SNESIM and the role of training images

SNESIM (Single Normal Equation SIMulation, despite the misleading name) is Strebelle's 2002 operationalisation of multipoint geostatistics. The algorithm replaces the variogram-driven kriging system of SISIM with a PATTERN LOOKUP from a training image (TI). At each unsampled cell, gather the conditioning values at template offsets, find the matching patterns in the TI, and draw the central value from the empirical conditional distribution. The training image plays the dual role of (a) the multipoint statistics catalogue and (b) the spatial PRIOR.

The SNESIM algorithm in five steps

• Choose a template T: a set of $k$ relative offsets around a central cell. In 1D you might use T = {-2, -1, +1, +2}; in 2D a + pattern of 5 cells, or a 3×3 box of 9 cells, or larger.
• Scan the TI with T: at every position where the full template fits in the TI, record the pattern of $k+1$ values (template offsets + central cell). This populates the empirical multipoint distribution.
• Build a search tree (production codes): index the patterns by their $k$ template values so the conditional probability lookup is O(log) instead of O(N_TI). Pedagogical codes can use a flat lookup table over all $2^k$ patterns for small $k$.
• Sequential simulation: visit unsampled cells in random order. At each cell, gather the currently-conditioned values at the template offsets; some may be hard data, some previously simulated, some still undefined.
• Conditional draw: look up the pattern in the table; for undefined offsets, MARGINALISE by summing over all patterns consistent with the defined offsets. Draw the central cell from the resulting conditional distribution. Add the cell to the conditioning set for downstream cells.

The training image is the prior

The TI is to SNESIM what the variogram model is to SISIM: the implicit specification of the spatial structure. Common TI sources:

• Modern analogues: digitised aerial photos of a present-day fluvial system, deltaic outflow, dune field. The geological assumption is that the analog is structurally similar to the unknown buried target.
• Outcrop digitisations: ground-truth maps of exposed equivalents in nearby (or geologically similar) basins.
• Object-based simulations: run a quick object-based model (sinuous channels in shale) at high resolution and use the output as the TI for a finer-scale or differently-conditioned MPS run.
• Expert sketches: a geologist's hand-drawn cartoon of the expected facies geometry, digitised onto a regular grid.

The choice of TI is the single most important modelling decision in an MPS workflow. A wrong TI gives geologically wrong realisations — the variogram-derived guarantees of SISIM are replaced with "garbage in, garbage out" for MPS.

Search-tree and multigrid extensions

Production SNESIM uses two extensions absent from the pedagogical 1D demo:

• Search tree: instead of exhaustively comparing each cell's conditioning to all TI patterns, store the patterns in a tree indexed by template offsets. Lookup is O(log N_TI) and reuses structure for common prefixes. Memory cost: substantial; modern codes (IMPALA, DeeSse) optimise this.
• Multigrid simulation: simulate on a coarse subgrid first (every 4th cell, then every 2nd, then every cell). Coarse-grid simulation captures large-scale structure using a coarse template; finer grids refine the details. Without multigrid, large-scale structures often get fragmented because the template is too small to span them.

Practical pitfalls of SNESIM

• TI quality: a bad TI is irrecoverable. The TI must be representative; if your target reservoir is dominated by channels with sinuosity ratio 1.8 and your TI has sinuosity 1.2, your realisations will inherit the wrong sinuosity.
• Stationarity assumption: SNESIM assumes the TI's multipoint statistics are spatially invariant. Trends or zones in the target require either multiple TIs (one per zone) or AUXILIARY VARIABLES that bias the lookup at each cell.
• Template size choice: too small — misses large-scale structure (channels look fragmented); too large — combinatorial explosion of patterns, sparse counts per pattern, weak inference.
• Computational cost: SNESIM is slower per cell than SISIM. With multigrid + search-tree it scales to ~10⁶ cells in minutes, but very large grids may require simplified algorithms (FILTERSIM, IMPALA, MS-CCSIM).
• Sparse pattern counts: rare patterns in the TI yield noisy conditional estimates. SNESIM extensions (TI replication, prior smoothing) mitigate this.

Try it

• Defaults: TI = "Long bands (4×15 + gap 5)". The top canvas shows the 200-cell training image. The pattern lookup table (bottom canvas) shows P(center = 1 | conditioning) for each of the 16 4-bit patterns; bars in green show the conditional probability inferred from the TI. The middle canvas shows 6 SNESIM realisations (with 8 hard data); each one reproduces the band structure of the TI.
• Switch the TI to "Periodic (period 8)". The pattern table changes — only certain patterns are common in the TI, so most table entries collapse to extreme values (0 or 1). The realisations now look periodic too: 4 cells of 1, 4 cells of 0. The hard data slightly perturb the period locally, but the global structure follows the TI.
• Switch to "Random run-lengths". The pattern table shows more uniform probabilities across patterns; the realisations look less structured but still respect run-length statistics from the TI.
• Switch to "Random Bernoulli p=0.6". This TI has NO structure beyond the marginal; the pattern table is approximately constant at 0.6 across all patterns. SNESIM realisations look like Bernoulli noise — effectively the same as SISIM. The TI's richness controls the simulation's richness.
• Click Resample with the long-bands TI. The realisations vary because of the random visit order + Bernoulli draws, but all reproduce the band structure. Hard data are honoured exactly.
• Compare the pattern counts (numbers under the bars). Common patterns have many counts and reliable probabilities; rare patterns (e.g., 1010 in the long-bands TI) have few counts and noisy probabilities. This is the sparse-pattern-count pitfall in miniature.

A consultant proposes SNESIM with a TI from a modern Mississippi-style braided river system for a Pliocene deltaic-distributary reservoir. Why might this TI be inappropriate? What three questions would you ask before accepting it?

What you now know

SNESIM replaces the variogram-driven kriging system with a pattern-lookup table built from a training image. The TI is the empirical multipoint statistics catalogue AND the spatial prior. Realisations reproduce the TI's multipoint structure, not just its variogram. The algorithm scales to 2D/3D via search trees and template multigrid. The dominant practical pitfall is TI quality: a wrong or unrepresentative TI gives geologically wrong realisations. The next section (§9.3) introduces FILTERSIM, which generalises SNESIM to continuous patterns via filter-bank pattern matching.

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 SNESIM paper.)
• Caers, J., Zhang, T. (2004). "Multiple-point geostatistics: a quantitative vehicle for integrating geologic analogues into multiple reservoir models." In AAPG Memoir 80. (Practical SNESIM workflows.)
• Boucher, A. (2009). "Considering complex training images with search tree partitioning." Computers & Geosciences 35, 1151–1158. (Search-tree implementation details.)
• Hu, L.Y., Chugunova, T. (2008). "Multiple-point geostatistics for modeling subsurface heterogeneity: A comprehensive review." Water Resources Research 44, W11413.
• Mariethoz, G., Caers, J. (2014). Multiple-point Geostatistics. Wiley-Blackwell. (Definitive reference; full SNESIM treatment.)