# Interpolation and anisotropy

###### By Kirk Spragg

## Introduction

In addition to interpolation, Leapfrog provides two tools that give the user control over the continuity of grade in their interpolants. These are the “Global Trend” and the more advanced “Structural Trend”.

The Global Trend can be effectively used to alter the results of an interpolant. The Global Trend is suitable to use in situations where the underlying geology implies that grade is continuous in a planar direction over large distances. If this is not the case, and the underlying geology implies that direction of grade continuity varies over space, then Leapfrog’s Structural Trend is a more appropriate tool to use when modelling your deposit or ore body.

## The Global Trend

The Global Trend enables the user to set a constant trend that will be used by the interpolant. This constant trend will favour grade continuity in one direction, “Maximum”, over two others, “Intermed” (intermediate) and “Minimum”. The extent to which one direction is favoured over the others is defined by the relative sizes of the “Ellipsoid Ratios”, i.e. the direction with the largest ratio is favoured more than the others, the direction with the smallest is favoured least.

The trend directions are set from the moving plane with the maximum direction along the pitch line shown as a green arrow. The intermediate and minimum directions are depicted as purple and blue arrows.

These directions in conjunction with the ellipsoid ratios define an anisotropy ellipsoid:

**NOTE:**

The Adjust Surface dialog is used to apply anisotropy to intrusions, deposit and erosional contact surfaces. This dialog uses a different convention:

“Along Pitch” is equivalent to maximum, “Out of Plane” is equivalent to minimum and the intermediate ratio is (invisibly) held at 1. This is not a restriction as you have the same flexibility, you simply have to scale your numbers so that the Intermediate ratio is 1.

## Explanation

The inner workings of the Global Trend are simple but counter intuitive. If you remember how the interpolant works, data points that are closer to the point “X” where we want to calculate an estimate, will have more influence on the value of the estimate. Therefore, if we want to increase grade continuity in a given direction, we need away to make points that are aligned to this direction appear to be closer to “X”.

This is done using the trend or anisotropy ellipsoid to scale real distances to “anisotropic distances”. The anisotropic distances of points aligned to the maximum direction are smaller than their real distances – so they will have a larger influence on the estimated value. Since these points have a larger influence, the estimated grade that is produced by the interpolant will be more continuous in this direction. In contrast, the apparent distances of points that are aligned with the minimum direction are larger than their real distances – so they will have a smaller influence on the estimated value. Thus the estimated grade will appear to be less continuous in this direction.

## Examples:

For the purposes of illustration we consider several simple 2d examples:

- No global trend i.e. the trend ellipse is a circle.
- Global trend aligned with the X axis.
- Global trend aligned with the Y axis, i.e. the anisotropy ellipse from example 2 is rotated 90 degrees.
- Global trend aligned 45 degrees clockwise from the Y axis.

### Example 1 – No global trend

This is our basic example in 2d. The maximum and minimum ellipse ratios are set to 1, so in this case the anisotropy ellipse is in fact a circle so no anisotropy is used.

Point | X | A | B | C | D |

Grade | 5.8 | 10 | 2 | 1 | 8 |

Real distance from X | 0 | 178 | 222 | 397 | 548 |

We have a number of data points labelled A, B, C and D. We wish to estimate the grade at another point X using interpolation. The resulting estimated grade value at X is 5.8.

To determine the estimated grade value at X, interpolation gives more influence to the data that is close to X than to that which is further away. The importance assigned to all the data is points plotted below:

The points with the most influence are points A and B which have grades of 10 and 2 respectively. Both have a similar influence. It is thus not surprising that the estimated grade value of 5.8 is close to the average grade value of points A and B, which is 6.

### Example 2 – Global trend aligned with the X axis

This is our first example where anisotropy is used. The maximum and minimum ellipse ratios are set to 2 and 1 respectively. The maximum ellipse axis is aligned with the X axis.

Now instead of showing a 2d map of the positions of our data point with respect to X, we show a map of the positions of our data point with respect to X in anisotropic space. i.e. we have stretched and squashed our map so that:

- Data points aligned along the maximum ellipse axis (green arrow) are moved closer to X
- Data points aligned along the minimum ellipse axis (blue arrow) are moved further away from X

Point | X | A | B | C | D |

Grade | 7.3 | 10 | 2 | 1 | 8 |

Real distance from X | 0 | 178 | 222 | 397 | 548 |

Anisotropic distance from X | 0 | 127 | 288 | 368 | 623 |

You can see from the anisotropic map that this has the effect of squeezing in the X axis whilst stretching out the Y axis. This is the counter intuitive aspect of what happens when anisotropy is used; it is reasonable (though wrong) to assume the anisotropic space would be stretched wider along the X axis due to the shape of the ellipse. What in fact happens is the opposite, the X axis is compressed inwards to bring points that are aligned with the maximum ellipse axis closer together. Bringing these points in closer has the effect of giving them more influence over the interpolation, therefore, when a grade value is estimated at our unknown point X, the points are given more influence on the estimated grade at X.

The effect of this is illustrated in the figure below. Instead of using the real distances of data points from X to determine which points have more influence, interpolation uses the anisotropic distances. These differ somewhat from the real distance as shown in the table above. As a result, interpolation will change the influence assigned to each data point and thus the estimated grade value at X will be different as well.

Comparing the influence that interpolation has assigned the data points based on anisotropic distance versus real distance, we see that in this example, point A has more influence, but point B now has much less. As a result, the estimate tends to a value closer to the value of A, so the estimated grade value at X increases from 5.8 to 7.3.

### Example 3 – Global trend aligned with the Y axis

This example is similar to example 2, but now the maximum ellipse axis is aligned with the Y axis. The maximum and minimum ellipse ratios are set to 2 and 1.

As in the previous example, we show a map of the positions of our data point with respect to X in anisotropic space:

• Data points aligned along the maximum ellipse axis (green arrow) are moved closer to X

• Data points aligned along the minimum ellipse axis (blue arrow) are moved further away from X

Point | X | A | B | C | D |

Grade | 4.1 | 10 | 2 | 1 | 8 |

Real distance from X | 0 | 178 | 222 | 397 | 548 |

Anisotropic distance from X | 0 | 249 | 191 | 502 | 591 |

This has the effect of moving points that were aligned vertically with X closer to it. In this case there is only one such point, point B. All the other points have been moved away and their anisotropic distances are larger than their real ones.

The effect of this can be seen in the figure above. Point B is moved inwards and now has more influence while all the other points have moved out and have less influence. As a result, point B now has the most influence followed by point A. The estimated grade value at X drops from 5.8 to a lower value of 4.1 to reflect the increase in influence of the low grade at point B.

### Example 4 – Global trend aligned diagonally

Finally, let’s consider an example where the anisotropy ellipse has been rotated so that neither the maximum and minimum directions are aligned with the X or Y axis. As in the previous two examples, the maximum and minimum ellipse ratios are set to 2 and 1 respectively.

This time, the map of the positions of our points in anisotropic space is more complicated. As the anisotropy ellipse is not aligned with the X and Y axes, our map has become skewed. This effect can be seen on the grid lines, on anisotropic map, the gridlines no longer intersect at right angles.

• The maximum direction is now pointing diagonally at an angle of 45 degrees to the X axis.

• All the data points that are aligned with the maximum ellipse axis (green arrows) are moved inwards towards X.

• All the data points that are aligned along the minimum direction (blue arrow) are moved further away from X.

Point | X | A | B | C | D |

Grade | 6.4 | 10 | 2 | 1 | 8 |

Real distance from X | 0 | 178 | 222 | 397 | 548 |

Anisotropic distance from X | 0 | 191 | 296 | 309 | 390 |

The effect this has on interpolation is plotted below:

This choice of anisotropy has caused two significant changes: The high grade at point D has moved in a significant amount and thus has been given more influence; additionally the low grade at point B has moved outwards and now has less influence. The overall effect is to give the high grade points A and D more influence compared to the low grade points B and C, so compared to the first example, the estimated grade value at X increases from 5.8 to 6.4.

## Conclusion

The Global Trend is one of Leapfrog’s many tools that allow users to control the behaviour of their interpolant models. It is suitable in situations where the underlying geology implies that grade is continuous in a planar direction over large distances.

In situations where the underlying geology implies that grade continuity follows a more complex trend that varies over space, it is more appropriate to use Leapfrog’s Structural Trend to control grade continuity.