Estimating availability from multiple surface and dive interval datasets
Jason Roberts
It is common in marine mammal density modeling to use the Laake et al. (1997) method to estimate the probability an animal will be available at the surface for visual detection. As inputs, this method takes mean surface and dive intervals for the species in question and also the amount of time an animal on the surface will remain visible as the observer passes by. I will omit the details for now but am happy to discuss them if needed. I attached a copy of the paper. Equation 4 seems to be the most common formulation applied in marine mammal modeling.
Once the probability of availability has been estimated, it can be divided into density and abundance estimates to boost them to account for animals that were present but not available to be detected. In traditional distance sampling, when the objective is to make a design-based abundance estimate, this step can be done at the very end, adjusting the final abundance estimate upward. In two-stage density surface modeling (Miller et al. 2013), this step is often done after detection functions have been fitted (stage 1) and transects have been split into segments, to correct the abundance of each survey segment prior to fitting the spatial model (stage 2).
My question concerns a common situation: how do you best obtain a mean probability of availability from a collection of dive data? As an example, consider the attached excerpt of Table 1 from Hain et al. (1999), who reported some dive data for North Atlantic right whale mother/calf pairs. This shows mean surface and dive intervals—exactly what is needed as input for the Laake et al. (1999) availability method. (For simplicity, I did not extract the whole table, just the "Mother/calf pairs (both considered)" section.)
However, it reports results for five mother-calf pairs. I can see there's a lot of variance between them, especially in Mean Surface Time. Given that, I do not want to base my availability estimate on a single record. What is the best way to obtain an availability estimate that combines them?
Options include:
1. First average the mean surface time across all five records, and the mean dive time, and then perform the Laake availability computation.
2. Compute Laake availability five times, once for each record, then average those results.
Note that for either option, the averaging could be performed with each record receiving equal weight, or by weighting them by Analysis Time. For now, whether or not to weight them is a secondary question.
What I can't really do is:
3. Compute Laake availability five times as in #2, but rather than averaging the results, conduct a sensitivity analysis on how these five estimates affect the final density model, and maybe average those final density estimates. My analysis pools pools segments from several different surveys, each with their own detection models, and then fits a single density surface model to the combined segments. Given my resources, it is infeasible to bootstrap g(0)s across these, or otherwise explore how uncertainty in g(0) affects final results (e.g. via variance propagation). So for each program, I need to come to a single g(0) to apply to its segments.
Is #1 or #2 better, or some other method? What I'm concerned about is how the point at which averaging is done might affect how abundance is rescaled. This might just be a basic math question—if so, I apologize for my mathematical incompetence!—but because of the complexity of the Laake equation, particularly its second term, I'm having trouble working it out.
Thanks very much for any advice you can provide,
Jason
Eric Rexstad
-
Borchers, D.L., Zucchini, W., Heide-Jørgensen, M.P., Cañadas, A. and Langrock, R. (2013), Using Hidden Markov Models to Deal with Availability Bias on Line Transect Surveys. Biom, 69: 703-713. https://doi.org/10.1111/biom.12049
Sent: 27 August 2025 16:39
To: distance-sampling <distance...@googlegroups.com>
Subject: [distance-sampling] Estimating availability from multiple surface and dive interval datasets
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Jason Roberts
You anticipated where I was going to go next, which was to ask for suggestions for when we have time-depth profile data for each animal (e.g. from depth-logging tags) rather than mean surface and dive intervals. Because those are not always available, I wanted to start with the "simpler" case of just having the mean intervals...
Borchers et al. (2013) provides a more comprehensive and complete model for availability, but it requires both perpendicular and forward distances, at least as implemented in the supplementary example and in the hmltm package (as far as I can tell). Almost none of the aerial surveys I have collected forward distances, so I'm not sure this approach would be tractable for my data. Also, this page says of hmltm "please note that this is a research-level output, not designed for general use."
So I must stick with Laake et al. (1997), at least for the near future, or perhaps a simulation-based approach (e.g. when time-depth profiles are available) that does not require forward distances. Does anyone have further suggestions?
Thanks,
Jason
Tiago Marques
To: Eric Rexstad <Eric.R...@st-andrews.ac.uk>; distance-sampling <distance...@googlegroups.com>
Subject: Re: [distance-sampling] Estimating availability from multiple surface and dive interval datasets
Jason Roberts
> I believe your first approach is the same as your second approach assuming that you are doing a weighted average where the weights are the tag durations.
But you explored another important issue, which is how to best generalize a collection of tag data into an availability correction for the entire population, given that tag durations may be quite different from each other. I totally agree that the within-whale vs. between-whale variability is important here. So is having a sample of whales that are sufficiently representative of the population in question. As you say, it is rare that you ever have enough to be fully satisfied...
Anyway, thanks for your comments!
Jason
Rachel Fewster
I think your question boils down to asking which of your two options has better intrinsic precision.
When your estimator of interest is a non-linear function of inputs, as it is here, your instincts are correct that it can make a difference to precision whether you average first and then apply the function to the averages (your option 1), or apply the function to the individual observations and then take the average of the results (your option 2).
There's no general rule, as it depends on the specific function you're working with; but it is often true that option 1 has better precision than option 2, and I'd hazard a guess that this will be the case here. For example if you're interested in a ratio of the format X/Y, it's often better to estimate this as mean(X) / mean(Y) rather than mean(X/Y). This is because there's scope for small Y-values to give large variability in individual observations of X/Y, creating high variance in mean(X/Y); whereas by taking the means first as in mean(X) / mean(Y) you stabilize the denominator and get better precision.
The precision that I'm talking about here is to do with the intrinsic variance - what it really is, rather than what you estimate it to be. Estimating the precision is a whole other thing; but regardless of your ability to estimate it, picking an estimator with low intrinsic variance is generally a good thing.
The answer above doesn't consider bias either; but you're probably reasonably safe to assume that any bias due to option 1 would be compensated for by an improvement in precision. In general, an estimator with a small bias and good precision will get you closer to the right answer, on average, than an unbiased estimator with poor precision.
If you want to be sure, you could set up a simple simulation. Pick some reasonable parameters for mu and lambda, generate data from the corresponding exponential distributions, calculate the Laake estimator both ways for each set of data, and calculate something suitable like the root-mean-square-error to assess each one. I'd hazard a guess that your option 1 will win, but if it were me I'd also want to check it by simulation before committing.
Best wishes,
Rachel Fewster
--
Rachel Fewster (r.fe...@auckland.ac.nz)
Department of Statistics, University of Auckland,
Private Bag 92019, Auckland, New Zealand.
https://www.stat.auckland.ac.nz/~fewster/
________________________________________
From: distance...@googlegroups.com <distance...@googlegroups.com> on behalf of Jason Roberts <jjr...@gmail.com>
Sent: Thursday, September 11, 2025 9:07 AM
To: Tiago Marques; Eric Rexstad; distance-sampling
Subject: Re: [distance-sampling] Estimating availability from multiple surface and dive interval datasets
Thanks, Tiago. Regarding:
> I believe your first approach is the same as your second approach assuming that you are doing a weighted average where the weights are the tag durations.
I'm not sure that's entirely true, as the availability from Laake et al. (1997) does not scale in a simplistic way according to the surface and dive intervals. Because it that, it might matter whether you average the inputs and then compute a single output, or don't average the inputs and but instead average multiple outputs, if you see what I mean.
But you explored another important issue, which is how to best generalize a collection of tag data into an availability correction for the entire population, given that tag durations may be quite different from each other. I totally agree that the within-whale vs. between-whale variability is important here. So is having a sample of whales that are sufficiently representative of the population in question. As you say, it is rare that you ever have enough to be fully satisfied...
Anyway, thanks for your comments!
Jason
On 9/8/25 03:57, Tiago Marques wrote:
Hi Jason,
I am coming late to the party but I'll say a few things which come to mind from the work I have done looking at acoustic cue rate production from tags. This is essentially the same problem with a different target measure.
I believe your first approach is the same as your second approach assuming that you are doing a weighted average where the weights are the tag durations. If that's the case you're better off using the individual tags with weights being the tag durations and then you actually get a precision measure for your availability.
But...
It is actually not quite clear to me wether in general one should use a weighted average or an unweighted average in these contexts. One can argue both ways. Ideally they would both be about the same and then you do not need to worry.
However...
If the weighted mean and unweighted mean result in very different values the question that arises is related to the variability within whale versus the variability across whales.
If the variability within whale is very small, but the variability across whales is large, then weighting for tag duration might be dangerous. Making an extreme example if one tag has a thousand hours and nine tags have just one hour you would be severely biasing the result towards that one long duration tag. Because the variability across wales was large you'd be very unlikely to be unbiased to the overall mean.
On the other hand if the variability across whales is small, but variability within whales is large, then the weighted average implies that you give more weight to whales you sampled for longer, so you estimate their mean more precisely, and hence it would be reasonable they should contribute to a weighted mean with duration as weights. In other words, if variability across whales is indeed small, but within whales large, then a weighted average might be better, as you are now giving more weight to tags with longer duration, averaging across the variability within whale. In such a case, whales with small durations, hence for which the cue rate was potentially estimated with bias, contribute less to the (weighted) mean.
In the ideal scenario, you sample long enough all whales such you get an unbiased estimate per whale, and then doing weighted versus unweighted should be about the same. Unfortunately, realistically, one likely never samples long enough for some whales...
I hope this helps in the spirit of brainstorming, ping on me if it does not or if further discussion might be helpful.
Cheers
Tiago
Enviado de Outlook para Android<https://aka.ms/AAb9ysg>
________________________________
From: distance...@googlegroups.com<mailto:distance...@googlegroups.com> <distance...@googlegroups.com><mailto:distance...@googlegroups.com> on behalf of Jason Roberts <jjr...@gmail.com><mailto:jjr...@gmail.com>
Sent: Thursday, August 28, 2025 3:29:12 PM
To: Eric Rexstad <Eric.R...@st-andrews.ac.uk><mailto:Eric.R...@st-andrews.ac.uk>; distance-sampling <distance...@googlegroups.com><mailto:distance...@googlegroups.com>
Subject: Re: [distance-sampling] Estimating availability from multiple surface and dive interval datasets
Thanks, Eric. It is a messy problem!
You anticipated where I was going to go next, which was to ask for suggestions for when we have time-depth profile data for each animal (e.g. from depth-logging tags) rather than mean surface and dive intervals. Because those are not always available, I wanted to start with the "simpler" case of just having the mean intervals...
Borchers et al. (2013) provides a more comprehensive and complete model for availability, but it requires both perpendicular and forward distances, at least as implemented in the supplementary example and in the hmltm package (as far as I can tell). Almost none of the aerial surveys I have collected forward distances, so I'm not sure this approach would be tractable for my data. Also, this page<https://distancesampling.org/online-course/11-mrds/g0_otherapproaches.html> says of hmltm "please note that this is a research-level output, not designed for general use."
So I must stick with Laake et al. (1997), at least for the near future, or perhaps a simulation-based approach (e.g. when time-depth profiles are available) that does not require forward distances. Does anyone have further suggestions?
Thanks,
Jason
On 8/28/25 03:26, Eric Rexstad wrote:
Jason
You raise a worthwhile (but messy problem) for which I don't have an answer.
As Table 1 of Hain et al. (1999) shows, there is variability in proportion of time available not only between pairs, but also within pairs. The values presented in the final column of the table are already averages over repeated dive cycles within pairs. You noted the large variability in availability estimates between pairs; you could at least measure this between-pair variability in availability if you implemented your method #2 (it would be lost using method #1). Hence large amounts of variability in this estimated correction factor that is unaccounted for in most analyses.
I expect your question "average of estimates or aggregation of data for single estimate" has analogies in other contexts. Perhaps others on the list have addressed the problem in their situation and can offer suggestions.
Finally, I note in passing, the availability measure of Laake et al. (1997) can be considerably biased in the situation where animals are at the surface for a prolonged period; Hain et al.'s data suggest pairs spend 2-10 minutes on the surface. I draw your attention to Figure 4 of Borchers et al. (2013)
*
Borchers, D.L., Zucchini, W., Heide-Jørgensen, M.P., Cañadas, A. and Langrock, R. (2013), Using Hidden Markov Models to Deal with Availability Bias on Line Transect Surveys. Biom, 69: 703-713. https://doi.org/10.1111/biom.12049
________________________________
From: distance...@googlegroups.com<mailto:distance...@googlegroups.com> <distance...@googlegroups.com><mailto:distance...@googlegroups.com> on behalf of Jason Roberts <jjr...@gmail.com><mailto:jjr...@gmail.com>
Sent: 27 August 2025 16:39
To: distance-sampling <distance...@googlegroups.com><mailto:distance...@googlegroups.com>
Subject: [distance-sampling] Estimating availability from multiple surface and dive interval datasets
Greetings everyone,
It is common in marine mammal density modeling to use the Laake et al. (1997) method to estimate the probability an animal will be available at the surface for visual detection. As inputs, this method takes mean surface and dive intervals for the species in question and also the amount of time an animal on the surface will remain visible as the observer passes by. I will omit the details for now but am happy to discuss them if needed. I attached a copy of the paper. Equation 4 seems to be the most common formulation applied in marine mammal modeling.
Once the probability of availability has been estimated, it can be divided into density and abundance estimates to boost them to account for animals that were present but not available to be detected. In traditional distance sampling, when the objective is to make a design-based abundance estimate, this step can be done at the very end, adjusting the final abundance estimate upward. In two-stage density surface modeling (Miller et al. 2013), this step is often done after detection functions have been fitted (stage 1) and transects have been split into segments, to correct the abundance of each survey segment prior to fitting the spatial model (stage 2).
My question concerns a common situation: how do you best obtain a mean probability of availability from a collection of dive data? As an example, consider the attached excerpt of Table 1 from Hain et al. (1999), who reported some dive data for North Atlantic right whale mother/calf pairs. This shows mean surface and dive intervals—exactly what is needed as input for the Laake et al. (1999) availability method. (For simplicity, I did not extract the whole table, just the "Mother/calf pairs (both considered)" section.)
However, it reports results for five mother-calf pairs. I can see there's a lot of variance between them, especially in Mean Surface Time. Given that, I do not want to base my availability estimate on a single record. What is the best way to obtain an availability estimate that combines them?
Options include:
1. First average the mean surface time across all five records, and the mean dive time, and then perform the Laake availability computation.
2. Compute Laake availability five times, once for each record, then average those results.
Note that for either option, the averaging could be performed with each record receiving equal weight, or by weighting them by Analysis Time. For now, whether or not to weight them is a secondary question.
What I can't really do is:
3. Compute Laake availability five times as in #2, but rather than averaging the results, conduct a sensitivity analysis on how these five estimates affect the final density model, and maybe average those final density estimates. My analysis pools pools segments from several different surveys, each with their own detection models, and then fits a single density surface model to the combined segments. Given my resources, it is infeasible to bootstrap g(0)s across these, or otherwise explore how uncertainty in g(0) affects final results (e.g. via variance propagation). So for each program, I need to come to a single g(0) to apply to its segments.
Is #1 or #2 better, or some other method? What I'm concerned about is how the point at which averaging is done might affect how abundance is rescaled. This might just be a basic math question—if so, I apologize for my mathematical incompetence!—but because of the complexity of the Laake equation, particularly its second term, I'm having trouble working it out.
Thanks very much for any advice you can provide,
Jason
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