rely on predictable mathematical relationships between body measurements and body mass. You measure a bunch of living critters, plot the results, find your regression line, and use that to estimate the masses of extinct things based on their measurements. Allometric methods have a couple of problems. One is that they are absolutely horrible for extrapolating to animals outside the size range of the modern sample, which ain’t so great for us sauropod workers. The other is that they’re pretty imprecise even within the size range of the modern sample, because real data are messy and there is often substantial scatter around the regression line, which if faithfully carried through the calculations produces large uncertainties in the output. The obvious conclusion is that anyone calculating extinct-animal masses by extrapolating an allometric regression ought to calculate the 95% confidence intervals (e.g. “Argentinosaurus massed 70000 kg, with a 95% confidence interval of 25000-140000 kg), but, oddly, no-one seems to do this.

Volumetric methods rely on creating a physical, digital, or mathematical model of an extinct animal, determining the volume of the model, multiplying by a scale factor to get the volume of the animal in life, and multiplying that by the presumed density of the living animal to get its mass. Volumetric methods have three problems: (1) many extinct vertebrates are known from insufficient material to make a good 3D model of the skeleton; (2) even if you have a complete skeleton, the method is very sensitive to how you articulate the bones–especially the ribcage–and the amount of flesh you decide to pack on, and there are few good guidelines for doing this correctly; and (3) relatively small changes in the scale factor of the model can produce big changes in the output, because mass goes with the cube of the linear measurement. If your scale factor is off by 10%, you mass will be off by 33% (1.1^3=1.33).

On the plus side, volumetric mass estimates are cheap and easy. You don’t need hundreds or thousands of measurements and body masses taken from living animals; you can do the whole thing in your kitchen or on your laptop in the space of an afternoon, or even less. In the old days you’d build a physical model, or buy a toy dinosaur, and use a sandbox or a dunk tank to measure the volume of sand or water that the model displaced, and go from there. Then in the 90s people started building digital 3D models of extinct animals and measuring the volumes of those.

But you don’t need a physical model or a dunk tank or even a laptop to do volumetric modeling. Thanks to a method called graphic double integration or GDI, which is explained in detail in the next section, you can go through the whole process with nothing more than pen and paper, although a computer helps.

Volumetric methods in general, and GDI in particular, have one more huge advantage over allometric methods: they’re more precise and more accurate. In the only published study that compares the accuracy of various methods on extant animals of known mass, Hurlburt (1999) found that GDI estimates were sometimes off by as much as 20%, but that allometric estimates were much worse, with several off by 90-100% and one off by more than 800%. GDI estimates were not only closer to the right answers, they also varied much less than allometric methods. On one hand, this is good news for GDI afficionados, since it is the cheapest and easiest of all the mass estimation methods out there. On the other hand, it should give us pause that on samples of known mass, the best available method can still be off by as much as a fifth even when working with complete bodies, including the flesh. We should account for every source of error that we can, and still treat our results with appropriate skepticism.

Graphic Double Integration

GDI was invented by Jerison (1973) to estimate the volumes of cranial endocasts. Hurlburt (1999) was the first to apply it to whole animals, and since then it has been used by Murray and Vickers-Rich (2004) for mihirungs and other extinct flightless birds, yours truly for small basal saurischians (Wedel 2007), Mike for Brachiosaurus and Giraffatitan (Taylor 2009), and probably many others that I’ve missed.

GDI is conceptually simple, and easy to do. Using orthogonal views of a life restoration of an extinct animal, you divide the body into slices, treat each slice as an ellipse whose dimensions are determined from two perspectives, compute the average cross-sectional area of each body part, multiply that by the length of the body part in question, and add up the results. Here’s a figure from Murray and Vickers-Rich (2004) that should clarify things:

One of the cool things about GDI is that it is not just easy to separate out the relative contributions of each body region (i.e., head, neck, torso, limbs) to the total body volume, it’s usually unavoidable. This not only lets you compare body volume distributions among animals, it also lets you tinker with assigning different densities to different body parts.

An Example: Plateosaurus

Naturally I’m not going to introduce GDI without taking it for a test drive, and given my proclivities, that test drive is naturally going to be on a sauropodomorph. All we need is an accurate reconstruction of the test subject from at least two directions, and preferably three. You could get these images in several ways. You could take photographs of physical models (or toy dinosaurs) from the front, side, and top–that could be a cool science fair project for the dino-obsessed youngster in your life. You could use the white-bones-on-black-silhouette skeletal reconstructions that have become the unofficial industry standard. You could also use orthogonal photographs of mounted skeletons, although you’d have to make sure that they were taken from far enough away to avoid introducing perspective effects.

For this example, I’m going to use the digital skeletal reconstruction of the GPIT1 individual of Plateosaurus published by virtual dino-wrangler and frequent SV-POW! commenter Heinrich Mallison (Mallison et al 2009, fig. 14). I’m using this skeleton for several reasons: it’s almost complete, very little distorted, and I trust that Heinrich has all the bits in the right places. I don’t know if the ribcage articulation is perfect but it looks reasonable, and as we saw last time that is a major consideration. Since Heinrich built the digital skeleton in digital space, he knows precisely how big each piece actually is, so for once we have scale bars we can trust. Finally, this skeleton is well known and has been used in other mass estimate studies, so when I’m done we’ll have some other values to compare with and some grist for discussion. (To avoid accidental bias, I’m not looking at those other estimates until I’ve done mine.)

Of course, this is just a skeleton, and for GDI I need the body outline with the flesh on. So I opened the image in GIMP (still free, still awesome) and drew on some flesh. Here we necessarily enter the realm of speculation and opinion. I stuck pretty close to the skeletal outline, with the only major departures being for the soft tissues ventral to the vertebrae in the neck and for the bulk of the hip muscles. As movie Boromir said, there are other paths we might take, and we’ll get to a couple of alternatives at the end of the post.

This third image is the one I used for actually taking measurements. You need to lop off the arms and legs and tote them up separately from the body axis. I also filled in the body outlines and got rid of the background so I wouldn’t have any distracting visual clutter when I was taking measurements. I took the measurements using the measuring tool in GIMP (compass icon in the toolbar), in orthogonal directions (i.e., straight up/down and left/right), at regular intervals–every 20 pixels in this case.

One thing you’ll have to decide is how many slices to make. Ideally you’d do one slice per pixel, and then your mathematical model would be fairly smooth. There are programs out there that will do this for you; if you have a 3D digital model you can just measure the voxels (= pixels cubed) directly, and even if all you have is 2D images there are programs that will crank the GDI math for you and measure every pixel-width slice (Motani 2001). But if you’re just rolling with GIMP and OpenOffice Calc (or Photoshop and Excel, or calipers and a calculator), you need to have enough slices to capture most of the information in the model without becoming unwieldy to measure and calculate. I usually go with 40-50 slices through the body axis and 9 or 10 per limb.

The area of a circle is pi*r^2, and the area of an ellipse is pi*r*R, where r and R are the radii of the minor and major axes. So enter the widths and heights of the body segments in pixels in two columns (we’ll call them A and B) in your spreadsheet, and create a third column with the function 3.14*A1*B1/4. Divide by four because the pixel counts you measured on the image are diameters and the formula requires radii. If you forget to do that, you are going to get some wacky numbers.

One obvious departure from reality is that the method assumes that all of the body segments of an animal have elliptical cross-sections, when that is often not exactly true. But it’s usually close enough for the coarse level of detail that any mass estimation method is going to provide, and if it’s really eating you, there are ways to deal with it without assuming elliptical cross-sections (Motani 2001).

For each body region, average the resulting areas of the individual slices and multiply the resulting average areas by the lengths of the body regions to get volumes. Remember to measure the lengths at right angles to your diameter measurements, even when the body part in question is curved, as is the tail of Heinrich’s Plateosaurus.

For sauropods you can usually treat the limbs as cylinders and just enter the lateral view diameter twice, unless you are fortunate enough to have fore and aft views. It’s not a perfect solution but it’s probably better than agonizing over the exact cross sectional shape of each limb segment, since that will be highly dependent on how much flesh you (or some other artist) put on the model, and the limbs contribute so little to the final result. For Plateosaurus I made the arm circular, the forearm and hand half as wide as tall, the thigh twice as long as wide, and the leg and foot round. Don’t forget to double the volumes of the limbs since they’re paired!

We’re not done, because so far all our measurements are in pixels (and pixels cubed). But already we know something cool, which is what proportion each part of the body contributes to the total volume. In my model based on Heinrich’s digital skeleton, segmented as shown above, the relative contributions are as follows:

  • Head: 1%
  • Neck: 3%
  • Trunk: 70%
  • Tail: 11%
  • Forelimbs (pair): 3%
  • Hindlimbs (pair): 12%

Already one of the great truths of volumetric mass estimates is revealed: we tend to notice the extremities first, but really it is the dimensions of the trunk that drive everything. You could double the size of any given extremity and the impact on the result would be noticeable, but small. Consequently, modeling the torso accurately is crucial, which is why we get worried about the preservation of ribs and the slop inherent in complex joints.

Scale factor

The 170 cm scale bar in Heinrich’s figure measures 292 pixels, or 0.582 cm per pixel. The volume of each body segment must be multiplied by 0.582 cubed to convert to cubic cm, and then divided by 1000 to convert to liters, which are the lingua franca of volumetric measurement. If you’re a math n00b, your function should look like this: volume in liters = volume in pixels*SF*SF*SF/1000, where SF is the scale factor in units of cm/pixel. Don’t screw up and use pixels/cm, or if you do, remember to divide by the scale factor instead of multiplying. Just keep track of your units and everything will come out right.

If you’re not working from an example as perfect as Heinrich’s digital (and digitally measured) skeleton, you’ll have to find something else to use for a scale bar. Something big and reasonably impervious to error is good. I like the femur, if nothing else is available. Any sort of multi-segment dimension like shoulder height or trunk length is going to be very sensitive to how much gloop someone thought should go between the bones. Total length is especially bad because it depends not only on the intervertebral spacing but also on the number of vertebrae, and even most well-known dinos do not have complete vertebral series.


Finally, multiply the volume in liters by the assumed density to get the mass of each body segment. Lots of people just go with the density of water, 1.0 kg/L, which is the same as saying a specific gravity (SG) of 1. Depending on what kind of animal you’re talking about, that may be a little bit off or it may be fairly calamitous. Colbert (1962) found SGs of 0.81 and 0.89 for an extant lizard and croc, which means an SG of 1.0 is off by between 11% and 19%. Nineteen percent–almost a fifth! For birds, it’s even worse; Hazlehurst and Rayner (1992) found an SG of 0.73.

Now, scroll back up to the diagram of the giant moa, which had a mass of 257.5 kg “assuming a specific gravity of 1”. If the moa was as light as an extant bird–and its skeleton is highly pneumatic–then it might have had a mass of only 188 kg (257.5*0.73). Or perhaps its density was higher, like that of a lizard or a croc. Without a living moa to play with, we may never know. Two points here: first, the common assumption of whole-body densities of 1.0 is demonstrably incorrect* for many animals, and second, since it’s hard to be certain about the densities of extinct animals, maybe the best thing is to try the calculation with several densities and see what results we get. (My thoughts on the plausible densities of sauropods are here.)

* Does anyone know of actual published data indicating a density of 1.0 for a terrestrial vertebrate? Or is the oft-quoted “bodies have the same density as water” basically bunk? (Note: I’m not disputing that flesh has a density close to that of water, but bones are denser and lungs and air spaces are lighter, and I want to know the mean density of the whole organism.)

Back to Plateosaurus. Using the measurements and calculations presented above, the total volume of the restored animal is 636 liters. Here are the whole body masses (in kg) we get using several different densities:

  • SG=1.0 (water), 636 kg
  • SG=0.89 (reptile high), 566 kg
  • SG=0.81 (reptile low), 515 kg
  • SG=0.73 (bird), 464 kg

I got numbers. Now what?

I’m going to describe three possible things you could do with the results once you have them. In my opinion, two of them are the wrong the thing to do and one is the right thing to do.

DON’T mistake the result of your calculation for The Right Answer. You haven’t stumbled on any universal truth. Assuming you measured enough slices and didn’t screw up the math, you know the volume of a mathematical model of an organism. If you crank all the way through the method you will always get a result, but that result is only an estimate of the volume of the real animal the model was based on. There are numerous sources of error that could plague your results, including: incomplete skeletal material, poorly articulated bones, wrong scale factor, wrong density, wrong amount of soft tissue on the skeleton. I saved density and gloop for last because you can’t do much about them; here the strength of your estimate relies on educated guesses that could themselves be wrong. In short, you don’t even know how wrong your estimate might be.

Pretty dismal, eh?

DON’T assume that the results are meaningless because you don’t know the actual fatness or the density of the animal, or because your results don’t match what you expected or what someone else got. I see this a LOT in people that have just run their first phylogenetic analysis. “Why, I could get any result I wanted just by tinkering with the input!” Well, duh! Like I said, the method will always give you an answer, and it won’t tell you whether the answer is right or not. The greatest advantage of explicit methods like cladistics and GDI is that you know what the input is, and so does everyone else if you are honest about reporting it. So if someone disagrees with your character coding or with how much the belly sags on your model sauropod, you can have a constructive discussion and hopefully science as a whole gets closer to the right answer (even if we have no way of knowing if or when we arrive, and even if your pet hypothesis gets trampled along the way).

DO be appropriately skeptical of your own results without either accepting them as gospel or throwing them out as worthless. The fact that the answer changes as you vary the parameters is a feature, not a bug. Investigate a range of possibilities, report all of those results, and feel free to argue why you think some of the results are better than others. Give people enough information to replicate your results, and compare your results to those of other workers. Figure out where yours differ and why.

Try to think of more interesting things you could do with your results. Don Henderson went from digitally slicing critters (Henderson 1999) to investigating floating sauropods (Henderson 2004) to literally putting sauropods through their paces (Henderson 2006)–not to mention working on pterosaur flight and swimming giraffes and other cool stuff. I’m not saying you should run out and do those exact things, but rather that you’re more likely to come up with something interesting if you think about what you could do with your GDI results instead of treating them as an end in themselves.

How massive was GPIT1, really?

Beats me. I’m not the only one who has done a mass estimate based on that skeleton. Gunga et al. (2007) did not one but two volumetric mass estimates based on GPIT1, and Mallison (2010) did a whole series, and they published their models so we can see how they got there. (In fact, many of you have probably been reading this post in slack-jawed horror, wondering why I was ignoring those papers and redoing the mass estimate the hard way. Now you know!) I’m going to discuss the results of Gunga et al. (2007) first, and come back to Mallison (2010) at the end.

Here’s the “slender” model of Gunga et al. 2007 (their fig. 3):

and here’s their “robust” model (Gunga et al. 2007:fig. 4):

(These look a bit…inelegant, let’s say…because they are based on the way the physical skeleton is currently mounted; Heinrich’s model looks much nicer because of his virtual remount.)

For both mass estimates they used a density of 0.8, which I think is probably on the low end of the range for prosauropods but not beyond the bounds of possibility. They got a mass of 630 kg for the slender model and 912 kg for the robust one.

Their 630-kg estimate for the slender model is deceptively close to the upper end of my range; deceptive because their 630-kg estimate assumes a density of 0.8 and my 636-kg one assumes a density of 1.0. The volumes are more directly comparable: 636 L for mine, 790 L for their slender one, and 1140 L for their robust one. I think that’s pretty good correspondence, and the differences are easily explained. My version is even more skinnier than their slender version; I made it about as svelte as it could possibly have been. I did that deliberately, because it’s always possible to pack on more soft tissue but at some point the dimensions of the skeleton establish a lower bound for how voluminous a healthy (i.e., non-starving) animal could have been. The slender model of Gunga et al. (2007) looks healthier than mine, whereas their robust version looks, to my eye, downright corpulent. But not unrealistically so; fat animals are less common than skinny ones but they are out there to be found, at least in some times and places. It pays to remember that the mass of a single individual can fluctuate wildly depending on seasonal food availability and exercise level.

For GPIT1, I think something like 500 kg is probably a realistic lower bound and 900 kg is a realistic upper bound, and the actual mass of an average individual Plateosaurus of that size was somewhere in the middle. That’s a big range–900 kg is almost twice 500 kg. It’s hard to narrow down because I really don’t know how fleshy Plateosaurus was or what it’s density might have been, and I feel less comfortable making guesses because I’ve spent much less time working on prosauropods than on sauropods. If someone put a gun to my head, I’d say that in my opinion, a bulk somewhere between that of my model and the slender model of Gunga et al. is most believable, and a density of perhaps 0.85, for a result in the neighborhood of 600 kg. But those are opinions, not hypotheses, certainly not facts.

I’m happy to see that my results are pretty close to those of Mallison (2010), who got 740 L, which is also not far off from the slender model of Gunga et al. (2007). So we’ve had at least three independent attempts at this and gotten comparable results, which hopefully means we’re at least in the right ballpark (and pessimistically means we’re all making mistakes of equal magnitude!). Heinrich’s paper is a goldmine, with loads of interesting stuff on how the skeleton articulates, what poses the animal might have been capable of, and how varying the density of different body segments affects the estimated mass and center of mass. It’s a model study and I’d happily tell you all about it but you should really read it for yourself. Since it’s freely available (yay open access!), there’s no barrier to you doing so.


So: use GDI with caution, but do use it. It’s easy, it’s cool, it’s explicit, it will give you lots to think about and give us lots to talk about. Stay tuned for related posts in the not-too-distant future.


81 Responses to “Tutorial 11: Graphic Double Integration, or, Weighing dinosaurs on the cheap”

  1. Andy Farke Says:

    I agree that this is certainly better than trying to plug measurements into a formula using femoral circumference, but I get really twitchy about any method that hasn’t been rigorously ground-truthed. Hurlburt came close (using 7 extant fleshy specimens of various sauropsids), but that sample was still far too small to know how much we can trust it. So, if anyone is looking for a project, go out and apply this method to 100 specimens of a single taxon, do the stats, and publish the results! And then go and do it with 100 taxa, do the stats, and publish the results!

  2. Andy Farke Says:

    (and I should clarify that by “specimen” I mean ‘real fleshy one’, not another fossil)

  3. ech Says:

    “lungs and air spaces are lighter” – indeed, for humans, if the lungs are as empty as is comfortable, then SG is >1, and if you inhale as much as is comfortable, it is <1. When talking about SG of animals, do people assume you mean in one state or the other?

  4. Jura Says:

    Great post Matt. I would love to see more GDI in mass estimates for extinct critters, as it does appear to be the least prone to error (at least of all the options available). I liked that you pointed out the importance of sticking error bars in one’s calculations. This seems to be all too often forgotten (or glossed over by folks citing papers).

    I think ech is on to something with the human estimate. I’m wondering if the SG = 1.0 statements come from mammal (or even just human) measurements only.

  5. Matt Wedel Says:

    indeed, for humans, if the lungs are as empty as is comfortable, then SG is >1, and if you inhale as much as is comfortable, it is <1.

    Really? I know from experimentation that even if I empty my lungs as much as possible, I can’t make myself sink to the bottom of a swimming pool. I always bob up to the surface, indicating an SG that is still less than 1. But then again, I’m fat. You state this confidently–is it something firmly established that I just missed along the way? If so, please point me to a reference.

    When talking about SG of animals, do people assume you mean in one state or the other?

    Good question. I assume neither, but rather a midpoint, but I don’t know how much attention that has gotten, or indeed whether it’s gotten any attention at all.

  6. ech Says:

    If I exhale as much as I can, I sink and can sit on the bottom of the pool without moving at all, and if I inhale as much as possible I can float without moving at all. Maybe I’m good at exhaling? Here’s a not entirely-flaky yet not entirely-solid source to back me up: http://www.newton.dep.anl.gov/askasci/bio99/bio99503.htm Sorry if I sounded too confident.

  7. Nathan Myers Says:

    I guess one or other of you is atypical. Or both.

  8. Asier Says:


    Thank you very much for the tutorial!

  9. Heinrich Mallison Says:

    Hi Matt,

    Awesome post – it is absolutely great that you take the time to write up such thorough and easy-to-understand tutorials – and, as always, even the „experts“ can learn something from them (I did! I had totally forgotten about Hurlburt 1999). Also, it is very nice to see that at least one person noticed that beautiful Plateosaurus figure in our paper on mechanical digitizing.

    And now we get to the “however” part.

    HOWEVER, I would have done two things differently.

    Minor gripe: If you measure a scale bar, always use the longest one provided (in this case the total length, 468.50 m), because any measuring error will be proportionally smaller the larger the distance you measure.

    Major gripe: why do you produce a Paulian anorexic dinosaur? Is the poor animal near death? Starved to the point where the muscles have atrophied to the point where walking becomes nearly impossible? I am fine with most of the animal: skull, neck, trunk, arms (though the shapes of the hands differ much from your assumed diameter relations, the overall error will be minimal), legs (again, feet are weird, flat thingies) – but the tail!!! EEK!!!!

    Persons & Currie (2010) has just been T-rex-hyped through the media, and I wish they had stuck to dinosaurs in general. However, the dissection work Scott did was great, and they show nicely that the bones in the tail are far from a good guide for determining the outline. The biggy paper (one that I also did not give its proper due for a long time, because I had focused on an entirely different aspect when I first read it) is Allen et al. 2009! Choice quotes:

    – “All fleshy outlines [of tails] were found to be substantially larger than those suggested by the skeletons (typically 140%–200%).”

    – [average tail shape in extant animals compared to bone shape extended] “158% mediolaterally, 133% dorsally, 186% ventrally, 91% dorsal diagonally, and 112% ventral diagonally”

    Wow, as I have been preaching for quite a while, them tails be fat!
    I guess this is where the difference in volume between our two models comes from. Mine has a proper tail, while yours has a wispy little thingy ;)

    On density: I fully agree with you that the 1.0 kg/L value is too high – especially considering that Plateosaurus had an avian-style flow through lung ventilated by air sacs, in all likelihood. I use it because I do not want to run afoul by underestimating an animal’s weight, and then directly overestimating its capabilities when CAE modeling its motions. There are high values published, the highest I know is Bellman et al. 2005 – but that’s a highly unreliable value due to the methods used (measuring the volume of a stuffed animal). Personally, I think something near 0.8 kg/L for most dinosaurs, and maybe a bit lower for those with very large pneumaticity is fine.

    About the Gunga et al. models:

    (These look a bit…inelegant, let’s say…because they are based on the way the physical skeleton is currently mounted; Heinrich’s model looks much nicer because of his virtual remount.)

    No, not really. The mount is just fine, only the hands are a bit screwed. See here:

    OK, a running pose is more elegant, but that’s not the problem with the Gunga models. Their history is telling: The group, none of them a biologist or palaeontologist (Prof. Gunga himself is, but had no part in the process), created a model over the laser scans of the mount, THEN came to get my input. I told them to junk it. They called it the “fat” model.
    They made a new one, which incorporated much of my input, but it still was pretty horrible. Technical difficulties and the lack of sufficient experience in modeling animals meant that much got screwed up. Oh well – I tried to do better:
    the first version, intentionally fatter:

    Click to access Mallison-H-2007_Virtual-Dinosaurs-FIGURES.pdf

    scroll down to figures 60 and 61.

    Overall, 500 kg is way too low for the GPIT Skelett 1 individual! 550-700 kg is more like it, which allows for proper tail muscles. More than that…. Possible, but I’d cut that off at 850 kg, that’s the likely max. Remember that we know a lot about the animal’s lifestyle, about the metabolism and lung physiology, about the way the bones articulate properly….. those are good guides for narrowing the soft tissues down. More, however, to come, I promise, once I get SIMM up and running. Viv Allen and I will build a musculoskeletal SIMM model of the tail base, hips and hind limbs, which will give us good input on the amount of muscle we can pack onto them.

    I have, btw, another open-access paper on Plateosaurus out:
    which deals with range of motion and previous reconstruction.


    Allen, V., Paxton, H., and Hutchinson, J.R. (2009) Variation in Center of Mass Estimates for Extant Sauropsids and its Importance for Reconstructing Inertial Properties of Extinct Archosaurs. The Anatomical Record: Advances in Integrative Anatomy and Evolutionary Biology 292:9, 1442-1461

    Bellmann, A., Suthau, T., Stoinski, S., Friedrich, A., Hellwich, O., and Gunga, H. (2005). 3D modelling of dinosaurs. In Grün, A. and Kahmen, H. (eds.), Optical 3-D Measurement Techniques VII (Proceedings of the 7th Conference) Part 1. Vienna.

    Persons, W. Scott, and Philip J. Currie. (2010) The Tail of Tyrannosaurus: Reassessing the Size and Locomotive Importance of the M. caudofemoralis in Non-Avian Theropods. The Anatomical Record: Advances in Integrative Anatomy and Evolutionary Biology; DOI: 10.1002/ar.21290

  10. Mike Taylor Says:

    ech says: “If I exhale as much as I can, I sink and can sit on the bottom of the pool without moving at all, and if I inhale as much as possible I can float without moving at all.”

    Same here. I assumed everyone could do this. When I was a kid, my friends and I used to have “underwater tea parties”, sitting in a little circle on the bottom of the swimming pool. Admittedly they were very short tea parties, and didn’t involve any tea, but they were definitely on the bottom of the pool.

    I can only conclude that, despite all other evidence to the contrary, Matt is less dense than the rest of us.

  11. Asier Says:

    Heinrich and Matt,

    Matt´s restoration is a too slim? Don´t think so. The results are too high.

    Think on this, (Assuming 0.95/1 density,) a really large Holstein cattle of 150 cm at the shoulder weighs less than 600 kg, and a 470 cm length semi-bipedal-tailed Plateosaurus should be quite smaller.

    And don´t think that in the torso of a small Plaetosaurus with a volume of 445 litres (636*0.7) fit an average sized Holstein cattle.

  12. Heinrich Mallison Says:


    you’re making a fundamental error here: going from a trunk of a mammal to the tail of a dinosaur! That’s a big no-no. Please read the sources I provided for the tail reconstruction (Allen et al. 2009 and Persons and Currie 2010) – that will make you think again!

    Also, you’re prejudiced against the reconstruction because you use the result to judge it – wrong way round! If you adapt a reconstruction until ti fits the weight you “feel” is correct, you’re doing it wrong! The bones are the only evidence we have, you must start with these. Bimechanical properties of muscles, tendons and bones being identical you can then add soft tissues based on bone shape. And the final result is much better than a “oh, I can’t imagine that, so it must have weighed less”.

    Furthermore, your Holstein cattle is anorexic; solid sources give:

    680 kg (1500 pounds – hope I did the conversion right) for a mature individual with a
    height of 147 cm at the shoulder. That’s ~15% more that you claim, if I take “less than 600 kg” to mean
    590 kg.

    Here’s a German source (student’s study from agricultural school) saying 750 kg for “high performance meat cow” of the Holstein breed with a height of 145 cm.

    Click to access poppen.pdf

    plus 21%, and for a slightly less tall animal as well.

    Lastly, let me point out that there is no way Plateosaurus was anything but an obligatory biped. Sources:
    Bonnan, M. F. and Senter, P. 2007. Were the basal sauropodomorph dinosaurs Plateosaurus and Massospondylus habitual quadrupeds? In Barrett, P.M. and Batten, D.J. (eds.), Evolution and Palaeobiology of early sauropodomorph dinosaurs. Special Papers in Palaeontology 77:139-155.
    as well as my papers:

  13. Heinrich Mallison Says:

    oh, and while we are at it, Asier, 150 cm is NOT very large fora a Holstein cattle – as the cited sources show, 145-147 cm is the breed aim right now! 3 to 5 cm taller than that is nothing special.

  14. Asier Says:

    Dear Heinrich,

    First of all thank you very much for the sources.

    I don´t know if you have seen a Holstein cattle in front of you. You have to know that around my city there are a lot of farmhouses and when I was younger I use to visit someones. From my experience, in my country these cows rarely exceeds 500 kg and 140 cm at the shoulders and a 680 kg specimen must be gigantic. Also you have to know that the bulls that runs in San Fermines never use to weight more than 600 kg.


    Ok, sorry I thought that you were telling that the animal torso was slim too.

    About restorations, I´m not talking about “feelings” I´m talking about my experience becasue I´ve restored prehistoric animals (which is very difficult for all you pointed above) including dinosaurs and I have calculated their masses by physical models.

    Gregoy Paul, which is an expert on the materia, have restored a 5 meters long Plateosaurus (may be the same specimen) at 300 kg, this weighs is more reasonable than 500 or 600 kgs.

  15. Matt Wedel Says:

    Gregoy Paul, which is an expert on the materia, have restored a 5 meters long Plateosaurus (may be the same specimen) at 300 kg, this weighs is more reasonable than 500 or 600 kgs.

    Sorry, that’s an assertion, not evidence. Why is Paul’s reconstruction more reasonable? Have you actually read Mallison (2010)? He provided far more information on how he derived the volume and mass of Plateosaurus than Paul has in any publication of which I am aware. So you should have no problem telling us exactly why Paul’s reconstruction is better than Heinrich’s.

    Whether you agree about the masses of cattle or Plateosaurus or whatever, so far you are missing the larger point of the post, which is that if you don’t show your work, no-one has any obligation to take you seriously.

    Heinrich, thanks for the correction on the tails. I know my Plateosaurus is freakishly thin–as I said in the post, I was trying to stake out a lower bound. Your reconstruction and the slender one of Gunga et al. (2007) are much more reasonable. Sorry I didn’t say so more explicitly!

  16. Asier Says:


    About the Gregory Paul, in my opinion is a expert because he has been 25 years restoring dinosaurs and other prehistoric animals and publishing papers on how to do it.

    Ok, I have to read Heinrich papers, then I will see (in my opinion) whose restoration is better. I don´t have any problem on giving the reason to Heinrich.

    About the cattle, is only to make you think. Is nearly impossible that in an elongated animal of only 5 meters in length with a long neck tail included torso´s can fit an adult cattle.

  17. Mathew Wedel Says:

    About the Gregory Paul, in my opinion is a expert because he has been 25 years restoring dinosaurs and other prehistoric animals and publishing papers on how to do it.

    You’re right, that is an opinion. It’s also called the argument from authority, and it’s not evidence of anything. I asked what aspects of Greg Paul’s model were demonstrably superior to Heinrich’s, based on actual evidence, not whether he is (or is not) The Man.

  18. Mathew Wedel Says:

    If I exhale as much as I can, I sink and can sit on the bottom of the pool without moving at all, and if I inhale as much as possible I can float without moving at all. Maybe I’m good at exhaling? Here’s a not entirely-flaky yet not entirely-solid source to back me up: http://www.newton.dep.anl.gov/askasci/bio99/bio99503.htm Sorry if I sounded too confident.

    Not at all–sorry if I sounded too skeptical. Nothing wrong with being confident if you’ve got evidence (indeed, it’s the only time one should be confident). Unfortunately, the most obvious conclusion I can draw from all of this is…not very flattering to me. Let’s get back to Plateosaurus!

  19. Heinrich Mallison Says:


    as Matt has pointed out, just because Greg Paul has been doing reconstructions for 25 years doesn’t mean he hasn’t been screwing them up for a quarter of a century! In fact, in the case of Plateosaurus, he has screwed them up, and if you had read my papers and that by Bonnan and Senter I cited above, you’d know this.

    Next, have you ever weighed a Holstein cow? “I saw loads of animals when I was a kid” is not very solid evidence! In contrast to you, I have actually de-mounted and re-mounted the GPIT 1 mount; I know how big the animal is from personal experience with the skeleton (not just looking at it), a from the digital work I did.

    Running the bulls – maybe it is a good idea not to take the biggest, baddest bulls for that, hu?

    But in order to show you that a 700 kg estimate is rather realistic I have taken a wikipedia drawing showing a Holstein cow and scaled it to fit the Plateosaurus picture (149 cm is shown in the pic, and that’s supposed to be the shoulder height of the cow).

    you can find it here:

    and superimposed the two:

    my sources said that a Holstein of slightly smaller size weighs around 680 kg. A roughly similar weight for Plateosaurus is pretty reasonable, given the similarity in size.

  20. dmaas Says:

    I felt this discussion deserved a more dramatic intro graphic than Heinrich’s cowsaurus pics:

    And… very cool article.
    It’s a bit scary to see how little substance is in many of the popular images about these creatures, and encouraging to see how much new techniques can perhaps change that.

    Its also interesting to see how the perception of this dinosaur shifts when we use a cow as scale instead of the waving guy or running woman. Can I vote to establish that as the new standard?

  21. Asier Says:


    I have to read all the papers you mention.

    Ok, assuming that a 150 cm at the shoulder cow is 700 kgs, it can say looking to your montage (http://img717.imageshack.us/i/cowplateo.jpg/) , that the cow should be considerably heavier than plaetosaurus.

    The image shows that cow has much larger body, taller and longer (although we don´t have the top view) and the body is the heaviest part of both animals by far. The dinosaur has a long tail, cows hasn´t, but last one compensates this with a large skull and neck.

    Based on Matt restoration, I compared a 430 kg cow (+-127 cm at the shoulder) with Plateosaurus torso.

    In the image can be see that many parts of the animal fall outside of the siluette (torso), parts of the body, nearly whole neck, whole skull, most of the legs… but again we don´t have the top view.

    With all these, although it must be a deeper analysis, for me by now the accurateness of GDI method is questionable .

  22. Mike Taylor Says:

    David Maas noted: “Its also interesting to see how the perception of this dinosaur shifts when we use a cow as scale instead of the waving guy or running woman. Can I vote to establish that as the new standard?”

    Well, I introduced what I hoped would become the new bloke-walking-his-dog scalebar standard scalebar in my Brachiosaurus paper. Can I help it if the world has, so far, not followed suit?

  23. Matt Wedel Says:

    With all these, although it must be a deeper analysis, for me by now the accurateness of GDI method is questionable.

    Really? Really? You admit that you don’t have the top views–which are absolutely required for GDI to work (or, if not top, some other orthogonal view)–but you’re ready to write off the whole method based on the rock-solid overlapping-a-cow-with-Plateosaurus technique? Have you considered the possibility that most cattle, especially those raised for beef, are more rotund than Plateosaurus? Check out the torso cross-section here.

  24. FWIW: http://skeletaldrawing.com/Master_Skeletal_File/plateosaurus_high_stance.jpg

    The base of the tail in my reconstruction is too slender, and as indicated in Heinrich’s paper the MTs are a bit too short, but otherwise using totally different techniques we’ve independently restored the same specimen in much the same way (although we have some minor quibbles on the exact position of the anterior ribs and pectoral girdle).

    My problem with the Gunga et al mass estimates is they don’t look healthy, they look like balloons. And I don’t mean they should be uniformly more svelte, but rather that none of their models follows a plausible set of anatomical models for reconstructing the soft tissue.

    GDI and digital volumetric modelling are both superior to femoral cross-section estimates in terms of accuracy, and they should have higher precision than measuring the volumetric displacement of physical models (since they should reduce measurement error) but ultimately it seems to me that the accuracy of all volume-based methods depend on the set of assumptions the researchers make about soft tissue, and failing to make explicit assumptions (i.e. the sort of digital shrink-wrapping of volumes onto skeletons ala the Gunga paper and some other recent attempts) merely hides the error bars.

    Matt of course did make one explicit assumption, to make the animal as thin as felt he could, and I think conceptually that these sorts of minimum mass estimates might be a pretty useful concept (“assuming a specific gravity of X, the animal must way at least Y”). Constraining a minimum estimate would let you then build up different degree of fat, musclulature, other soft tissue structures, etc., and see how those inferences impact the results.

  25. dmaas Says:

    @Mike; maybe if the bloke’s walking a cow we’ll be able to muster enough consensus to box it through ;-)

    @Asier: I feel you’re stepping into two traps. The first is viewing the forms two-dimensionally – which obviously excludes a full dimension of information.
    The other is the mammal-bias. I’m only starting to appreciate the impact of this myself, but archosaurs have wildly divergent characteristics – a major one being the structure of the tail in locomotion. It is really a huge volume. I speed-painted that image over skeletons scaled correctly to each other.
    Cow: big, masticating head and fly-swatter tail
    Plateo: tiny, swallow-it head and massive tail volume

    I strongly suggest you read the papers before setting your opinion in concrete.

  26. Asier Says:

    Matt, I´m not saying that GDI method is wrong. I said that “for me” and “by know” this method doesn´t show me really high accurateness.

    I´ve found a top view of a 3D cow:

    As you see in the photo the cow is greatly more massive than the dinosaur. May be you can think that may be is too fat, but even if you get narrower the animal by 10 to 15% still broader than the dinosaur.

    3D source:



    Heinrich, as you have seen the plateosaurus skeleton personally, you are the best person to make an idea of how big was the animal, I recommended you to visit a farmhouse and try to find a 700 kg cow and take measures, I can assure you that you will be impressed with the animal size, the volume of such a big cow is enormous.

  27. Mark Robinson Says:

    Comparing the masses of Plateosaurus and Bos is all very well, but which one would have tasted better?

    I’m surprised that Mike’s idea of using a scale silhouette of a man going dogging hasn’t caught on more. I think I’d be more inclined to read a paper that featured this.

  28. Heinrich Mallison Says:


    have you any clue what method I used to arrive at the weight estimate I arrived at? Any clue at all?

    “it can say looking to your montage[…] , that the cow should be considerably heavier than plaetosaurus.”

    First of all, be precise, please. What is “considerably” to mean? 50 kg? 300 kg? The former is possible, the latter is utter nonsense.

    Let’s have a look: I traced those parts in blue where the cow is bigger than Plateosaurus, and those in read where Plateosaurus is bigger than the cow. Obviously, you overestimated the cow, significantly to boot.

    the body of the cow is marginally longer, we do not know if it is taller (I assume so, in my comparison, but Plateosaurus was herbivorous, and may well have had a big soft belly, too), it has a bigger head and thicker but much shorter neck. In sum, the animals are comparable. End of “cow is heavier” claim.

    “Based on Matt restoration, I compared a 430 kg cow (+-127 cm at the shoulder) with Plateosaurus torso.”
    So a starved, dead Plateosaurus has a trunk as big as that of a smaller cow – and then you go ignoring the massive tail completely? That’s – sorry to be so blunt – useless nonsense! You claim that
    “that many parts of the animal fall outside of the siluette (torso), parts of the body, nearly whole neck, whole skull, most of the legs”, but if you would be honest, you’d also list which parts of Plateosaurus fall outside the cow: the entire tail, most of the hind limbs, the whole skull and part of the neck.

    See, this is useless: you need to quantify things properly, not just wave your hands!

    and, btw, note that your claim “whole skull” is wrong: the neck of P. covers nearly half the skull. Be exact or shut up – making vague and inaccurate stuff up is not the proper way to criticize a scientific method.

    So, I’ll now save you the bother of actually reading the things you dismiss as wrong without having any evidence, I’ll tell you what method I used to estimate the weight:

    I used the 3D digital mount (so yes, I have the trunk width!) to create a 3D NURBS body around it. There’s NO WAY you can fit a 300, 400 or even 500 L body around it, even 600 L is a stretch. 800 L is comfy – multiply with 0.8 kg/L and you get a weight of roughly 640 kg. See here:

    now try claiming that DGI is wrong, when it gives the same weight range as 3D volume measurements based on the real, 3D skeleton fully mounted :p

    Oh, and as for cows being broader than herbivorous sauropodomorphs: nonsense again. Check Figure 4F here

    Click to access app20090075.pdf

    or this:

    as you can see, the shoulder girdle is relatively narrow (as in cows), and further back the ribcage gets very wide.

  29. Heinrich Mallison Says:


    your comment became visible only now, so I didn’t know it when I wrote the above.

    that said: trust me, I have seen cows. Large cows. 850 kg record holders. 900 kg bulls. I know how big they are – certainly bigger than Plateosaurus. But I have also seen people tip cows that were certainly not tiny, but, well, I’d say they were typical Holsteins, and should thus have weighed some 600 to 700 kg. I am always surprised how slender cows are except for the forehead and the belly. Everything else is just slender bones with a tiny bit of flesh.

    So I feel quite confident that the mass estimates for Plateosaurus are reasonable. Not those by Paul, who doesn’t show the model he based it on, nor that by Henderson (2006), who used a wrong scale bar, but those based on accurate 3D reconstructions of the skeleton.

    I appreciate that you question methods and results; all people should do that. However, there’s a point where one should look at the evidence, look at how the data was derived, and then just stop.

    @Scott: There’s a good reason why your pic (of AMNH 6810) and mine (of GPIT Skelett 1) are so similar: same species, same size, same thoroughness in the articulation of the elements. You really earned the commendation I gave you ;)

    You’re entirely right that the Gunga recons do not follow biomechanical constraints, and that’s why we should ignore them.

    Henderson, D.M. 2006. Burly gaits: centers of mass, stability and the trackways of sauropod dinosaurs. Journal of Vertebrate Paleontology 26:907-921.

  30. Asier Says:


    Thank you for the comments, I´m not going to continue with plateosaurus vs cow, because I have to read your paper. I also think that GP estimate is too low, in part for the tail.

    I´ve question your results because I have restored some dinosaurs, and from my experience your estimate is high, otherwise I can be wrong, may be once I read your paper I have to modify my models.


  31. @Heinrich: I just like that we used different methodology and replicated the same results. Of course I haven’t (publicly) laid out my technique as explicitly as you have in your paper…and now that I think about it, I really need to do that at some point. Darn it, I’m gonna get back to work on these skeletal drawings and pretend this conversation never happened.

    Excellent post Matt.

  32. Asier Says:


    I´m going to explain you why in my opinion GDI method don´t give us a so accurate results.

    First of all, as in your Plateosaurus example as with the Moa, there are used only 2 views for calculations, and third view must be required, otherwise you loose vital information of the body shape. Even with a third view, some information is loosing.

    The top view only gives us the maximum breadths of the body, so without additional view we don´t know if the chest part is as broad as the top back and vice versa. So only with 2 views the results will be always inflated.

    For example if you apply the GDI method in that way in a Spinosaurus, you will get a result with several extra tons of volume, because the large part above the back can´t difference if it is a slim sail of 1 cm in breadth or a massive hump 1 meter broad.

    With physical models or 3D restorations, all body shape details can be solved, but with black silhouettes some information will be always missing.



    I have read your papers, congratulations for them. I have to say you that I still think that your restoration is too “fat” and I´m going to say why.

    First of all your skeletal restoration is perfect, as all bones have been scanned from the real ones. About the “life” restoration, I´m totally agree with you that dinosaurs had great musculature in the upper hindlimbs and tail, as we saw in the hadrosaur mummy, and many past restorations including Paul´s ones should be modified.

    Otherwise I check in your volumetric restoration and you avoid many details of the dinosaur body shape, is too uniform, no musculature and some bone structure can be appreciated, the arms and legs (pes and ankle) are too inflated. In general the whole body is like someone has blow in it (like a ballon) because as pointed above some details of body are missing, so the appearance of the animal is a bit artificial. As it was a bipedal animal, should have more “athletic” appearance.

    In other hand, I think that front part of the body where the scapulas are situated, is too broad (nearly as broad as the broadest part of the body). First I think that you add too much flesh above the scapulas, and for second as this dinosaur doesn´t have collar bone like in humans way, the scapulas use to tend to joint each other, you can check this in any extant mammal.

  33. Heinrich Mallison Says:


    thank you for providing specific criticism – even if you’re patently wrong about many points. Please re-check this figure:
    Here is an enlarged version:

    As you can see, there is very little space between the coracoids. You are totally wrong (and if you had really read my papers carefully, you would know that) about “this dinosaurs doesn’t have collar bone like in humans” – I specifically cite the paper by Huene that describes how the SMNS13200 specimen was found with medially touching collarbones, and the one by Yates and Vasconcleos (IIRC), who describe the same thing in Massospondylus. I have articulated the shoulder girdle to conform to this evidence, thank you for missing it.

    Additionally, check how tightly the 3D surface is placed over the pectoral girdle. You complain that the width is nearly the same as in the hips – but why is that a valid argument? In mammals, the hips are broad, but in dinosaurs they are usually narrow. Therefore, it is not surprising if the shoulder girdle is as wide as or even wider than the hips. Please look at the top view in the cited figure. Also, please note that the arm of Plateosaurus is roughly the length of my arm – the bone cross sections, however, are about 4 times as big. That indicates 4x the force acting on the arm, requiring, for equal control, 4x the muscles. I only put about three times as much on them, so if anything my model has too thin arms and undermuscled shoulders. Same for the legs and feet: The bone cross sections were my guide for the forces, and thus amount of musculature. As opposed to derived birds and mammals, we do not see a strong tendency in prosauropods to “remote-control” the distal joints through long tendons, and we do not see an emphasis on limb extension to produce gaits with unsupported phases (running). Thus, it is unlikely that they had un-fleshy feet (like birds and most mammals), and more closely resembled alligators, who have fleshy hands and feet.

    You also complain that the animal looks like a balloon – based on what? All you can really complain of is that there is no exact definition of muscle groups. You complain that this looks artificial – but we are talking volume, not minute details of the surface. Also, let’s see if that exists in large mammals……

    hmmmmm, aside from the missing surface texture, my model doesn’t look balloon-y in any way, compared to very large mammalian herbivores.

    You also complain that: “As it was a bipedal animal, should have more “athletic” appearance.” – why is that? What evidence do you base this opinion on? and what exactly is “athletic” supposed to mean? more slender? Why should it be more slender than the bone sizes and cross sections indicate? You seem to forget that Plateosaurus was a plant-eater, they tend to have big bellies, and that is especially important in animals that relied on fermentation, not oral processing, to digest food (read: dinosaurs more than mammals).

  34. Asier Says:


    Thanks for uploading a high resolution image, now I can see it in more detail. First of all I didn´t say that front part should be as less broad than pelvis, is perfectly possible. As we are talking about volume, missing minute details is this quite large animal, finally you are adding several liters of volume.

    I study proboscidea, the elephants don´t look like your restoration, they have different body shape all around the body with curves and concave forms.

    I definitely can tell you that you add too much flesh all around the back, the neural spines are just under the skin so you have too much flesh on the
    top of the back. For your info, the skin of a mature cow asian elephant in its thickest part of the back is 3.2 cm (Shoshaniet al., 1982). Plateosaurus is much more smaller animal so you should add around 1-1.5 cm of flesh at the back.

    All this added flesh at the back increases the whole volume in a significant percent.


    Shoshani J + 76 coauthors. 1982. On the dissection of a
    female Asian elephant Elephas maximus maximus (Linnaeus
    1758). Elephant 2: 3–93.

    Paul, Gregory S. (1997). “Dinosaur models: the good, the bad, and using them to estimate the mass of dinosaurs”. Dinofest International 1997: 129-154.

  35. Heinrich Mallison Says:


    you before complained that GDI is bad because, as you suggested, it gives way to high results. You said that a 300 kg estimate for Plateosaurus was much better. Then, you acknowledged that this was too low, so what do you think is right? 400 kg?

    And NOW you say that I am
    “adding several liters of volume”.
    That’s absurd: on the one hand, you say I am hundreds of liters wrong, now you insist that your claim was correct because I am a few liters off? Make up your mind, please.

    The next issue is that you conclude that prosauropods had hides similar to certain large mammals. That’s a very long stretch, because we know that prosauropods are archosaurs. Please go measure a crocodile….. I have, btw, looked at alligator anatomy, and I agree that I should shave some 10 l off the back of my model. That’s a grand ~1.5% error. Similarly, re-shaping the body contours slightly may detract another 20 kg. Error all in all, induced by not making the model look nice: under 5%. Wow. Shocking!

    Do you realize that what you complain about here is way below the error bar acknowledged to be present in both methods?

  36. Asier Says:


    As I tell you above the whole body of your restoration look me too compact, uniform and inflated, as I explained you the elephant morphology, and with severals, I mean many liters all around the body.

    Yes, Paul´s mass estimate is quite low, I will say that 400 kg is more accurate and anything up 500 kg will be too much.

    At this point we are stuck, you have your opinion with arguments and I have also mines, we are not going to change our ideas.


  37. Heinrich Mallison Says:

    well, Asier, I can tell you that there is very little substance over the ribs, so most of the belly is as thin as it can be. Also, as explained before, if you try for a 500 kg volume, you will have bones sticking out of the volume. And that’s wrong.

    Fruthermore, you constantly make generalized claims, and bring nothing but “I have reconstructed animals” as source – not very convincing.

    You are fully entitled to your opinion, be it right or wrong, but you should
    a) be more polite
    b) be more exact and specific
    c) refrain from making generalized dismissive statements without support
    d) acknowledge when shown wrong (e.g., clavicles).

    Why don’t YOU build a 400 kg (500 l) 3D model in a CAD program (e.g., Blender is free) and we see if the bones fit is?

    oh, and please show me concave body forms on an Alligator. Please?

  38. Heinrich Mallison Says:

    a nice tidbit for those who wonder just how heavy them archosaur tails are:

    in Nile crocodiles, body length of 1300 mm, seven individuals, the tail on average made up 18% of the live weight, and 33% of the empty carcass (i.e., gutted) weight.

    Hoffmann, L. C., Fisher, P.C., and Sales, J. 2000. Carcass and meat characteristics of the Nile crocodile (Crocodylus niloticus). Journal of the Science of Food and Agriculture, 80:390-396

    Depending on where exactly Matt cut the tail off his Plateo, his tail is too thin (I suspect) or the trunk too fat (I think not, because the ribcage delimits the trunk better than the vertebrae delimit the tail).

  39. Mathew Wedel Says:

    Depending on where exactly Matt cut the tail off his Plateo, his tail is too thin (I suspect) or the trunk too fat (I think not, because the ribcage delimits the trunk better than the vertebrae delimit the tail).

    Interesting. I redid my GDI using a torso rescaled according to your model and a fattened tail base, and got numbers more in line with yours. In particular, the fatter tail now makes up 18% of the volume of the animal.

    Thanks, everyone, for the further discussion, which IMHO adds considerably to the post.

  40. dmaas Says:

    The cow and plateo cut out of each other. Heinrich has the numbers, but the two are very comparable in volume.

  41. Heinrich Mallison Says:

    here’s the two again, this time with the top of the back aligned. This shows the differences better.

    I must point out that this cow model is horribly fat; it has a volume of 1000 L, correctly scaled.

    Also, don’t get confused: Plateosaurus didn’t grow, compared to the picture Matt posted above. it is just posed perfectly straight-backed, -tailed and -necked, and the limbs are less flexed.

  42. Heinrich Mallison Says:

    I should have posted the link, too.


  43. Asier Says:

    I´ve compared Heinrich`restoration with Scott`s one (same scale, same femur length):

    They are based on the same specimen, but in body size there is a big difference, this can be due the Heinrich´s restoration has longer ribs and in my opinion it has too inflated torso.

    As pointed above, Scott reconstruction should has more robust tail.

    Anyway, It seems to me that Scott restoration should be around my estimate of less than 500kg.

  44. Heinrich Mallison Says:


    sorry for being very blunt: quite obviously you are not being thorough enough for this. You haven’t even read the comments here properly!

    You write:
    “They are based on the same specimen”
    when in fact it is clearly stated above in my post:
    “your pic (of AMNH 6810) and mine (of GPIT Skelett 1)”. Don’t you think I know what Scott used, when in fact he made this version of the drawing at my personal request, as is stated in my paper? Or that Scott would have corrected me here if I was wrong?

    Also, there is no
    “Heinrich´s restoration has longer ribs”.
    As you stated above (so don’t come claiming you didn’t know):
    “your skeletal restoration is perfect, as all bones have been scanned from the real ones.”

    Scott made a restoration drawing, I didn’t make a restoration with ribs, much as you call it so. I assembled and posed a digital skeleton mount. Nitpicking, maybe, but there is nothing at all “restored” in it.
    Also, the rib lengths are facts, so you can’t argue that I am in error using them (as you seem to do).

    You are of the opinion that my 3D model has too big a belly – fine. Show how it should be. I have suggested before that you make a model of your own, and then we see if you can fit the skeleton in a <500kg model. Why don't you back up your words with deeds and facts?

  45. Asier Says:

    Heinrich, I giving you the reason, don´t you see? I said that your ribs are longer! and I as you pointed I told you that is is “perfect” skeletal reconstruction because you have scanned it.

    Ok sorry, they are different specimens, but they are scaled at the same size (femur length).

    With “restoration” I mean reconstruction, sorry.

    Otherwise, I never said that your model is less than 700 kg, in fact it is. I always said that a 5 meters plaeto… and I always made the comparisons with Matt´s reconstruction, because in my opinion is more accurate due it isn´t so inflated (He posted above that he find out similar body mass as yours, but as I explained above, GDI method can give us erroneous results).

    I´m accustomed to 2D reconstructions as Scott´s ones, I also use that technique.

    Don´t you think that Scott´s reconstruction should be around 500 kg?

    At this point I´ve to admit that GPIT Skelett 1 is more than 500 kg, because its large ribs.

  46. Heinrich Mallison Says:


    sorry, I now see better where you come from. Please accept my apologies.

    OK, my 3D model has a VOLUME of ~740 L. Using a reasonable density of 0.8 kg/L to 0.9 kg/L this gives as a total weight of between 590 kg and 666 kg.

    This models was NOT meant to be a minimum model, but an average. Some body parts may have been slimmer, some more muscular. So maybe the animal was only 500 kg, maybe it was 800 kg. Less than 500 and the bones stick out, more than 800 and I’d call it obese.

    Yes, GDI is not perfect. However, Plateosaurus has a “good” body shape for it, because so many body parts are in fact elliptical. The ribcage is, the tail is, the neck is (unless there were huge air sacs, but they do not add weight), the thighs are, the shanks are, the upper and lower arms are, even the feet and hands!
    The skull deviates some from an ellipsoid, but is tiny.

    So Plateosaurus is well suited for GDI, and GDI gives us, give or take some 10%, the same result as a very different method, as long as it is based on a highly accurate basis. Use figures of my model, use Scott’s very accurate drawing, use my model and do things in 3D – you always get roughly the same range.

    So you dismiss GDI too quickly here: it has problems, but they can be corrected for, and in this case they are likely small. The main factor remains the amount of soft tissues on various body parts.

    I am glad that we finally got this cleared up :)

  47. Asier Says:


    You don´t have to apologize, but me yes.

    Please, apologize me. I´m young and impatient, I should think more the things before acting.

    I´m with you about GDI it vary depending the animal shape. In this moment I´m restoring some prehistoric mammals, and I´m in suspect (because their body shape) that GDI will not be the best method to find out their body mass. Anyway I will make physical models and also apply GDI to contrast.

  48. Heinrich Mallison Says:

    well, I think of myself as young, too, and I am very impatient, too ;)

    one trick about GDI you may want to use is to segment the animal more than Matt did. In bipedal dinosaurs, the “necktrunktail” is more or less one body, with the biggest change a sudden narrowing between trunk and neck. In mammals, it may be well worth sectioning them apart, and cutting the trunk into parts: pectoral, belly, pelvic. Then, define the ellipses accordingly, or even use more complex shapes. I think Motani detailed how to correct for non-ellipses; Matt probably cited that paper above.

  49. Asier Says:

    Ok, thank you for the suggestions.

  50. Cary Woodruff Says:

    Great post! Although I can see where this method is not entirely foolproof, the basis is sound and the ease of use makes this an application that more people should follow. As with women, weight is a touchy subject. However more work (particularly sauropods) is becoming biomechanically inclined, and needless to say weight is critical to biomechanics. GDI quite literally allows for multiple individuals to nearly on the spot calculate body mass in a standard uniform application (that’s free!). I’m sure critics would scoff at the “ball park” variance, but for a group of extinct animals that cannot physically fit our allometric measurements what better alternative is there? I think that this subject would make a great SVP poster. A simple breakdown of the technique, followed by several examples from different Dinosaurian groups. I would hope that more biomechanics embrace this technique and use it, but let’s face it folks still have trouble with scale bars…

  51. Jura Says:

    Returning to lungs and buoyancy, I just came across this: http://www.youtube.com/watch?v=MgRpwESWPLM&feature=player_embedded All the more impressive given that this was in even denser seawater. Apparently one’s mileage may vary.

  52. ech Says:

    Jura Says: “Returning to lungs and buoyancy, I just came across this: http://www.youtube.com/watch?v=MgRpwESWPLM&feature=player_embedded All the more impressive given that this was in even denser seawater. Apparently one’s mileage may vary.”

    He is “cheating”, of course; he keeps moving his hand in a “come here” gesture to keep himself on the floor. Totally amazing feat, obviously, but demonstrating negative buoyancy was not his goal; to do that, one should remain motionless.

  53. Heinrich Mallison Says:

    what’s really ridiculously stupid is how the commentator brings up the aquatic ape nonsense again. :aargh:

  54. I just wanted to point out that my skeletal reconstruction only weighs a gram or so. Less if you mean the digital incarnation…

    In all seriousness, Heinrich and I actually are using the same rib length, it’s their articulation and the position of the pectoral girdle where the differences in torso volume come in. Still, we’d be talking about perhaps 10% of torso volume, so what, 2-3% difference in overall volume? My own weight can change that much in a couple weeks, and it’s wellllllll within the margin of error for GDI.

    One thing that would be interesting to do would be to base a sculpture (digital of physical) on the same specimen as rigorously as possible, and then compare GDI to it (I’d prefer to laser scan said sculpture, but displacement would work too). I’m curious if the “elipticalizing” of an animal consistently results in volumes that vary in predictable ways from a model.

  55. Heinrich Mallison Says:

    Scott, I am not sure that AMNH6810 and GPIT/RE/7288 (GPIT 1) have exactly the same rib length. But yes, most of the diff comes from articulation – I totally forgot about that!

  56. Mathew Wedel Says:

    Still, we’d be talking about perhaps 10% of torso volume, so what, 2-3% difference in overall volume?

    Probably more like 7%; in all of the GDIs I’ve seen and done on sauropodomorps, the torso is pretty consistently 70-73% of the overall volume.

    You are correct that 2-3% is a negligible difference. Although there are no hard and fast rules for this, 7% is over the threshold where I start thinking, “Hold on, this is something I should be paying attention to.”

    One thing that would be interesting to do would be to base a sculpture (digital of physical) on the same specimen as rigorously as possible, and then compare GDI to it (I’d prefer to laser scan said sculpture, but displacement would work too). I’m curious if the “elipticalizing” of an animal consistently results in volumes that vary in predictable ways from a model.

    Yeah, that would be a very doable and useful project. Ideally it would be a digital model with actual volume known to the last voxel, and GDI based on orthogonal renders free of perspective effects. I think Motani has cracked this problem for torpedo-shaped critters, but AFAIK no one has tried it for dinos or other terrestrial animals, most of which have more complex body shapes and cross-sections.

  57. Scott Says:

    >>>Probably more like 7%; in all of the GDIs I’ve seen and done on sauropodomorps, the torso is pretty consistently 70-73% of the overall volume.<<<

    Well, that's what I get for making the numbers up. In all seriousness, while I agree that 7% is worth investigating to tease out the differences in methodology, at the same time it's well within the range of variation for an individual over the course of adulthood. Heck, my own mass has varied by more like 20% the last 4 years, without any appreciable difference in the availability of food…

  58. Matt Wedel Says:

    In all seriousness, while I agree that 7% is worth investigating to tease out the differences in methodology, at the same time it’s well within the range of variation for an individual over the course of adulthood. Heck, my own mass has varied by more like 20% the last 4 years, without any appreciable difference in the availability of food…

    :-) You raise an important point that I didn’t emphasize enough in the post, which is keeping what we know about the model straight from what we know about the animal. Our models are always going to be imperfect approximations of the living creatures, and what’s worse, we’ll probably never have a good handle on just how imperfect they might be. OTOH, in light of that uncertainty I think it behooves us to do the best we can with our models, document them obsessively, and make sure that we at least know what assumptions we’re making with them.

  59. Mike Taylor Says:



  60. Mathew Wedel Says:

    Yes, srsly. I suggest you take some rest and recover your strength, Master Dwarf.

  61. […] find a problem that occupies your mind similarly, at least for part of the day. I wrote the GDI tutorial when I was doing a lot of mass estimation for a couple of upcoming projects, and I found that I was […]

  62. […] progress smoothly from hips to tail, so I wanted to see that in Brontomerus.  At the same time, the tail needed to have more flesh on it and the ischium should have been producing a visible bulge in the ventral margin behind the […]

  63. Fishleaper Says:

    Although I have minimal knowledge in the field of paleontology, this is very interesting and understandably written. Is there a specific program that computes GDI for mass estimates in animals?

  64. […] goat. It’s not hard, really… just put a mammalian tail on a Kentrosaur, or get all touchy-feely about your cow data. Tags: heinrich Mallison / […]

  65. dobermunk Says:

    Blink. Why’d that come through only now?
    very old news.

  66. […] critter, different views. If anyone wants to GDI this baby, you now have everything you need. Many thanks to Bob for permission to post these and the […]

  67. […] masses. Matt: What about ‘em? Mike: Taylor (2009:803) measured the neck of Giraffatitan by GDI as 4117 liters. Matt: k Mike: I didn’t convert that to a mass, but I guess density of 0.5 is […]

  68. […] result has cast a long shadow over discussions of sauropod masses, as in this paper and these posts, so it’s nice to see similar results from an independent analysis.  If you’re […]

  69. […] Knochen vorzustellen. Hier sieht man, wie sie zusammenpassen. Und mittel dieser Fotos kann man  Graphic Double Integration (GDI) machen, eine tolle Methode um das Volumen und mithin das Gewicht eines Tieres abzuschätzen. Man […]

  70. […] from 2D images of individual bones. Here, you can see how they fit. And these photos allow doing Graphic Double Integration (GDI), a very neat way of estimating an animal’s volume and indirectly mass. They also allow […]

  71. […] in 2008, when I did the GDI of Giraffatitan and Brachiosaurus for my 2009 paper on those genera, I came out with estimates of […]

  72. […] using photogrammetrical measurements of the actual skeletal mount. And so should the latter because Hurlburt (1999) showed that GDI is generally the least inaccurate of our mass-estimation techniques. That still gives us a […]

  73. […] pterosaur mass estimates. I was fully on board, since estimating the masses of extinct animals is a minor obsession of mine. But what really caught my attention is that Liz and Colin had full stacks of slices […]

  74. […] decided to see if 59 metric tons was plausible for Dreadnoughtus by doing Graphic Double Integration (GDI) on the published skeletal reconstruction and body outline (Lacovara et al. 2014: fig. 2). The […]

  75. P.f Says:

    Greetings Dr. Wedel,

    I thank you for this excellent treatise on GDI. However, I still have a few queries that I hope you would deign to answer, if it would not terribly inconvenience you.

    You mention that ” […]You could use the white-bones-on-black-silhouette skeletal reconstructions that have become the unofficial industry standard.[…] ” However, most of these reconstructions are invariably of a single orthographic view of the animal. How would one use said reconstructions to perform GDI, which, if I read this article correctly, requires more than one directional viewpoint of a reconstruction?

    What materials would one require to do this entirely by hand?
    You mention that ” […]you can go through the whole process with nothing more than pen and paper, […] “, but you mention here that ” […]if you’re just rolling with GIMP and OpenOffice Calc (or Photoshop and Excel, or calipers and a calculator), you need to have enough slices to capture most of the information in the model without becoming unwieldy to measure and calculate.[…] “. This seeming contradiction regarding the requisite materials was what confused me.

  76. Matt Wedel Says:

    How would one use said reconstructions to perform GDI, which, if I read this article correctly, requires more than one directional viewpoint of a reconstruction?

    Basically there are two options. Ideally, you’d want reconstructions done in at least two views, like Nima Sassani’s Puertasaurus shown in this post. Failing that, you could make some assumptions about the likely cross-sectional geometry of the animal–like should the torso be circular in cross-section, or a 2×3 ellipse, or some other shape, like I did in this post on Dreadnoughtus.

    What materials would one require to do this entirely by hand?
    …This seeming contradiction regarding the requisite materials was what confused me.

    GIMP, Photoshop, or other image-editing software are handy because they let you measure pixel counts for raster-based images, and OpenOffice Calc, Excel, or other spreadsheet programs are helpful for cranking the numbers for you, but it is certainly possible to do GDI without them.

    To roll entirely by hand, you need:
    1. the reconstructions (or anything else that allows you to capture information about body slice geometry–you could do it on a toy dinosaur, for example;
    2. some kind of measuring device, like a ruler, tape, or caliper;
    3. some paper and a writing utensil for doing the calculations by hand.

    BUT if you have an internet connection to read this, you can probably get the free softward mentioned above, in which case doing it by hand would not be necessary, although it might be a diverting challenge.

  77. P.f Says:

    I thank you profusely for the surprising expediency with which you answered my question. I must now copiously apologize for inquiring yet again about the materials required to perform GDI by hand; what measuring device would you recommend as the most convenient?

  78. Matt Wedel Says:

    A mouse. Seriously. If you can get online to read SV-POW!–and you’re not either reading on a smartphone or a borrowed computer at the public library–you can download GIMP for free and use that.

    If you’re truly committed to doing GDI without shuttling electrons around, then like I said, you can use a ruler, tape, or caliper. I can’t tell you which of those is most convenient for you, you’ll have to figure that out on your own, based on what you have available and which tool you prefer.

  79. […] perfectly. The total length of Archaeoceratops, including tail, is almost exactly one meter. Using graphic double integration, I got a volume of 8.88L total for a 1m Archaeoceratops. That would come down to 8.0L if the lungs […]

  80. […] There are a couple of ways to get to the answer here. One is to use Graphic Double Integration, which is explained in this post. […]

  81. […] SV-POW – Tutorial 11: Graphic Double Integration, or, weighing dinosaurs on the cheap […]

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