From two-point to multipoint: motivation

Parts 3–8 built a powerful framework around the VARIOGRAM — a two-point statistic measuring average squared differences between PAIRS of cells at a given lag. The kriging family (OK, SK, UK, KED, co-kriging) and the simulation family (SGS, SISIM, TGSIM, plurigaussian) are all variations on the theme of "use the variogram to drive estimation/simulation". This is the TWO-POINT WORLD. It captures everything that pair correlations can encode — and nothing more. The motivation for Part 9 is to expose the limits of this world and introduce the multipoint framework that breaks past it.

The two-point ceiling, demonstrated

Consider two binary patterns on the same 1D grid with the same global proportion of 1s:

• Pattern A: structured bands — four wide stripes of 1s separated by gaps of 0s.
• Pattern B: a constrained permutation of A's cells, scrambled but with variogram approximately matched to A.

An empirical variogram check WILL NOT DISTINGUISH them. Yet to the eye they look totally different: A has obvious structure, B looks random. A 3-bit pattern histogram (count how often each of the 8 patterns 000, 001, ..., 111 appears in 3 consecutive cells) immediately shows the difference: A has many 111 and 000 runs (band interiors), while B has many alternating 010, 101 patterns.

The lesson: any spatial structure that depends on TRIPLETS or higher tuples of cells is invisible to a variogram. This is not a bug in the variogram; it is the variogram's fundamental information content.

Multipoint statistics — the right tool for the right job

A MULTIPOINT STATISTIC is the frequency distribution of patterns observed in a multipoint template T (a set of $k$ cells with relative offsets). For a binary field with template of size $k$, the pattern catalogue has $2^k$ possible patterns. The empirical multipoint statistic at template T is the histogram of observed patterns in the field at all valid offsets of T.

The variogram is the special case $k=2$, restricted to a single offset. Multipoint statistics with $k=3, 4, 5, \ldots$ progressively capture finer geometric structure: 3-point reveals run-length patterns; 5-point reveals short curvature; 9-point reveals shape templates; 25-point reveals object templates.

The training image as catalogue

For a multipoint statistic to be USABLE, you need a source of pattern frequencies. With small grids and very rich data you could try to estimate them from the data itself, but in practice you need a TRAINING IMAGE (TI) — a fully specified field (digitised modern analog, outcrop, simulation, expert sketch) whose pattern frequencies stand in for the unknown frequencies of the unknown truth. The TI plays the role the variogram model plays in two-point methods: it is the spatial PRIOR.

Strebelle (2002) operationalised this idea with the SNESIM algorithm (§9.2): use the TI to populate a SEARCH TREE of patterns, then sequentially simulate by finding compatible patterns at each cell. FILTERSIM (Zhang 2006, §9.3) extends to continuous patterns via filter banks. Modern generative ML approaches (GANs, diffusion, §9.4) bypass the explicit pattern catalogue and learn the distribution implicitly.

When do you NEED multipoint?

The cost of multipoint

Three things change when you move from two-point to MPS:

• A training image is required. No TI ⇒ no MPS. Sourcing or constructing a quality TI is itself a modelling task that can dominate the workflow effort.
• Stationarity assumptions are stronger. The TI's multipoint statistics must be assumed representative of the unknown truth — equivalent to assuming both stationarity AND a fixed multipoint model. Trends and non-stationarity require explicit handling (multiple TIs by zone, or proportion auxiliary fields).
• Computational cost grows. Pattern matching at each cell, search-tree construction, and template-size choices all add cost. Modern MPS codes are heavily optimised but still slower than SISIM at the same grid size.

Try it

• Defaults: 800 variogram-matching iterations, seed 13. The TOP canvas shows the structured truth (4 bands of width 8 separated by gaps of 6) and a constrained shuffle. The BOTTOM LEFT plot shows the indicator variograms of both patterns — they are nearly identical. The BOTTOM RIGHT shows the 3-bit pattern histograms — completely different.
• Increase variogram-matching iterations to 3000. The shuffle becomes a tighter match to the truth's variogram (variogram mismatch readout decreases), but the 3-bit pattern histogram mismatch remains large. No matter how many iterations you run, the shuffle cannot recover the structured pattern from the variogram alone.
• Set iterations to 0. The shuffle is now a fully random permutation — variogram mismatch is highest. As iterations grow, variogram mismatch drops but pattern mismatch stays large. This is the central pedagogical point.
• Click Resample. The shuffle changes but the qualitative story holds: variogram alone is insufficient.
• Look at the "long-run count" readout: the truth has 4 long runs of width ≥4. The shuffle has 0. The variogram check passed; the long-run count diagnostic immediately reveals the structural mismatch.
• Now imagine extending this 1D demo to 2D: the truth might be a sinuous fluvial channel; the shuffle might be a noisy field with the same variogram. Visually the channel structure would be entirely absent in the shuffle. This is the actual problem MPS solves.

A consultant gives you a SISIM-realisation deck for a fluvial reservoir and shows that the empirical variograms match the model. You stare at the realisations and the channels look wrong — broken into fragments instead of continuous sinuous bodies. Before recommending MPS, what THREE diagnostics would you compute on the realisations to confirm the failure is multipoint (not, say, an order-relation issue)?

What you now know

The two-point world is bounded by what pair correlations can encode. Multipoint structure (curvature, connectivity, channel meanders, dendritic patterns) requires multipoint statistics, which in practice means MPS conditioned on a training image. The next sections explore SNESIM (§9.2, search-tree pattern matching), FILTERSIM (§9.3, filter-bank continuous patterns), generative ML (§9.4, GANs and diffusion), and the method-choice decision tree (§9.5).

Mark section complete →

References

• Strebelle, S. (2002). "Conditional simulation of complex geological structures using multiple-point statistics." Mathematical Geology 34(1), 1–21. (The foundational MPS paper.)
• Guardiano, F.B., Srivastava, R.M. (1993). "Multivariate geostatistics: beyond bivariate moments." In Geostatistics Tróia '92. Kluwer. (Multipoint statistics for the first time in geostatistics.)
• Caers, J. (2011). Modeling Uncertainty in the Earth Sciences. Wiley. (Beyond two-point statistics.)
• Mariethoz, G., Caers, J. (2014). Multiple-point Geostatistics: Stochastic Modeling with Training Images. Wiley-Blackwell. (Comprehensive MPS reference.)
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford. (When to use MPS vs two-point methods.)