Introduction  Merklinger's Approach  Special Case of Merklinger's Approach: Setting Distance to Infinity  Everything Sharp from "Here" to Infinity...  Merklinger's "Rules of Thumb"  Why Is Making Merklinger's Approach not Used More Widely?  A Practical Example  Conclusions  References
On this page, I would like to investigate Harold M. Merklinger's approach to estimating depthoffield in landscape photography, which can be seen as an alternative to using the standard depth of field approach. A special case of Merklinger's approach, namely setting the focus to infinity can also be regarded as an alternative to setting the focus to the hyperfocal distance in order to achieve sharpness from a certain distance up to infinity.
Note: On page Merklinger's Approach to Estimating Depth of Field (Short Version), I cover this topic without calculations and more concise. The page you are currently viewing is for the people who need the proofs and more details...
A Sony RX100 M1 user from Russia made me aware of Harold M. Merklinger's approach to estimating depthoffield in landscape photography. Admittedly, I had never heard of this approach before. He also pointed me to Kevin Boone's article Hyperfocal distances and Merklinger's method in landscape photography. In this article, Boone, among others, compares the practice of setting the focus to the hyperfocal distance in order to have everything sharp from a certain distance (which is, in fact, half the hyperfocal distance) up to infinity with setting the distance to infinity. This is actually a longstanding practice for photographers, but it can also be regarded as a special case of Merklinger's approach. Merklinger himself discusses the relationship between the two methods as well, and provides some numbers and information about the "infinity case". Thus, even if we would prefer not to subsume this case under the label "Merklinger approach", he at least, contributed useful information about it.
For those, who just want to use Merklinger's approach, Boone's article provides already the most important things that you need to know, although I do not find the paper easy to understand in every aspect... Nevertheless, you can find tables that help you employ the method (including added minimum distances = hyperfocal distances, see here), as well as a discussion about the traditional approach to depth of field and hyperfocal distance versus Merklinger's approach, including a discussion of the simple approach to just set aperture to f/8 and distance to infinity.
Merklinger also published a shorter article Depth of Field Revisited about his approach with less mathematics, but easier to understand for me.
Merklinger presents his approach in the paper The INs and OUTs of Focus  An Alternative Way to Estimate DepthofField and Sharpness in the Photographic Image and calls it an object field approach (see side note) to depth of field  in contrast to the conventional image field approach based on the circle of confusion (CoC). His counterpart to the CoC in the object field is the disk of confusion (DoC). Merklinger explains:
Merklinger's "acrosstheboard image quality standard" mentioned above is represented by the circle of confusion (or CoC), which plays an important role in depth of field and hyperfocal distance calculations. For the following discussion, it is important to note that the circle of confusion lies on the film  that is, in the image field. The CoC is based on assumptions about how much detail the human eye can resolve when people view a photo at arm's length (about 25 cm), resulting in the CoC convention of 1/30 mm for 35 mm film (for different film/sensor formats adopted values are in use). For more information about the CoC, see the glossary.
The disk of confusion (DoC; diameter d) is, according to Merklinger, an exact analog of the circle of confusion (CoC; diameter c) to describe depth of field. The disk lies, however, in the object field, that is, in the scene to be photographed. Merklinger explains it as follows:
The Merklinger paper confused my a bit, because the arguments go somehow "upside down". In practice, you decide on the size (or diameter) of the objects you want to resolve (the disk of confusion), and all you need is an fnumber for the focal length that you want to use. The theory, however, delivers a formula that gives you the size of the disk of confusion, depending on distances and the aperture opening, which again depends on the focal length and the fnumber. Thus, you need to rearrange Merklinger's formula to get what you need...
For arriving at the diameter of the disk of confusion Sx (or Sy), Merklinger asks the question (with additions by me):
I will not present Merklinger's mathematical derivation of the result here, but would rather like to refer you to his paper. He seems to do a lot of algebra at the end of his derivation*, whereas I found that the intercept theorem leads to the same solution in a simpler way. Anyway, here is Merklinger's result:
Addition: If you express the object distance from the point of focus in fractions or multiples x or y of the focus distance D, you get even simpler formulae and "rules of thumb"*:
Legend: Sx = size of disk of confusion (object between lens and point of focus D); Sy = size of disk of confusion (object between point of focus D and infinity); D = distance of point of focus; d = f / N = diameter of working aperture; f = focal length; N = fnumber (fstop)
*) Merklinger writes in another paper: "The rule for determining what objects are resolved is ultrasimple. We focus our lens exactly at some distance, D, from our lens. An object onetenth of the way back from D, towards our camera (that is, at a distance of 0.9 D), will be resolved if it is at least onetenth as big as the opening in our lens diaphragm. If the object is onequarter of the way from the point of exact focus to the camera lens, it will have to be onequarter as big as our lens to be resolved. And so on. The same rule will hold on the far side of that point of exact focus also. An object twice as far from the lens as the point of exact focus will have to be as large as the lens aperture to be resolved."
*) I was able to reproduce Merklinger's calculations, but still do not understand, how he arrived at the starting formula Sx = c * (X / B) for the diameter of the disk of confusion (he does not explain this; B = distance between lens and film plane, X = distance between lens and object x). Just using the interception theorem lead me directly to the above formulae  without any assumptions about the relationship between c and d and any use of the lens formula...
Merklinger summarizes his result as follows:
Thus, whenever you specify a DoC criterion for the size of objects to be resolved (for example, 4 mm for blades of grass), you do it for distance X (not for D, there Sx = 0...)! You specify such a criterion also for a distance Y beyond the point of focus D. In the case that both criteria are the same (as we have for the CoC), we get:
That is, if the criteria are the same, the far point (or plane) Y is at the same distance behind the point of focus D as the near point (or plane) X is before D.
A special, but important case is that the lens is focused at infinity. I deal with it further down.
According to Merklinger, "the disk of confusion is about the size of the smallest object which will be recorded distinctly in our image. Smaller objects will be smeared together; larger objects will be outlined clearly  though the edges may be a bit soft."
The size of the disk of confusion is easily estimated for two specific distances:
The latter two statements can be easily derived from Merklinger's formulae for the disk of confusion:
Merklinger starts his calculations of the size of the disk of confusion from a relationship between it and the size of the circle of confusion:
In the course of the calculations, however, he eliminates the CoC from the formula by substituting it with the DoC. Therefore, the formulae tell us nothing about the CoC and its relation to the DoC.
Since the relationship formula above contains an unknown distance B (distance between lens and film plane*), the CoC cannot be calculated directly. I therefore asked myself whether it would be possible to arrive at a relationship formula that contains only wellknown parameters. I found that B can be calculated from the lens formula (1 / f = 1 / D + 1 / B) and got: B = f * D / (D  f). Substituting B with the righthand expression lead to the following formulae (no guarantee for its correctness, the final form of the formulae depends an taste...):
For the case that D = infinity (i.e., focus on infinity), we get the following simplified formulae:
Thus, for the infinity versions of the formulae, the object distance X is more or less "normalized" for multiples of the focal length. At an object distance of the focal length, the diameters of the circle of confusion and the disk of confusion are identical.
I assumed that this formula might also be used to calculate the minimum distance, based on focal length of the lens, a criterion for the diameter of the CoC (image field) and one for the diameter of the DoC (object field). This would be fairly easy for the case that distance is set to infinity because of the simpler formula. To verify this, I created an example, and since it seemed to work, I also calculated a complete distance table in Excel for the Sony RX100 M1  only to find out that it was nearly identical to the table of hyperfocal distances for this camera. Only then, I discovered that indeed the hyperfocal distance is just the minimum distance based on the CoC criterion, when distance is set to infinity (see here for a derivation). So here we know at least, both of them. At least, this makes plausible that my formulae above are correct... I deal with the "infinity" case further down.
Traditional depthoffield formulas and tables assume equal circles of confusion for near and far objects. Some authors, such as Merklinger (1992), have suggested that distant objects often need to be much sharper to be clearly recognizable, whereas closer objects, being larger on the film, do not need to be so sharp. The loss of detail in distant objects may be particularly noticeable with extreme enlargements. Achieving this additional sharpness in distant objects usually requires focusing beyond the hyperfocal distance, sometimes almost at infinity. For example, if photographing a cityscape with a traffic bollard in the foreground, this approach, termed the object field method by Merklinger, would recommend focusing very close to infinity, and stopping down to make the bollard sharp enough. With this approach, foreground objects cannot always be made perfectly sharp, but the loss of sharpness in near objects may be acceptable if recognizability of distant objects is paramount.
Other authors (Adams 1980) have taken the opposite position, maintaining that slight unsharpness in foreground objects is usually more disturbing than slight unsharpness in distant parts of a scene.
Moritz von Rohr also used an object field method, but unlike Merklinger, he
used the conventional criterion of a maximum circle of confusion diameter in
the image plane, leading to unequal front and rear depths of field.
(From Wikipedia: en.wikipedia.org/wiki/Depth_of_field )
The method of setting distance to infinity is a longstanding photographic practice, but can also be regarded as a "special case" of Merklinger's approach (see also Merklinger p. 29ff).
Setting distance to infinity, simplifies Merklinger's formulae and thus, makes calculations of the disk of confusion easier. For the infinity condition, you can simply set the object distance X (or Y) to zero (in comparison to D = infinity) and get the following result (in bold):
Thus, under this condition, the disk of confusion S assumes, regardless of the distance to the object, the constant size d of the diameter of the working aperture, which can be calculated from the focal length f and the working fnumber N for any given lens. This can be condensed into a simple rule of thumb (from Michael Reichmann at The Luminous Landscape):
For example, if you are using a 50 mm lens focused at infinity, and the aperture is f/8, then 50 divided by 8 is about 6 millimeters, meaning that objects of 6 mm size and more will be identifiable within the resolution capabilities of the lens and the sensor. 6 mm objects may be a bit soft, but are still identifiable (from The Luminous Landscape and Merklinger, adapted).
Merklinger makes the following comments on this condition:
Both statements are direct consequences of the formula S = d presented above.
In practice, this may mean that you have to use the formula "the other way round" because you started from specifying the disk of confusion. According to Merklinger, all you need in this special case is to calculate the fnumber N for a given focal length f and size of the disk of confusion d (which is your criterion of what is to be resolved in the image), that is, N = f / d. You can calculate this easily in your head.
What is still missing here, is a criterion for the near limit, albeit Merklinger probably would state that the disk of confusion is the near limit, according to his approach. But since his methods does not tell anything about the CoC at closer distances, you might get a bit nervous about this... I can assure you that you are not completely "left in the dark". As I derived elsewhere, the hyperfocal distance (HFD) is just the near limit if you adopt the conventional CoC as your "sharpness "criterion. I also show below (and Merklinger also mentions this) that at half the HFD, the CoC is twice the "acceptable" or "standard" size.
Instead of doing the DoC math in your head, you can also calculate tables of dvalues. I published such tables for the disk of confusion for my cameras (except for the Ricoh GXR) on this site  see the sections for the individual cameras. For my tables, however, I did it the "original" way and used fixed fnumbers, as given by cameras, and the most important focal lengths of my cameras to calculate tables of dvalues. When you use these tables, you first have to decide on a value of d, based on the scene to be photographed and a focal length f . Then search the table for a suitable dvalue and extract the corresponding fnumber for that focal length. Often, you will not find the exact dvalue in the table and have to select one that comes close. Boone proposes to choose the fnumber conservatively in this case, that is, to select the next larger fnumber to be "on the safe side".
Here is a stepbystep procedure for using the tables (there are two variants of it, one for cameras/lenses without a DOF scale and one for lenses with a DOF scale  they differ in the last step 4):
*) For my tables, I used fixed fnumbers, as given by the cameras,
and the most important focal lengths of my cameras to calculate tables
of dvalues. When you use these tables, you first have to decide
on a value of d based on the scene to be photographed and a focal length.
Then search the table for a suitable dvalue and extract the corresponding
fnumber for that focal length. Often, you will not find the exact
dvalue in the table and have to decide for one that comes close. Choose the
fnumber conservatively in this case, that is, select the next larger fnumber
to be "on the safe side".
**) Choose the fnumber conservatively, that is, select the next larger fnumber
to be "on the safe side".
As a "specialist for tables" I try to capture the essence of Merklinger's approach for a limited focus distance D versus the special case focus at infinity" in a small table:
Focus at D, Object
at X, Y 
Focus at Infinity 

Distance  CoC  DoC  DoC Formulae  CoC  DoC  DoC Formula 
Near Limit*  n.a.*  n.a.  Y > D: Sy = n * d, with: X < D: Sx = d / n, with: 
n.a.**  d  S = d 
D / n  n.a.  d / n => d  
D / 2  n.a.  d / 2  
D (focus)  0  0  
2* D  n.a.  d  
11*D  n.a.  10 * d  
n * D  n.a.  n * d => inf  
Far Limit*  n.a.*  n.a.  
Infinity  n.a.  inf  0  
Further useful formulae >  d = f / N  H (approx.) = f * f / (N * c) = f * d / c 
*) Not dealt with by Merklinger's method
*) To
be calculated from DOF formulae; **) for X = H
(HFD), the CoC assumes the "acceptable" value
Or in "compressed" format:
Focus at D, Object
at X, Y 
Focus at Infinity 

Distance  CoC  DoC  DoC Formulae  CoC  DoC  DoC Formula 
D / n  n.a.  d / n => d  Y > D: X < D: 
n.a.*  d  S = d 
D (focus)  0  0  
n * D  n.a.  n * d => inf  
Infinity  n.a.  inf  0  
Further useful formulae >  d = f / N  H (approx.) = f * f / (N * c) = f * d / c 
*) For X = H (HFD), the CoC assumes the "acceptable" value
Legend: Sx = size of disk of confusion (object between lens and point of focus D); Sy = diameter of disk of confusion (object between point of focus D and infinity); D = distance of point of focus; X, Y = distance of object (X<D, Y>D); n = abs (focusing distance  object distance)/(focusing distance); H = hyperfocal distance (H); d = f/N = diameter of working aperture; f = focal length; N = fnumber (fstop); c = diameter of circle of confusion
Often, we want everything sharp from "here" to infinity. There are two alternative methods for dealing with this requirement: Either ...
In the following, I attempt to compare both methods.
Above, I mentioned that Merklinger's simplified approach is an alternative to setting distance to the hyperfocal distance. Merklinger (p. 31ff) therefore discusses "what we gain and what we lose when we focus at infinity instead of the triedandtrue hyperfocal distance" and makes the following comments on this:
To verify these statements for focusing at the hyperfocal distance, I used the Merklinger formulae and set X (smaller than D) or Y (larger than D) to the hyperfocal distance H, fractions of it, or multiples of it:
When we, instead, focus at infinity, we get Sx = d at any distance, which is half the resolution at HFD/2, the same at double the HFD, and better for any other multiple of the HFD compared with focusing at the hyperfocal distance.
Merklinger concludes "Thus, if we are using a good lens, good film, and careful technique, we potentially have a lot to lose in the resolution of distant subjects by focusing the lens at the hyperfocal distance. In practice, by focusing instead at infinity, we will lose a factor of two in subject resolution at the near limit of depthoffield but gain about a factor of six in the resolution of distant subjects! It's often worth the trade."
Another question would be, which diameter the CoC will have at half the HFD (X = H/2) when we focus at infinity (at HFD, that is, for X = H, it is the "nominal" value). I start from the formula for c from above:
X can be expressed in fractions or multiples of H (X = H/x for fractions):
Now, we should be able to replace H using its simplified definition H = f * f / (N * c):
It looks as if the CoC is simply a fraction or multiple in size at HFD/x when focused at infinity  in contrast to focusing at the HFD (x = 1), where it is just "normal" size. I guess I have to verify this...
Note that Kevin Boone also includes an intensive discussion of hyperfocal distance versus Merklinger's approach in his article Hyperfocal distances and Merklinger's method in landscape photography. He also shows an example of how the "far limit" is negatively affected by setting the focus point too close when using the hyperfocal distance (due to settings errors or insufficient CoC). The far limit may move dramatically closer in such a case, which might ruin a shot. Boone therefore cautions the readers:
I prepared some "quickanddirty" test photos with some of our cameras, where I compare the following conditions: (1) focus set at infinity and (2) focus set at the hyperfocal distance. I also used several different apertures for each camera. I used all cameras at the same equivalent focal length of 28 mm. In hindsight, I realized that only the focal length is relevant. Thus, I could as well have taken the photos with one camera at different focal lengths.
Here are links to the respective pages (located in the sections of the cameras):
While the photos could be of better quality, they more or less confirm the above statements.
Many examples in Merklinger's paper deal with the hyperfocal distance (HFD), and I asked myself, "What is so special about a distance that probably 95% of all photographers do not even know of?" First, I thought, that perhaps the hyperfocal distance is special in that certain relations are valid for its fractions and multiples so that is especially easy to deal with. But then I found that this is valid for any distance that you focus on. Thus, if you express the object distance X or Y in fractions or multiples x or y of the focus distance D, you get:
In these formulae only the ratio of the focus and the object distance is relevant, but not that the fact that the focus distance is hyperfocal. Above, I listed already Merklinger's statement on this:
But over all the fuss with the hyperfocal distance, I forgot this gem...
So, I think the only relevant aspect from a calculation point of view is that the for the HFD, half the HFD, and infinity, we know the diameter of the circle of confusion (CoC), when we focus at HFD. From a practical point of view, it is, of course, also relevant by providing us with a distance setting that delivers a depth of field (DOF) that extends from a near point up to infinity (according to the CoC criterion, that is, with "acceptable sharpness" at the near and far limits).
In the following, I attempt to capture the differences between the hyperfocal distance approach and Merklinger's approach in an overview table:
Criterion  Hyperfocal Distance (HFD) 
Merklinger

Comment  
Special Case: Focus at Infinity 
Focus at Object at Finite
Distance 

Distance set to...  HFD  Infinity  Distance of most relevant objects in scene (<< infinity)  Typically farther away when using Merklinger's method 
Near Limit  HFD/2 (CoC criterion)  HFD (CoC criterion); can be easily read from lenses
with a DOF scale
The size of the DoC = aperture opening (f/N) is constant across the whole distance range up to infinity 
Given by the size of the DoC for the near limit (distance X) specified by the photographer  Infinity: Merklinger's method does not specify a near limit, only a "minimum resolved" object size that is constant up to infinity 
Far Limit  Infinity (CoC criterion)  Infinity (optimal sharpness)  Given by the size of the DoC for the far limit (distance Y) specified by the photographer  Infinity: Sharpness is optimal at infinity 
Best Sharpness at...  Hyperfocal distance  Infinity  Object that is in focus  Merklinger's method puts the emphasis on objects farther away or at infinity; the HFD is typically fairly close to the photographer 
DOF  Between half the HFD and infinity Determined by the choice of aperture, CoC, and focal length of lens 
Between HFD and infinity (the near limit is not specified
by Merklinger's method, but via the CoC) Determined by the choice of aperture, CoC, and focal length of lens 
From X to Y, according to the DoC criteria for the near limit X and the far limit Y  You can specify different DoC criteria for near and far objects and look for a "compromise" fnumber 
Role of Aperture  Determines DOF (near limit; far limit is infinity)  End result of the calculations  End result of the calculations  The fnumber is the "final result" of Merklinger's method 
Choice of Aperture  On the basis of HFD tables or calculations (or DOF scales)  On the basis of the DoC tables or individual calculations (which is easy)  On the basis of individual DoC calculations ("infinity" tables can also be used)  
Formulae  HFD formula (or approximation)  Special case DoC formulae with object distance X or Y = infinity (these formulae are much simpler than the regular ones and can be used for calculating DoC tables, from which the fnumber can be chosen)  "Regular" DoC formulae  For focus at infinity (D = infinity) the formulae can be simplified 
Formulae for DoC   

With d = f / N you get:

Note that Sy and Sy differ from D in the general case (for focus at infinity, S and d are the same) 
Legend: Sx = size of disk of confusion (object at distance X between lens and point of focus D); Sy = diameter of disk of confusion (object at distance Y between point of focus D and infinity); D = distance of point of focus; d = f/N = diameter of working aperture; f = focal length; N = fnumber (fstop); c = diameter of circle of confusion
The following table summarizes the results from this section:
Focus at H (HFD),
Object at X, Y 
Focus at Infinity 

Distance  CoC  DoC  DoC Formulae  CoC  DoC  DoC Formula 
H / 2  c (near DOF limit)  d / 2  Y > H: X < H: 
2 * c  d  S = d 
H  0 (focus)  0  c (near DOF limit)  
2 * H  < c  d  c / 2  
Infinity  c (far DOF limit)  inf  0 (focus)  
Further useful formulae > 
d = f / N  H (approx.) = f * f / (N * c) = f * d / c 
Legend: Sx = size of disk of confusion (object between lens and point of focus H); Sy = diameter of disk of confusion (object between point of focus H and infinity); H = hyperfocal distance (HFD); X, Y = distance of object (X<H, Y>H); n = multiple or fraction of ratio between focusing distance and object distance; d = f/N = diameter of working aperture; f = focal length; N = fnumber (fstop); c = diameter of circle of confusion
Merklinger published a number of "rules of thumb" (chapter 10, p. 69ff), which are direct consequences of the formulae that he derives in his paper. In the following, I cite some of them (I shortened rules, if this seemed appropriate), namely the ones that I find myself most important for my work:
Ad 1 and 2) Explanation to be provided; but see my test photos (Sony RX100 M1 (28 mm equiv.), Leica X Vario (28 mm equiv.), Leica M (Typ 240) with MRokkor 28 mm)
Ad 3) This is a consequence of Merklinger's formulae for the disk of confusion.
Ad 4) The consequences of this strategy are discussed above.
Ad 5) The focus rule can be derived from Merklinger's formulae, but needs to be explicated here (the DoC value is probably arbitrary in this case)...
Ad 6) This is derived in Merklinger's paper, and I list it here, because this is useful information in my view.
A Russian RX100 M1 user who had initially pointed me to Merklinger's approach also asked me: "What is the PRINCIPAL PRACTICAL usage (help) of this theory and why very few people use it in practice?" Before I haven't put the approach into practice, I cannot answer the first question. I also cannot give definitive answers to the second question, but at least, I can offer a few guesses here...
One reason for the lack of acceptance might be that the method is "simply" too complex for realworld use, particularly, if you want to take photos "spontaneously", as my friends once characterized their style of photographing. Merklinger only needs a few sketches and does most of the calculations in his head. For me, it looks at the moment as if you need a computer at hand to do the calculations (actually, no issues in the times of smartphones...).
And, of course, you must also be willing to think about the structure of the scene that you want to photograph, particularly about the size of the details that you want to capture (resolve). You may also have to decide, whether you set distance simply to infinity or to another value, which would require more calculations.
And last, but not least, the theory makes calculations easier if you think in fractions and multiples of the hyperfocal distance. First of all, I fear that only few photographers know of this concepts, my friends do not.
The other issue that I see is that while we are used to estimate and think in distances, the HFD is more difficult to deal with, because it is different for each fnumber. The larger, the fnumber, the smaller the HFD and the larger the DOF. That is, what most advanced photographers understand. But a larger fnumber also decreases the disk of confusion if you focus closer than infinity. Now you have two effects together: the objects can get closer and the disk of confusion gets smaller and thus, smaller objects can be resolved. Now the question is, "Is the better resolution at larger fnumbers just the result of being able to get closer to the object, or is there actually a 'practical' increase?" When I looked at my test photos, this question came up, and I still have to find an answer to it. Maybe, it is a simple as that: The larger fnumber allows you to get closer to an object (at the same "conventional" resolution) and thus, you can resolve more details. In this case, no "Merklinger theory" would be needed...
The same RX100 M1 user asked me what Merklinger's approach would deliver for the following example (or better, which would be the optimal aperture value for this): The camera should resolve blades of grass, for which a diameter of the disk of confusion of 5 mm (0.5 cm) is assumed. The question is:
Actually, this is not the "correct Merklinger question", since the point of focus always delivers optimal sharpness. Merklinger's theory is based on geometric optics and does not say anything about this. In a short paper on his method, Merklinger writes: "True, diffraction effects won't let me resolve 11 mm at 5000 ft. ...." (the resolving power of the lens and the sensor may also limit what can be resolved). Thus, this is another story and not the point here. The point here is that we have to make decisions about where we want to resolve 5 mm, either in the foreground, the background, or both (which may make it hard to find a compromise...). These are our "Merklinger" near and far limits, corresponding the the CoCbased near and far limits, but defined in the object space.
Let's say that we want to resolve objects of 5 mm diameter at X = half the distance D at which we focus (that is, in the foreground).
I distinguish the following two cases:
Here are the basics for my calculations:
If we decide that our near limit X will be at half the focusing distance D (and the far limit X extends half the focusing distance D beyond it, meaning that DOF = D), we get
With d = f / N, we get:
When we think in fractions and multiples of the focus distance, the resulting dvalues are the same for the same fraction or multiple of the focus distance, regardless of its actual value (this can be derived from the formulae above). Therefore, I decides  without loss of generality  that the focus distance D be the hyperfocal distance H, because focusing at this distance is a competitive approach to Merklinger's. This also allows me, to present some numbers not only for the DoC but also for the CoC...
Thus, in the results table below I use the hyperfocal distance H as point of focus (D) and consider the near limit X = H / 2 (or D / 2 in the general case). The basics for my calculations can be found here. On this page, I just present the results, that is, the fnumbers for a DoC of 5 mm (note that the table does not include or need a "real" distance D because the near limit is half the focus distance):
Criterion: Sx = 5 mm (DoC)  Focus at H (HFD) 
Focus at Infinity 

Camera*  Focal Length (mm)  fNumber for Sx at X = H/2 
Focus 
Near Limit 
CoC at H/2 (mm)  fNumber (f/d) 
Near Limit (CoC)  Resolving Limit (0  inf)  CoC at H/2 (mm)  
Real  Equ.  Calcul.  Nearest*  H (m)  H/2 (m)  DoC** at H/2 (mm)  Calcul.  Nearest*  H (m)  DoC (mm)  
Sony RX100 M1 1" Sensor Diffr. Limit: f/8 
10.4  28  1.04  1      5  0.011  2.08  2  4.93  5  0.022  
13.0  35  1.3  1.5      5  0.011  2.60  2.8  5.44  5  0.022  
18.5  50  1.85  1.8  17.48  8.74  5  0.011  3.70  4  7.80  5  0.022  
25.9  70  2.59  2.8  21.59  10.80  5  0.011  5.18  5.15.6  10.81  5  0.022  
37.1  100  3.71  4  31.32  15.66  5  0.011  7.42  7.18  15.68  5  0.022  
Leica X Vario APSC Sensor Diffr. Limit: f/16 
18  28  1.8  1.8      5  0.019  3.6  3.54  4.28  5  0.038  
23  35  2.3  2.5      5  0.02  4.6  4.55.1  5.27  5  0.04  
33  50  3.3  3.5  4.80  2.40  5  0.02  6.6  6.37.1  7.67  5  0.04  
46  70  4.6  4.5  5.91  2.96  5  0.02  9.2  910  10.54  5  0.04  
Leica M Fullframe Sensor Diffr. Limit: f/22 
15  15  1.5  1.41.5  5.32  2.66  5  0.03  3.0  3.2  2.38  5  0.06  
21  21  2.1  2.02.4  5.85  2.93  5  0.03  4.2  4.5  3.30  5  0.06  
25  25  2.5  2.42.8  7.39  3.70  5  0.03  5.0  5.1 (5.04)  4.16  5  0.06  
28  28  2.8  2.8  9.27  4.64  5  0.03  5.6  5.6 (5.66)  4.65  5  0.06  
35  35  3.5  3.5  11.49  5.75  5  0.03  7.0  7.18  5.76 (5.14)  5  0.06  
50  50  5.0  5.1  16.59  8.30  5  0.03  10.0  1011  8.32 (7.42)  5  0.06  
75  75  7.5  7.18  23.51  11.76  5  0.03  15.0  16  11.79  5  0.06  
90  90  9.0  9.0  30.16  15.08  5  0.03  18.0  18  15.12  5  0.06  
135  135  13.5  1416  38.10  19.05  5  0.03  27.0  28  21.44  5  0.06 
*) fnumber set to nearest "infinity" value. The fnumbers should
not be taken too seriously, in part they are "nominal", in part "exact" (sometimes,
the exact ones are given in parentheses).
**) At H/2, the DoC is d/2, at H, the DoC is 0, at 2*H, the DoC is d again,
and then it grows without limit.
*) For the calculations, only the real focal
length is relevant, not the sensor size or the camera type.
For focusing at infinity, the answer is straightforward: The aperture values are valid for any distance. If you increase the fnumber further, the disk of confusion gets smaller, and you can resolve even smaller structures. Be sure not to increase the fnumber beyond the diffraction limit.
The method also tells us that 5 mm can even be resolved if objects are hundreds of meters away (and more...). But then the angular separation should be so small that the blades are no longer be resolvable. Since the Merklinger method is based on geometric optics, it does not know anything about resolving limits caused by the media (film: grain; sensors: pixel size, Bayer filter, AA filter) and by the characteristics of real lenses  as it does not know anything about diffraction... So please take such results with a "grain of salt."
For focusing at a finite distance, we get the theoretical result that the diameter of the disk of confusion (DoC) will continually decrease from the lens to the point of focus D, where it is zero, and then increase again, until it grows above all limits. When we specify the near limit in multiples or fractions of the focus distance D, we can generalize our results and need not use "real" distances.
When I calculated the values for the finite distance, I was somewhat shocked at the beginning, because the fnumbers were so small, smaller than some cameras offer. What does this tell us? The answer is that any fnumber larger than the calculated one will do (below, the diffraction limit, of course...). One might say, the criterion is too lax, and there is room for making it more strict. If I would, for example, adopt a disk of confusion of half the size, that is, of 2.5 mm in diameter as my criterion, I could just use the fnumbers that I calculated for the infinity condition. This also showed me the way to how the tables for the infinity condition can also be used for finite distances: Just calculate or estimate (X  D) / D (it's even simpler for fractions of D) and multiply the fnumber for infinity with it  voila!
Returning to the starting question, a simple answer to a Sony RX100 M1 user might be:
Do not expect grass blades that are farther away to be resolved, because there is more to photography than geometric optics...
*) Note: Elsewhere on this Website you will find a discussion about whether "acceptably sharp" at infinity is sufficient in comparison with focusing on infinity. You will find that a decision on this question depends on your requirements for viewing images on the TV or computer screen and for printing (print size).
At the moment, I can only make some preliminary remarks. Merklinger's approach may be difficult to handle for some people, because, in the general case, it requires calculations or the use of sketches. The special (or simplified) case that distance is set to infinity is easier to handle and can be dealt with using individual calculations or using tables, much like the DOF and HFD tables.
For me is, however, more important that Merklinger's method cautions me of some of the dangers or difficulties (for example, on cameras without a distance scale) that are implied by using the HDF. The biggest danger seems to be for me that you set the distance (meant as HFD...) too short (which is easy with some lens's distance scales) so that objects at large distances (or infinity) get too fuzzy. Boone demonstrates in his article that it is easy to err "on the wrong side" (= near limit) and thus, arrive at unsatisfactory results. Erring on the far limit has much less impact, particularly for distant objects, where nothing changes. Only close objects may get a little bit fuzzier.
Merklinger's paper and Boone's article also show that the strategy to set distance at infinity and "waste" some DOF in the foreground is a "safe" strategy for landscape shots where foreground objects do not play a major role. In the end, I have used this strategy all over my photographic life, but always felt guilty of "not using the optimum", that is, the hyperfocal distance.
As both authors point out, with the hyperfocal distance approach, objects at infinity are just "acceptably" sharp. But what "acceptably" means, can vary from situation to situation. Better monitors and larger prints may require a smaller diameter of the CoC than is used in the camera (or on the distance scale of the lens) as standard. Under these conditions, your photos may look too fuzzy at infinity. Setting distance to infinity will ensure that you always get the optimum at large distances, whatever output device is being used.