Quantile Map: Cumulative Distribution Diagnostics, Outlier Detection, and Geostatistical QC
A quantile map is a display built from the quantile values of a data set, meaning the values are ranked and assigned a cumulative probability position rather than plotted on their raw measurement scale, so that the analyst can examine the shape and behaviour of a variable's distribution. In geostatistics it is used as a quality-control and exploratory tool to reveal problems with the distributions of the variables that go into spatial estimation and simulation, problems that a simple posting of raw values can hide. A quantile is a cut point that divides ordered data into intervals of equal probability: the median is the 0.5 quantile, the quartiles fall at 0.25 and 0.75, deciles at every 0.1, and percentiles at every 0.01. By transforming each measured value into its position on the cumulative distribution and mapping or plotting that position, the analyst can immediately see skewness, heavy tails, outliers, gaps, and multiple populations that would distort a variogram or bias a kriged estimate. This matters because the workhorse methods of reservoir geostatistics carry strong distributional assumptions. Ordinary kriging is a best linear unbiased estimator that performs poorly when a variable is strongly skewed, and sequential Gaussian simulation, the standard tool for generating equiprobable property realizations, requires that the variable first be converted to a standard normal distribution through a normal-score transform, which is itself a quantile-to-quantile mapping. Petrophysical properties in the Western Canadian Sedimentary Basin almost never arrive normally distributed. Porosity in the Cardium or Viking is often near-symmetric but truncated at physical limits, while permeability is classically log-normal, spanning several orders of magnitude, so a quantile view exposes the long right tail that demands a logarithmic or normal-score transform before modelling. A quantile map can also unmask a mixed population, where two facies, for example a clean sand and a shaly interbed, have been lumped into one variable and produce a bimodal or kinked cumulative curve; that diagnosis tells the geomodeller to split the data by facies before estimating, a decision that materially changes the resulting volumetrics and the uncertainty attached to a reserves estimate reported under Alberta Energy Regulator and National Instrument 51-101 frameworks. Used early in a study, the quantile map prevents the classic error of running a sophisticated simulation on data whose distribution silently violates the method's assumptions.
Key Takeaways
- Rank-based display: A quantile map plots values by their cumulative-probability position rather than their raw scale. The median is the 0.5 quantile, quartiles fall at 0.25 and 0.75, and percentiles at every 0.01. This ranking exposes distribution shape that raw value postings conceal, making it a front-line diagnostic in geostatistics.
- Reveals distribution problems: The primary purpose is quality control, surfacing skewness, heavy tails, gaps, outliers, and mixed populations before they corrupt a spatial model. A kink or step in the cumulative curve often signals two facies lumped into one variable, telling the modeller to split the data by rock type first.
- Feeds the normal-score transform: Sequential Gaussian simulation requires a standard normal input, achieved by mapping each value to its equivalent normal quantile. The quantile view is therefore not just diagnostic but the direct basis of the transform that makes simulation valid, and the back-transform that returns results to real units.
- Permeability is log-normal: WCSB permeability in Cardium, Viking, and Montney rocks spans several orders of magnitude with a long right tail. A quantile map makes that tail obvious, signalling that a logarithmic or normal-score transform is mandatory before variogram modelling or kriging to avoid a handful of high values dominating the estimate.
- Protects reserves estimates: Running kriging or simulation on data that silently violate distributional assumptions biases porosity and permeability fields and the volumetrics built on them. A quantile check early in a study guards the integrity of reserves reported under AER directives and National Instrument 51-101 disclosure rules.
Reading Skew and Outliers on a Quantile Display
On a quantile plot a perfectly normal variable traces a straight line against the theoretical normal quantiles, so any departure is meaningful. A concave-up curve indicates positive skew, the signature of permeability and of many net-pay thickness data sets, while points that lift away from the trend at the high end flag outliers that may be measurement errors, fractures, or a genuinely different rock type. A flat step in the middle reveals a gap where no data fall, and a distinct change in slope partway up marks the boundary between two populations. Each pattern points to a specific corrective action before estimation begins.
From Quantile Diagnosis to Normal-Score Simulation
Once a quantile map confirms a variable is non-normal, the geomodeller applies a normal-score transform, which is a quantile-matching operation: each datum is replaced by the value from a standard normal distribution that shares its cumulative probability. Variogram modelling and sequential Gaussian simulation then run in normal-score space, where the Gaussian assumptions hold, and every realization is back-transformed through the same quantile relationship to recover real porosity or permeability units. Skipping the quantile check risks transforming a hidden bimodal variable and propagating that error through hundreds of realizations and the P10, P50, and P90 volumes drawn from them.
Fast Facts
The quantile transform sits at the heart of why a single high permeability streak can wreck an untransformed estimate. Permeability in a tight WCSB reservoir can range from under 0.01 millidarcy in the matrix to several hundred millidarcies in a fractured or vuggy interval, a span of more than four orders of magnitude. On a raw arithmetic scale a handful of those high values dominate the mean and the variogram, but viewed as quantiles they occupy only the top few percent of the distribution, exactly the warning a quantile map is designed to make impossible to overlook.
Related Terms
A quantile map is one of several quality-control tools that precede Kriging, the linear estimator most sensitive to skewed input. It is the basis of the normal-score transform required by Sequential Gaussian Simulation. The distribution behaviour it reveals shapes the Variogram, the model of spatial continuity, and it most often flags problems in Permeability, the classically log-normal property that demands transformation before modelling.
Real-World WCSB Scenario: Viking Permeability QC in Central Alberta
A geomodelling team building a static model of a Viking pool in central Alberta posted core permeability into a quantile display before running simulation. The cumulative curve showed a sharp slope change near the 0.8 quantile, revealing two populations: a clean upper Viking sand and a tighter, bioturbated lower interval that had been merged in the input file. Modelling them together would have smeared high matrix permeability into the tight zone.
The team split the data by facies, transformed each population separately, and re-ran sequential Gaussian simulation. The revised model lowered the P50 connected pore volume in the tight interval and tightened the uncertainty band, changing the recommended infill well count and saving an estimated CAD 4 million in deferred drilling on rock that would not have produced as first modelled.