Variogram Maps

[Jeffrey Yarus]

I'm having trouble understanding the angles reported from the Variogram Map. Here is an example of one I created:

modelvm <-
vmap.auto(vm,
flag.noreduce = TRUE,
lower = c("M1V1=0.2"),
upper = c("M1V1=0.5"),
struct = melem.name(c(1, 2, 3)))

I added the "lower" and "upper" arguments to be consistent with my model created earlier. The image I am uploading is the result. Intuitively, the max direction of continuity seems correct. The "spike" may be an indication of a fault (?) (it seems to offset the principle ellipse). But I'm trying to identify what the Exponential component is that has an angle of 16.4 degrees. The principal contribution seems to come from the Spherical model (~ -160 degrees, although I am not sure why it reports it as a -160 degrees.
Here are the stats that came back:

Covariance Part
---------------
Nugget Effect
- Sill = 0.200
Exponential
- Sill = 2.127
- Ranges = 561.358 14983.560
- Theo. Ranges = 187.386 5001.636
- Angles = 16.444 0.000 <- HOW DO I INTERPRET THIS? I know this is a smaller structure, but is this something which I simply cannot see (too small)? Rotation Matrix
[, 0] [, 1]
[ 0,] 0.959 -0.283
[ 1,] 0.283 0.959
Spherical
- Sill = 2.922
- Ranges = 57706.480 17955.536
- Angles = -159.860 0.000 <- HOW TO INTERPRET THIS? This is consistent with the Max dir of continuity. Why the "-" sign, why not just +20 degrees???
- Rotation Matrix
[, 0] [, 1]
[ 0,] -0.939 0.344
[ 1,] -0.344 -0.939
Total Sill = 5.249

Drift Part

[Didier Renard]

From what I see, you have joined two figures: the left one seems to be an experimental variogram map, the right hand side one seems to be the fitted model represented in terms of variogram map.
Obviously I cannot judge the quality of the variogram map calculations as this depends on the data set that I don't have. Assuming this has been performed reasonably correctly (I am thinking to the choice of the calculation parameters), now the problem is to fit it.
This is achieved in a model whose contents is displayed next and this requires some complementary information.

The structure is composed of three basic elements; a nugget effect, a short range exponential and a large range spherical. As expected, the nugget effect component is not visible on a variogram map (it would be a discontinuity at the origin, i.e. in the center of the map).
We can discuss about the two other components.

The exponential component is displayed as follows:
- ranges (let us focus on practical ones, discussion would be the same with theoretical ... up to the conversion factor for the exponential).
Ranges = 561 / 14983
- rotation angles for the anisotropy ellipse: given as a list of 2 numbers (the second "angle" reported is dummy: it has just been put there in order to follow the rule that we need as many values as the dimension of the space to express a rotation angle. Obviously this not valid for 1D [where there is no rotation] and in 2D where a single value is sufficient)
angle = 16.4 degrees

For the spherical:
- ranges = 57706 / 17955
- angle = -159 degrees

How should we read this?

Let us describe this through the exponential case.
The ranges are expressed as a pair of values: the small one comes first, the large one is the second number. Let us say that the short range (first value) is 561 and the long range (second value) is 14983.
What we have to realize is that the first value (the small range in the exponential case) corresponds to the direction pointed by the angle (16.4 degrees). In other words, the short range is oriented in the direction 16.4. You will double check this if you remember that the angle definition is provided in trigonometrical convention (counted counter clockwise from East). You can check (more easily) that the long range (14983) is oriented along the direction orthogonal to the direction of the small axis (i.e. 16.4 + 90 = 106.4 degrees).

Now let us consider the spherical component to check our recent knowledge.
The ranges are expressed with the long range first (57706). Its direction corresponds to the angle, i.e. -159.8. As a matter of fact, we can check that the elongated structure (for the long scale component) is oriented mainly in east_west direction.

Note that the rotation convention is different from ISATIS one ... but there is no ambiguity as in the many possible geologist definitions (from North eastwards, from North Westwards, from South for southern hemisphere geologists, ...). On the contrary, there is only ONE trigonometric convention.

I agree that this description could have been improved with 3 main improvements:
- always start with the longest range (rather than letting the automatic procedure to choose its resulting expression)
- define angles in a different manner: today angles are expressed as lying between -180 and 180. It may be more relevant to express tham between 0 and 260
- as the angle specifies an orientation (not a direction), the angle -159.86 could be also expressed as 180-159=21 degrees (may be easier to spot).

This will be improved in future versions and certainly in gstlearn.

Regarding your comment on the lack of elliptic shape for the short range component, you must remember that this ellipse is represented on a grid basis (discretized). You could certainly improve its graphic expression by tuning a grid with a finer definition.

Hope this will help