RESERVE ESTIMATION
9.2 STATISTICAL APPLICATIONS
9.2.19 Trend Surface Analysis
Trend surface mapping is a mathematical technique for computing an empirical 2D plane or 3D curves, contours, and wire meshes derived by the regression method. It provides the overall structure of spatial variation present in the data set. It is a linear function of geographic coordinates of total scattered observations; it constructs global func- tional relations and estimates new values on regular grid points. The estimated values must minimize the deviations from the trend functioning as a wide range operator. It is a global pattern recognition technique in mineral exploration.
The mathematical equation expressing the functional rela- tionship is stated as:
Zn ¼ a0þa1xþa2yþe ðfirstdegree equationÞ Zn ¼ a0þa1xþa2yþa3x2þa4xyþa5y2þe
ðseconddegree equationÞ and by similar types of equations, where:
Zn ¼ dependent variableðsample valueÞ
X; Y and ZðelevationÞ ¼ independent variablesðlocation of sample coordinatesÞ e ¼ random error component;
a0.an ¼ unknown coefficients.
The number of coefficients is dependent on the degree (NDEG) of the polynomial used, and is computed as fðNDEGþ1Þ ðNDEGþ2Þg=2. The higher degree of polynomial attempts to reduce the noise level in data minimizes the sum of squares of residuals, and allows definition of the trend in a unique manner. However, it is observed that after reaching a certain degree, the linearfit becomes static due to error propagation and the noisy na- ture of the large matrices.
TABLE 9.8 % Pb Assay Value of Five Samples Analyzed at Six Standard Laboratories n[6
m[5
N[536[30 Replicate (Lab) Sample 1 Sample 2 Sample 3 Sample 4 Sample 5
1 1.92 1.87 1.25 2.03 1.99
2 1.87 1.43 1.43 2.25 2.43
3 2.13 2.02 0.87 1.76 1.76
4 1.65 1.76 1.14 1.84 2.02
5 1.73 1.93 0.95 1.59 1.84
6 2.24 1.61 1.65 1.90 1.91
Px 11.54 10.62 7.29 11.37 11.95 ¼52.77
Px2 22.45 19.03 9.29 21.80 24.07 ¼96.66
TABLE 9.9 Results of Analysis of Variance Source of
Variation
Sum of Squares
Degree of Freedom
Mean
Squares F-Test Among
samples
2.37 4 0.59 10.17
Within replicates
1.47 25 0.058
Total variation
3.84 29
FIGURE 9.10 Analysis of the F-test from the table value at a 5% level of confidence.
In afirst-degree equation we need three normal equa- tions tofind the three coefficients:
PZn ¼ a0nþa1Xxþa2Xy PxZn ¼ a0Xxþa1Xx2þa2Xxy PyZn ¼ a0Xyþa1Xxyþa2Xy2 The equation in matrix form is:
2 64
n P
x P
P y
x P
x2 P P xy
y P
xy P
y2 3 75
2 64
a0 a1 a2
3 75 ¼
2 64
PZn PxZn PyZn
3 75
The coefficients for the best-fitting linear trend surface are obtained by solving this set of simultaneous equations.
The trend surface technique is illustrated by a 2D case study. A zinc-lead orebody extends along N42WeS42E over a strike length of 90 m. It is lens shaped and widest at the center. The width varies between 2 and 16 m with an average of 7 m. It pinches at both ends. Mineralization dips to the west and plunges 55NW. The orebody has been drilled from underground at four levels at 15 m intervals.
The mineralized intersections are predominantly zinc rich with moderate iron and subordinate amounts of lead. The grades vary between 3.96 and 11.18 with an average of 6.87% Zn, 0.00 and 5.98 with an average of 0.40% Pb, and 2.44 and 15.25 with an average of 6.05% Fe. The drill intersection values are plotted along a longitudinal section.
The dependent variables, e.g., Pb, Zn, Fe, and width, are assumed as a numerical functional relation of criteria var- iables of x (strike) and y (level) coordinates. The trend surface maps of lead and zinc at the fourth degree are generated on a longitudinal vertical section (Figs. 9.11 and 9.12, respectively) considering local coordinates of x and y.
The zinc value has been computed at every observed drill intersection point of the orebody using the fourth- degree trend surface equation. The comparison between actual sample and computed values is given inTable 9.10.
The analysis and observations from actual samples and the trend surface equation (Table 9.10) are:
Total variance (TV)¼71.80 Unexplained variance (UV)¼12.10 Explained variance (EV)¼(TVUV)¼59.70 Goodness offit (GF)¼(EV/TV)100¼83.15%
GF is a measure of efficiency of the trend equation applied, and depends on input data and inherent structure.
The extreme values will lower the GF even with a large database. GF is a useful measure for a homogeneous population, but is not appropriate for skewed or mixed types.
Hence lack of significance or high significance needs to be supported by geological interpretation. It determines the functional relationship between predictor and criteria variates.
FIGURE 9.11 Fourth-degree trend surface map of lead content of orebody 7W, Balaria deposit, Rajasthan, India.
The accuracy of map generation is centered on the midpoint of the 2D grid. The computed trend value and actual values may show a strong divergence away from the midpoint.
The margins are the most inaccurate zones. Extrapolation of the trend equation beyond the control point will result in erroneous values.
The trend surface map can indicate (1) possible conti- nuity of mineralization in depth and strike directions, (2) inherent pattern of mineralization like metal zoning and association, and (3) the grain of longer and shorter axes of variations. It may often generate false anomalies since the separation of noise into positive and negative sets is closely
FIGURE 9.12 Fourth-degree trend surface map of zinc content of orebody 7W, Balaria deposit, Rajasthan, India.
TABLE 9.10 Computed Value of Zinc Content of Orebody 7W, Balaria Deposit, Rajasthan, India Level Latitude
% Zn (Sampled)
% Zn
(Computed) Level Latitude
% Zn (Sampled)
% Zn (Computed)
343 68 6.11 6.05 288 102 4.85 4.27
343 86 7.17 7.46 263 15 6.44 6.67
343 102 11.18 10.24 263 38 8.28 8.26
343 105 10.01 10.62 263 54 6.81 7.92
343 134 4.41 4.49 263 67 8.54 7.04
314 46 6.63 6.75 263 84 4.51 4.81
314 60 6.37 6.21 250 5 7.30 7.15
314 82 7.70 7.55 250 35 8.22 8.23
314 91 8.73 7.88 250 64 4.13 4.27
314 101 7.44 7.17 317 59 6.34 6.17
288 28 6.20 6.19 286 34 6.29 6.41
288 49 7.58 6.98 330 81 6.88 7.18
288 74 8.74 7.92 309 80 6.11 7.56
288 90 5.34 6.96
Unpublished Ph.D. Thesis, Haldar, 1982.
linked and together they sum to zero. Smoothing of higher values and enhancement of lower values are an inbuilt part of the procedure. Subset trend maps show no relation to composite trend maps. In spite of those constraints, these are utilized for a generalized pattern of global variation.