Sequential Indicator Simulation (SISIM)

Indicator kriging (§8.2) gives a probability map. Sequential Indicator Simulation (SISIM) is the SIMULATION-SIDE counterpart: at every unsampled cell, build the local conditional probability via IK, DRAW from it, write the draw into the cell, and ADD the drawn value to the conditioning set for downstream cells. The result is a set of equally-likely categorical realisations whose pointwise mean converges to the IK probability map (the central asymptotic guarantee of any sequential simulation).

The SISIM algorithm — binary case

For a binary facies indicator $I(x) \in {0,1}$ with hard data ${I_\alpha}$:

• Initialise: condition set $\mathcal{C}$ = hard data; visit list $\mathcal{V}$ = all unsampled cells in random order.
• For each cell $x_v \in \mathcal{V}$:

Solve the simple-IK system on $\mathcal{C}$ to obtain $\hat{P}(I(x_v) = 1 \mid \mathcal{C})$. - Draw $u \sim \text{Uniform}(0,1)$; set $I(x_v) = 1$ if $u < \hat{P}$, else $0$. - Add $(x_v, I(x_v))$ to $\mathcal{C}$.

- **Repeat** until $\mathcal{V}$ is empty — one realisation complete. - **Restart** with a new random seed for the next realisation.

Multi-facies SISIM

For $K$ facies, define one indicator per facies $I_k(x) = \mathbb{1}[\text{facies}(x) = k]$. At each cell:

• Run IK on each facies indicator to get $\hat{P}_k = \hat{P}(I_k(x_v) = 1 \mid \mathcal{C})$.
• Apply ORDER-RELATION CORRECTION (sweep or projection) to enforce $\hat{P}_k \in [0,1]$ and $\sum_k \hat{P}_k = 1$.
• Draw a category from the discrete distribution ${\hat{P}_1, \dots, \hat{P}$K}{P^1​,…,P^K​}: pick $u \sim \text{Uniform}(0,1)$ and select the $k$ such that $\sum${j < k} \hat{P}j \le u < \sum{j \le k} \hat{P}_j.
• Add the drawn facies to $\mathcal{C}$.

Convergence: SISIM mean → IK map

Let ${I^{(r)}}_{r=1}^R$ be R SISIM realisations and define the empirical pointwise probability:

As $R \to \infty$, $\bar{P}${\text{sim}}(x) \to \hat{P}{\text{IK}}(x) at every cell. This is the simulation-side check: if your SISIM realisations average to something OTHER than the IK map, your code has a bug or your visit order/neighbourhood is too restrictive.

Why SISIM and not just IK?

• Discrete worlds, not just probabilities: each realisation is one possible reality — usable as input to a flow simulator, mining-block model, contaminant-transport code.
• Joint uncertainty: cell-by-cell IK probabilities are MARGINAL; SISIM realisations capture JOINT structure (e.g., the probability that two cells are BOTH sand differs from the product of their marginals).
• Categorical / threshold variables: facies, ore vs waste, land-use classes — SGS's Gaussian assumption is inappropriate; SISIM handles them natively.
• Distribution-free: no Gaussian or lognormal assumption needed; the conditional CDF is built empirically from indicators.

Practical pitfalls

• Visit-order artifacts: a poorly designed visit path (e.g., raster scan) can produce systematic biases. Random paths with multiple-grid refinement are standard.
• Search-neighbourhood choice: too small ⇒ realisations look like noise around the prior; too large ⇒ correlated artifacts and computational cost. Typical: 16–32 conditioning points per cell within an ellipsoid covering 1–2 indicator-variogram ranges.
• Order-relation propagation: in multi-facies SISIM, violations at one cell propagate via the conditioning set; per-cell correction (not just at the end) is required.
• Connectivity reproduction: SISIM honours two-point statistics (indicator variograms) but NOT multipoint connectivity (e.g., channel meanders). For complex connectivity see MPS / SNESIM in §9.

Try it

• Defaults: indicator range = 5.0, 8 realisations. The TOP canvas shows 8 SISIM realisations stacked (one per row): red cells are sand, blue cells are shale. Dashed vertical lines mark the 8 hard-data positions. Note the realisations DIFFER from each other but ALL pass through the same hard-data points.
• Look at the BOTTOM canvas: solid blue is the IK probability map P(sand | data), red dashed is the empirical SISIM mean across the 8 realisations. They should be close but not identical.
• Increase realisations to 32. The red dashed curve (SISIM mean) snaps closer to the blue solid (IK probability). The reported mean |P_sim − P_IK| drops by roughly √(4) ≈ 2×.
• Increase the indicator-variogram range to 10.0. Realisations become MORE CONTINUOUS: longer connected sand and shale runs. With a short range (1.5), realisations look almost like random shuffling around the prior fraction.
• Click Resample (new seed). All realisations change, but the IK probability stays fixed (it does not depend on the random seed). This is the simulation-vs-estimation contrast in action.

A reservoir engineer needs 100 facies realisations conditioned to 30 well data to feed a flow simulator. Why is SISIM (and not just IK on the indicators) the right tool? Name one diagnostic you would compute on the 100 realisations before handing them off.

What you now know

SISIM extends indicator kriging from estimation (probability map) to simulation (categorical realisations) via the sequential paradigm: visit unsampled cells in random order, IK to get the local conditional CDF, draw from it, add to conditioning set. Each realisation is one possible reality; their average reproduces the IK map. SISIM is the standard tool for binary or multi-category facies modelling with two-point conditioning. For complex connectivity (channels, fractures) the multipoint methods of §9 are required.

Mark section complete →

References

• Journel, A.G., Alabert, F. (1989). "Non-Gaussian data expansion in the Earth Sciences." Terra Nova 1(2), 123–134. (Sequential indicator framework.)
• Deutsch, C.V., Journel, A.G. (1998). GSLIB, 2nd ed. Oxford. (sisim source code + algorithmic notes.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford. (SISIM applications.)
• Chilès, J.-P., Delfiner, P. (2012). Geostatistics: Modeling Spatial Uncertainty, 2nd ed. Wiley. (Comparative analysis of sequential simulators.)
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford. (SISIM in reservoir-modelling workflows.)