Lecture – 07
Distance Sampling – I
[FL] In today’s class we will have a look at Distance Sampling.
(Refer Slide Time: 00:19)
This is a sampling procedure that we normally use for counting of wildlife. So, in case of directly getting into distance sampling let us build up the concepts step by step.
(Refer Slide Time: 00:31)
(Refer Slide Time: 00:34)
(Refer Slide Time: 00:37)
Now, if we consider a forest then the density of animals D or the density is the number of animals divided by the total area of the forest or the total area of the region under study. (Refer Slide Time: 00:48)
So, now let us consider this forest in which these dots represent the animals. Now these animals are concentrated to and side and are sparsely on the other side probably because say, this second side is having a high mounted and this side is having more of grasslands.
So, this area is providing more food to the animals.
Now in this hypothetical situation; if we counted each and every animal in this forest and we divided it by the total area of the forest. So, in this case total area is 100 meters by 100 meters, which becomes one hector.
(Refer Slide Time: 01:25)
And suppose we found that there were 205 animals in the complete forest. So, 205 animals divided by 1 hectare gives us a value of 205 animals per hectare. So, this is our normal formula which is D is equal to N by A, but then doing a complete census in most cases is difficult.
N = number of animals: 205; A= Area: 1 hectare; D= density
= 205 ??????? ??? ℎ??????
Refer Slide Time: 01:46)
So, can we make use of some sampling procedure in place of getting the complete census. So, in the case of a sampling procedure we would not be using the complete number of animals that are there in the forest. So, we will not be counting the whole of these number of animals, but will be dividing this area into small plots and then we will be using the number of animals in those plots to get an estimate.
(Refer Slide Time: 02:13)
So, in this case we can make use of a strip plot. So, in the case of a strip plot we have taken these 3 strips and in all of these 3 strips, we count the number of animals that are here, we count the number of animals that are here and we count the number of animals that are here. We divide this figure of the total number of animals divided by the area of the strip plots.
So, the area of this strip plus the area of this strip plus the area of this strip; so, we will get an estimate of the density, and then we would say there that the density of our animals in this strips is the same as the density of the animals in the complete forest. So, by multiplying this density by the area of the forest we would be getting an estimate of the number of animals in the forest.
(Refer Slide Time: 02:58)
Similarly, we could make use of circular plots. So in this case we have these 3 circular plots and we perform the same procedure, we count the number of animals in all of these 3 plots divided by, we take the sum of animals in these 3 plots divided by the sum of these 3 areas to get a density and then we say that the density of our sample is the same as the density of the population multiply that with the area of the forest to get the number of animals in the forest.
Refer Slide Time: 03:30)
So, how would that look? So, to illustrate let us go with our strip plot example. So, in this case we had our forest and we had divided it into strips and we had taken 3 strips like this. Now, and we had animals both inside and outside the strip and we had only counted the animals that were inside the strip. Now, let us say that each of these strips was of length l.
So, how do we take a strip?
(Refer Slide Time: 04:07)
So, how do we make a strip plot? So, in a very simple example let us take the stick I hold it like this and then I start walking. What I am interested in is noting the number of pellets of animals that are there on the ground or say, noting the number of some particular plant species that is there on the ground.
So, in that case when I am walking on a straight line, then this is the half width on my left, this is the half width on my right and l is the distance that I am moving and I will only be considering any samples that are just below, just within this length of the rod. So, when I am moving, I am creating a strip through my movement. So, when I am walking on a straight line like this, I have a half width ‘w’ on my right side and a half width ‘w’ on my left side and by walking this distance, I am creating a strip that has a total width of 2 w and a total length of l.
So, the area of one strip is twice ‘w’ into ‘l’, because it is the width into length the area of a rectangle. Now, in place of taking these rods I could also use a computer and say that when I am walking on this line take this much distance to the right and take this much distance to the left, I and can always make use of a laser rangefinder to check if I have seen any animal and whether that animal falls within my chosen width half width of w or whether it is outside of that distance. So, the use of a rod is not always essential, but it is to simplify this concept and it is used in cases when we are doing a pellet count or when we are doing a hub count.
Now, this is the area of one trip. So, this is area of trip. Now suppose I took k number of strips. So, in our example we had 3 strips. So, k is equal to 3. So, the total area of strips is the area of 1 strip multiplied by the number of strips. So, in this case it becomes 2 w into l into k. Now coming back to the slides, so here we have it, the total area of all the strips is k into 2 w into l. Now, in our strip method suppose we counted n number of animals in this area small ‘a’ then the density of animals is the number of animals divided by ‘a’. So, it becomes n divided by k 2 w l.
????? ?????? ?? ??????? = ?; ????? ???? ?? ?ℎ? ????? = 2? × ? × ?
??????? ?? ?ℎ? ??????? =
Refer Slide Time: 07:18)
Now, when we are doing a strip plot we say that this density d which is n by a is the same as the density of the animals in the whole of the forest which we represented as capital D with a hat. So, capital D with a hat is the estimate of the density of animals in the whole forest, now just because it is an estimate we put a hat on top of it. Capital D, because it is representing the whole of the forest and this small d is because it is representing the strip plots. So, once we take this assumption that the density of animals in the forest is the same as the density of animals in our strips then we can use this formula to find out the total number of animals in the forest.
??????? ?? ??????? ?? ?ℎ??? ?????? = ?̂
So, total number of animals would be given by the density of animals multiplied by the area of the forest. So, capital A is the complete area of the forest. Now, because we took this assumption that D hat is equal to small d which is n by a. So, we have n by a, as in the previous slide we had seen that a is k which is the number of strips multiplied by twice of half width multiplied by l; so, k into 2 w into l. So, we put that here k into 2 w into l multiplied that with the area of the forest A to get this formula n small n capital A divided by 2 w l here this capital L is small k into small l.
A: complete area of forest
?̂ = ? = ?
? = ? × 2? × ?
?̂ = ?̂ × ?
So, small l was the length of 1 strip k is the total length that I have moved. So, now coming back to the board, when I had moved this distance of small l, the effort that I have put because I am not moving in the whole of the area I am only moving in a straight line. So, this line is of length l. So, when we say effort is the total distance moved by the observer.
So, total distance moved in this case is the length of 1 strip multiplied by the number of strips that we had; so, l into k which is also given by capital L. So, now, coming to the slides here we have small k that is the number of strips multiplied by the length of the strip is capital L. So, n A divided by 2 w capital L which is the estimate of the number of animals in the whole of the forest and capital L is the effort which is k into l. Now this much portion is simple.
Since, L= k x l
Refer Slide Time: 10:10)
And we use hats in capital N hat and capital D hat because both of these are estimates. (Refer Slide Time: 10:23)
So, when we wrote small n, coming back to the board; when we wrote small n. So, small n is the number of animals that we actually observed; when we wrote d that is the density of the animals that was actually calculated. Why we say actually because, d is n by small a we have actually observed the number of animals n and we have, we know precisely what the value of a is. But, when we say that the density of animals in the whole of the forest is equal to this small d. Now d is something that we know for 100 percent sure, but capital D we do not know that whether it is 100 percent equal to small d or whether there is a small fluctuation.
So, we make this estimate to say that capital D is the same as small d. So, just because this is an estimate we put a hat on top of it similarly when we calculate the number of animals in the whole of the forest we put a hat here because this again is an estimate because it makes use of d into a and d here is an estimate. So, when we have this estimate, capital A is something that we know 100 percent sure, but D hat is something that we do not know, this is again an estimate. So, estimate multiplied by some value will again be an estimate.
So, this much is simple. So, it is very easy to calculate the density of animals in a forest just by taking some plots and then we count the number of animals in those plots divide that by the area of those plots to get a density of animals in the plot and then we say that the density of animals in the forest is the same as the density of animals in our sample plots. So, we get density of animals in the forest multiply that with the area of the forest and to get the total number of animals in the forest. But then distance sampling is different from these simple plot samplings, because of this assumption.
(Refer Slide Time: 12:21)
Plot sampling assumes that all the animals in the sample plots are detected and counted, which may or may not be a correct thing in the case of a forest.
Refer Slide Time: 12:37)
Because to illustrate let us take the example of this rabbit, now if this rabbit is out there on the ground will be able to see it, but if this rabbit in a patch of grass it might be occluded.
So, we might miss out this rabbit and this is something that we need to put into our computations to get the correct figures, because in the case of forest there are a number of situations in which we miss out the animals, any animal that is behind a tree will be missed out, any animal that is not moving is just camouflaged inside say, some bushes will be missed out, any animal that gets into the grasses will be missed out and this is something that we need to compensate for in our computations.
Refer Slide Time: 13:22)
So, distance sampling takes this factor of non detection into account by considering that not all the animals in the transit are detected. Now what is our transect? So, when we move this path we call it a transect. So, transect is a path on which we are moving in the forest to get our data.
So, distance sampling coming back to the sides. So, distance sampling takes the factor of non detection into account by considering that not all the animals in the transect are detected, then it tries to estimate the number of animals that we missed out in the counting exercise.
Refer Slide Time: 13:57)
So, this estimate is given as n hat is equal to n divided by p hat. Now n is the number of animals that we counted, but our probability of detection is not 100 percent because we have missed out some animals. So, when we divide n by the probability of detection we get another estimate which is small n hat. So, small n hat is n divided by p hat, where p hat is the probability of detection.
(Refer Slide Time: 14:28)
What we want to say in this case is that, when we moved on these transects, when we took out these strip lots there were some animals that we missed out.
(Refer Slide Time: 14:38)
And suppose, out of every 4 animals, we were only able to see 3 animals and we missed out 1 animal. So, in that case our probability of detection that is p hat is given by the number of animals detected divided by the number of animals actually present.
?????? ?? ??????? ????????
??????? ???????? ???????
(Refer Slide Time: 14:49)
So, suppose we detected 3 animals when we had 4 animals actually there. So, we will have a p hat is equal to 3 by 4 or 0.75.
?̂ = = 0.75
Now suppose we have this data of p hat we know that our p hat is 0.75, then we did another exercise and we directed 30 animals. So, we will have number of animals detected as 30, number of animals present is x, which we do not know and this ratio number of animals detected by the number of animals actually present is p hat and we have somehow computed this p hat to be 0.75.
?̂ = 0.75
So, in this case we have 30 by x is 0.75 or x is 30 by 0.75 is 30 by 3 by 4 is 40. So, once we know our value of p hat and once we know the number of animals that we actually counted, which is small n, we can estimate the number of animals that would actually been there.
?????? ?? ??????? ???????? 30
?????? ?? ??????? ??????? ?
? = 40
Refer Slide Time: 16:28)
Now, another distinction between plot sampling and distance sampling is that of the assumption, in the case of plot sampling because, we said that we have detected each and every animal; so, our strips cannot be very wide because when the strips are wide enough. (Refer Slide Time: 16:50)
So, suppose we are moving in the forest and if there is an animal here, it will be easier to do to detect this animal as compared to an animal that was at a far off distance. So, when we say that our assumption is that we are detecting each and every animal, the width or the half width of the strip cannot be very large. So, this w would be acceptable, but this w will not be acceptable, in the case of a strip plot. But in the case of a distance sampling experiment because we are assuming that we are not detecting each and every animal, so in this case we will be having certain value of p hat let us call it p hat 1 in this case we will be having a certain value of p hat which is p hat 2.
Now, because in both the these situations we are saying that we are not detecting each and every animal and there is some p hat involved we can take any width of the strip as much as we want. So, basically this assumption is crucial in the case of forest, because like when you are moving in a forest and when you see an animal at a great distance, so whether you include it into your sample or not. So, when you see an elephant that is say, 800 or even a 800 meters or even say a kilometer away from you, you cannot include incorporate that into a strip plot, but you can always incorporate that as a data point in the case of your distance sampling and we can always compute p hat and use that in the figures. (Refer Slide Time: 18:35)
So, now coming back to the distance sampling formula we had estimated for the plot sampling that N hat is D hat into A and we had taken out this formula as we had seen in the previous slide in the case of distance sampling.
?̂ = ?̂ × ?
Refer Slide Time: 18:49)
We replace n with n hat or n by p hat. So, in the case of this, in the earlier slide we had this formula n A by 2 will and in this case it will become n A by 2 w L p hat. So, in place of n we have replaced it by n by p hat. So, now all that remains is to estimate the value of p hat.
So, how do we estimate the p hat.
? = ?̂ = ??? ?̂ =
?̂ = 2???̂
Refer Slide Time: 19:20)
So, to estimate p hat, what we do is when we are walking on a transect line and we are seeing animals we find out the distance of these animals from us and then we group it into a number of categories. So, suppose we saw 35 animals at a distance of 0 to 10 meters. We saw 35 animals at a distance of 10 to 20 meters.
Then at 20 to 30 meters we saw only 22 animals, at 60 to 70 meters we saw only 1 animal. Now our assumption was that the density of animals throughout the forest is constant. So, essentially we should be seeing equal number of animals between our distance of 0 to 10 and our distance of 60 to 70, but this is not happening in the case of actual detection of animals. So, we plot these animals as a chart.
Refer Slide Time: 20:10)
(Refer Slide Time: 20:15)
So, what we have done in this plot is that we have noted out our distances from the transect line on the x axis and the number of animals that we have detected at different distances on the y axis. So, from 0 to 10 meters we saw 35 animals from 10 to 20 we again saw 35 animals and at a large distance we saw only one animal.
So, this is our bar diagram that we got from all that. Now this bar diagram can then be converted into a curve by taking the midpoints and then drawing a curve that goes through all the midpoints of all the bars. So, this red line detects our curve. Now our assumption was that, the density of animals throughout the forest is constant. So, the number of animals that we have detected here should be the same of the number of animals that should be at this distance as well.
So, we take a straight line from the, from the top of the first curve till the end point and then we draw this rectangle. Now in this rectangle the number of animals that were actually present is given by the area of the rectangle, because we have this number of animals or the density of animals and which is constant at all the distances.
So, if we take the area of this rectangle it will be proportional to the total number of animals that were actually present in the forest. However, when we did our experiment we only saw these animals which is the area of the curve that is below this line. So, this is the number of animals are detected and this total is the number of animals that were present and if you look at this hashed portion this is the number of animals that were not detected. (Refer Slide Time: 22:09)
So, what we saw in our curve is that, we have animals that we have plotted as these bar charts and then, we used this bar chart to create a curve which is this red curve, which is a scooped out data line of the number of animals actually seen. Now the total number of animals that were there in the forest is given by or is proportional to the area of this rectangle whereas, the number of animals that we actually saw in the forest is given by the area of this figure under the curve and the number of animals that we missed out is given by this area.
Now, going back to the equation of p hat is the number of animals detected divided by the number of animals actually present. So, when we use this formula, we get p hat as the area under the curve given by these yellow lines divided by the total area of the rectangle. (Refer Slide Time: 23:35)
So, coming to the slides we have p hat is the area under the fitted curve divided by area under the rectangle. So, this is one way in which we can estimate p hat. (Refer Slide Time: 23:46)
And the number of animals that is left out is given by the area in the hashed lines in red, in this figure.
(Refer Slide Time: 23:53)
Now, when we are estimating p hat it will depend on a number of field situations. So, if the animals are large and clearly visible as in the case of elephants in a grassland the curve will be flat for a considerable distance and the curve will match the area of the rectangle giving p hat close to 1. What we mean in this case is that when we are plotting distance versus number of animals.
(Refer Slide Time: 24:17)
So, we are considering a situation in which we have grasslands for a very large distance and we have large sized animals like the elephants. Now, even if you have an elephant at a distance and you have short grasses there, you will be able to see the elephant, and basically because we have a flat terrain we are able to see it for a very large distance.
So, the density of elephants that we find at every distance would be nearly the same. So, essentially our curve would be close to this. Even if we are missing out some elephants at a distance the total number of animals that we are missing out, this much, is very small as compared to the area of the complete rectangle.
So, in this case we will be having p hat is close to 1 or p hat is tending towards 1. On the other hand if you have a situation in which the animals are barely visible; so, consider that you are moving in a grassland and the animal you are interested in is say a mouse that has found in that grasslands.
(Refer Slide Time: 25:53)
Now when you are walking on your transect if the mouse is there on the transect itself, you will be able to see it but as soon as your mouse gets into the grasslands you will not be able to see it. So, essentially when we draw the curve it will be like this.
(Refer Slide Time: 26:02)
So, you have distance and the number of animals detected. So, at a very close distance, at a distance of 0 when your animal is actually on your path you are able to see that animal, but as soon as you this animal has moved away from you it has gone into a bush of grasslands. So, as soon as this animal enters into the grass it will be occluded you will not be able to detect it.
So, coming back to the board your detection it might be a bit of detection when it is just entering to the grass, but then it will become close to 0. Now, when you draw your curve it will be something like this, and when you take the area under the curve and divide it by the area under the rectangle, you will have a p hat that is very close to 0 or a p hat that is tending towards 0. Similarly, p hat would also depend on a number of other factors. (Refer Slide Time: 27:16)
So, coming back to the slides p hat would depend on the characteristics of the terrain, if your terrain is flat, so you are able to see for a very long distance you will be having a greater probability of detection as compared to an area which is having an undulating topography.
(Refer Slide Time: 27:38)
So, if this is my area I am walking on a transect that goes like this, so any animal which is here will not be seen, whereas if I am walking on a flat line this is my transect, any animal at a very large distance also will I will be able to see that animal. It also depends on the nature of the transect exercise.
So, basically when you are walking this transect you can either walk on foot or you can take a vehicle. When you are walking; when you are moving on this transect in a vehicle which we call as a vehicular transect, you are at a much elevated height. So, from that elevated height you will be able to see for a much larger distance as compared to when you are walking on by yourself on foot.
(Refer Slide Time: 28:21)
Also in the case of a forest when you are moving in a vehicle the animals are much more comfortable because like, in most of our tiger reserves we have so many tourists that are that are already moving in the park areas. So, the animals are habituated to the sight of a vehicle moving because that vehicle does not do any harm to them. But the animals are not habituated to the to the situation of finding a human being on foot. So, the animals might be startled. Now this can have a good impact on our probability of detection or it can have a bad impact. A good impact because even animal is standing somewhere in the forest, it is difficult to detect that animal mostly because most of the animals are actually camouflaged they are already camouflaged.
So, camouflaging is consider the case of a tiger, a tiger has stripes it moves in the grasses, so it blends with the grasses and so, you are not able to see that animal. If you see any animal say chital. So, chital is normally found in areas that has brown colored soil. So, when you have a chital in the foreground brown colored soil in the background you do not see the chital because chital already has a brown color. So, brown and brown mixes well. But when this animal is moving you can very easily say that yeah, there is a chital that is moving.
So, just because of its movement you are able to detect it. Now, in the case of a vehicular transect, because the animals are comfortable they will just be doing their own activities. So, if they are grazing on the grasses they will be grazing on the grasses they will not be running. But when you are moving into that area the animal will run, when you are moving on foot.
So, when you when you are moving on foot, you will be able to detect that animal in a much greater probability as compared to when you are moving in a vehicle. Just considering the movement aspect, but then a typically it is observed that in the forest the detection of the animal of the human being is much greater. So, basically when you are moving in the forest the animal will see you much before you will see the animal.
So, with your at a distance from the animal and if that the animal has detected you and it has run away then you have already missed that animal. So, the p hat will depend a lot on the nature of the transact exercise, there will be some things that will help you, there will be some things that will not be helping you and these aspects will have to be put into your computations. The nature of the transect; so, transects on areas that are more frequented by animals will have larger detection probabilities. So, for instance, there are some animals that prefer walking on the roads. So, if you are walking on the roads you will have a much greater detection of animals just because you have the animals that are concentrated there. (Refer Slide Time: 31:18)
p hat also depends on the local traditions. So, if there are some areas in which the locals regularly feed animals and if you are walking in those areas then the animals are already habituated to the sight and spell of a human being. So, they will come closer to you and so they will facilitate their detection. It will also detect depend on the characteristics of the animals, some animals that are shy might freeze and if they freeze on seeing you will not be able to detect them. But on the other hand, if there are some animals that are shy and they run away, you will be able to detect them.
(Refer Slide Time: 31:57)
Similarly, the color of dress that is worn by you when you are doing a transect exercise, the perfume that has worn by you, the food that is eaten by you because it will have some smell and it will also impact the animals because, they will be able to detect you and then depending on the nature of the animals they might even just freeze at that location or they might start running away which will impact the detection probability. (Refer Slide Time: 32:20)
p hat also depends on the mental state and fatigue of the observer and on the size of the observer group because typically we have seen in the forest that whenever these exercises are carried out. In the beginning, the people are extremely excited to see the animals and note down the results. So, even small movements are noticed by the people, but after a while when people get fatigued their mental state also becomes dull and they do not take note of all the animals.
Similarly, the size of the observer group, if you have a larger group, you have more number of eyes that are looking out for the animals which should increase your detection probability, but typically what we have seen is that in a larger group people start chatting and this sound also alerts the animals. So, this again might be having a positive or a negative impact.
(Refer Slide Time: 33:11)
Next is the direction of the wind, if the wind is moving from the animal towards you the animal will not be able to detect you through the smell, but if the wind is moving from you towards the animal then the animal will be able to detect (Refer Time: 33:24) you from your at a very large distance. So, that will also impact the p hat. It also depends on the weather on the day of the observation because animals will typically be showing very different characteristics on different days of observation. So, for instance, if you are looking out for snakes in a cold season and you have sun outside.
So, the animal will typically be found basking on the rocks or probably even basking on the trail, whereas if you are looking out for the snakes on an overcast day then probably the animal would be hiding somewhere. So, the nature and behavior of the animal also depends a lot on the weather on the day of observation. So, typically it is easy to say that we can estimate p hat, but p hat because it depends on so many factors it becomes difficult to estimate and so has to be computed using numerical methods for each and every exercise. So, essentially we do not have a value of p hat that we can just put up in our equations to get the number of animals that should be there, it will have to be computed again and again for every exercise and we will look at it in greater detail in the next lecture.
Thank you for your attention, jai hind.
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