Anisotropic ellipsoids and the search ellipse

The §4.1 and §4.2 variogram models all share an ISOTROPY assumption: the variogram depends only on the scalar lag distance $h = |\mathbf{h}|$, not on the direction of the lag vector $\mathbf{h}$. Real earth-science fields rarely respect that assumption. A sedimentary deposit has a depositional dip — porosity varies more slowly along strike than across it. A fractured aquifer has a dominant joint orientation — permeability is directional. A glacial moraine has a flow direction. A pit-slope grade-control campaign in an alluvial gold deposit has stream-channel orientations preserved in the high-grade zones. In every case, the empirical variogram you computed in Part 3 looked different along different directions: shorter range along the minor direction, longer range along the major direction. §3.4 quantified that observation as a 2-D anisotropy ellipse with three numbers — $\theta$, $a_{\mathrm{maj}}$, $a_{\mathrm{min}}$ — by fitting six or eight directional variograms simultaneously.

§4.3 closes the loop: it takes those three (or, in 3-D, six) numbers and turns them into a permissible anisotropic variogram model $\gamma(\mathbf{h})$ that depends on the full lag VECTOR, not just its scalar length. The construction is geometrically elegant — wrap any isotropic family from §4.1–§4.2 inside a coordinate transform that maps the anisotropy ellipsoid to a unit sphere — and computationally cheap: a rotation matrix, a diagonal scaling, and one call to the iso-family code you already have. §4.3 also introduces a SECOND geometric object that lives downstream of the variogram but is intimately related to it: the search ellipse (in 2-D) or search ellipsoid (in 3-D) used by kriging in Part 5 to decide which neighbourhood samples enter the local linear system. The search ellipse shares its orientation and aspect ratio with the variogram ellipse but is sized differently. This section develops both, the geometric relationship between them, and the practical sizing rules that make kriging numerically stable and statistically efficient.

Building the anisotropic variogram from an isotropic family

Start with a permissible isotropic family $\gamma_{\mathrm{iso}}(h)$ from §4.1 — Spherical, Exponential, or Gaussian — written as a function of a scalar non-negative lag $h$. The anisotropic generalisation is

where $\mathbf{h} \in \mathbb{R}^d$ is the LAG VECTOR (not just its length) and $\mathbf{A}$ is a $d \times d$ rotation–scaling matrix that maps the anisotropy ellipsoid to a unit sphere. The composite quantity $h' = |\mathbf{A},\mathbf{h}|$ is called the reduced distance: in the original coordinates $\mathbf{h}$ ranges over the anisotropy ellipsoid; in the $\mathbf{A}$-transformed coordinates the same locus is the unit sphere $h' = 1$. The variogram is then effectively isotropic IN THE TRANSFORMED FRAME. This is the standard construction in Goovaerts 1997 §4.3, Deutsch & Journel 1998 §III.4, Chilès & Delfiner 2012 §2.5.

The construction preserves permissibility automatically. If $\gamma_{\mathrm{iso}}$ is conditionally negative definite, then so is $\gamma(\mathbf{h}) = \gamma_{\mathrm{iso}}(|\mathbf{A},\mathbf{h}|)$ — a coordinate transform of a permissible variogram is itself permissible (Christakos 1984; Cressie 1993 §2.5). You can use Spherical, Exponential, Gaussian (with the §4.1 caveats), or the nugget-augmented composites of §4.2, plug them into the anisotropic wrapper, and the result is still a valid permissible variogram. The Part 5 kriging machinery accepts it without modification — the only change is that every $\gamma$-evaluation inside the kriging matrix now passes through the $\mathbf{A}$ transform first.

The 2-D form

In 2-D the matrix $\mathbf{A}$ decomposes into a rotation and a scaling:

where $\theta$ is the azimuth of the major axis (measured from a chosen reference direction — usually east in mathematical convention; usually north in geological convention, with software-specific sign conventions; ALWAYS document explicitly). Apply $\mathbf{A}$ to a lag vector $\mathbf{h} = (h_x, h_y)$ in two steps. First rotate: $(u, v) = (\cos\theta,h_x + \sin\theta,h_y,;-\sin\theta,h_x + \cos\theta,h_y)$ — $u$ is the lag-vector component along the major direction, $v$ is the component along the minor direction. Then scale: divide $u$ by $a_{\mathrm{maj}}$ and $v$ by $a_{\mathrm{min}}$. The reduced distance is

The locus of lag vectors with $h' = 1$ is an ELLIPSE in the original $\mathbf{h}$ plane, centred at the origin, with semi-major axis $a_{\mathrm{maj}}$ pointing along $\theta$ and semi-minor axis $a_{\mathrm{min}}$ perpendicular. Take $\theta = 0$ to recover the axis-aligned case $h' = \sqrt{(h_x/a_{\mathrm{maj}})^2 + (h_y/a_{\mathrm{min}})^2}$ — exactly the standard ellipse equation. Take $a_{\mathrm{maj}} = a_{\mathrm{min}} = a$ to recover the isotropic case $h' = |\mathbf{h}|/a$ — the variogram depends only on the rescaled scalar distance, with the ellipse degenerating to a circle of radius $a$.

For the Spherical family — the most common practitioner choice — the anisotropic variogram is

The variogram reaches the sill EXACTLY on the ellipse $h' = 1$ — that is, at distance $a_{\mathrm{maj}}$ along the major axis, at distance $a_{\mathrm{min}}$ along the minor axis, and at intermediate distances along oblique directions following the directional-range formula $a(\phi) = 1/\sqrt{(\cos(\phi-\theta)/a_{\mathrm{maj}})^2 + (\sin(\phi-\theta)/a_{\mathrm{min}})^2}$. Inside the ellipse, the variogram climbs gently along the major direction (the cubic polynomial in $h'$ rises slowly along $u$ because $a_{\mathrm{maj}}$ is large) and climbs steeply along the minor direction (the same polynomial in $h'$ rises rapidly along $v$ because $a_{\mathrm{min}}$ is small).

Visualising the 2-D anisotropy ellipse — first widget

The first widget for §4.3 makes the geometry tangible. Set the major-axis azimuth $\theta$, the major range $a_{\mathrm{maj}}$, the minor range $a_{\mathrm{min}}$, and the sill $c$. The left panel draws the anisotropy ellipse with its major and minor axes labelled; the right panel plots two γ curves on a common axis — γ along the major direction (orange, gentle climb) and γ along the minor direction (pink, steep climb) — with a horizontal sill line and two vertical range markers at $h = a_{\mathrm{maj}}$ and $h = a_{\mathrm{min}}$.

Three things to do with this widget. First, set $\theta = 0$, $a_{\mathrm{maj}} = 0.5$, $a_{\mathrm{min}} = 0.1$ (a 5 : 1 anisotropic field oriented along the $+x$ axis). The ellipse looks like a flat horizontal rugby ball. The major γ curve climbs slowly and reaches the sill at $h = 0.5$; the minor γ curve climbs five times faster and reaches the sill at $h = 0.1$. The two curves bracket the entire family of directional γ curves: any oblique direction sits between them with a directional range somewhere between $a_{\mathrm{min}}$ and $a_{\mathrm{maj}}$.

Second, slide $\theta$ from $0$ to $60^{circ}$. The ellipse rotates rigidly — the major and minor RANGES are unchanged, only the AZIMUTH changes. The two γ curves on the right are unaffected: they measure γ along the major and minor PRINCIPAL DIRECTIONS, which rotate together with the ellipse. The shape of the variogram model is determined by $(a_{\mathrm{maj}}, a_{\mathrm{min}})$; the orientation $\theta$ is a separate parameter that the empirical variogram identifies from the ROSE-PLOT step of §3.4.

Third, set $a_{\mathrm{maj}} = a_{\mathrm{min}} = 0.3$. The ellipse degenerates to a circle; the two γ curves overlay exactly. This is the isotropic case — the variogram depends only on $|\mathbf{h}|$, no directional information. Slowly slide $a_{\mathrm{min}}$ down from 0.3 to 0.06. The ellipse stretches; the two curves separate; the anisotropy ratio readout climbs from 1 : 1 through 5 : 1. The single number "anisotropy ratio = $a_{\mathrm{maj}} / a_{\mathrm{min}}$" is what most practitioners report — it summarises the directional dependence in one number, and the mildness/moderation/strength of the anisotropy is classified by that ratio.

The 3-D extension — three Euler angles and three ranges

The 2-D construction generalises cleanly to 3-D. The matrix $\mathbf{A}$ is now $3 \times 3$ and decomposes into a 3-D rotation followed by a diagonal scaling:

where $\mathbf{R}_3$ is a 3-D rotation built from THREE Euler angles. In the GSLIB tradition (Deutsch & Journel 1998 §II.3), the three angles are called:

• Azimuth $\theta$: the horizontal angle of the major axis, measured CLOCKWISE from north (in plan view). Range $[0^{circ}, 360^{circ})$. Geological convention; differs from mathematical convention (CCW from east).
• Dip $\phi$: the angle by which the major axis dips below horizontal, measured downward. Range $[-90^{circ}, 90^{circ}]$; negative means the major axis tips upward.
• Plunge $\psi$: a rotation about the (now tilted) major axis, controlling the orientation of the medium and minor axes within the plane perpendicular to the major. Range $[0^{circ}, 360^{circ})$.

Multiplied out, $\mathbf{R}_3 = \mathbf{R}_z(\theta) , \mathbf{R}_y(\phi) , \mathbf{R}_x(\psi)$ (the standard "Tait–Bryan" composition; GSLIB documents the exact element-wise form in its variogram-modelling chapter). The reduced distance $h' = |\mathbf{A},\mathbf{h}|$ defines an ELLIPSOID rather than an ellipse: a 3-D oval surface with three orthogonal principal semi-axes. The variogram model needs SIX parameters in 3-D — three angles plus three ranges — on top of the iso-family parameters (sill, optional nugget). Add a few more for a nested structure (§4.4) and the parameter count grows. Real reservoir-characterisation variogram fits typically use 6–9 parameters per structural component, fitted across hundreds of directional pairs by the §4.5 methods.

The 3-D anisotropy ratios are reported as a PAIR: $a_{\mathrm{maj}} / a_{\mathrm{med}}$ (the "horizontal anisotropy" if the major and medium axes both lie close to horizontal) and $a_{\mathrm{maj}} / a_{\mathrm{min}}$ (the overall anisotropy). In sedimentary reservoirs, $a_{\mathrm{maj}} / a_{\mathrm{med}}$ commonly sits in 2 : 1 to 5 : 1 (a paleo-flow direction longer than its perpendicular within bedding), and $a_{\mathrm{maj}} / a_{\mathrm{min}}$ commonly sits in 10 : 1 to 100 : 1 because the vertical range (perpendicular to bedding) is much shorter than the horizontal ones — bedding interfaces decorrelate the field rapidly in $z$. Pyrcz & Deutsch 2014 §5 lists representative 3-D anisotropy parameters across many reservoir analogue datasets; the vertical-to-horizontal contrast of 10–100 is the modal observation.

The sign-and-convention zoo for the three Euler angles is the main 3-D pitfall. GSLIB uses one convention; Petrel uses another; Leapfrog uses another; commercial mining packages routinely disagree among themselves. The defensible practice in a publishable variogram report is to (1) name the convention, (2) cite the convention's reference, and (3) write the rotation matrix in full so a reader who is uncertain can implement it from scratch. Variogram reports that say "azimuth = 30°, dip = -15°" without specifying the sign convention are uninterpretable — the same numbers can mean four physically distinct orientations depending on whether azimuth is CW from north or CCW from east, and whether dip is downward-positive or upward-positive.

The search ellipse — kriging's neighbourhood-selection geometry

The variogram model — anisotropic or not, with or without nugget, single or nested — is a continuous function defined on the entire lag-vector space. Kriging at a target location $\mathbf{s}_0$ uses, in principle, EVERY sample in the dataset. In practice this is computationally infeasible (the kriging system at $n$ samples is $n \times n$) and statistically wasteful (distant samples carry essentially-zero kriging weight when their lag vectors are past the variogram range). The standard fix is to define a search neighbourhood: a geometric region around the target inside which samples are eligible for inclusion in the local kriging system, with samples outside the region excluded.

The search neighbourhood is itself an ellipse (in 2-D) or an ellipsoid (in 3-D) — the search ellipse. It shares orientation and shape with the variogram's anisotropy ellipse but is SIZED INDEPENDENTLY. Mathematically the search ellipse is defined by

with its own matrix $\mathbf{A}_s$. Standard practice (Goovaerts 1997 §5.6; Deutsch & Journel 1998 §III.3; Pyrcz & Deutsch 2014 §6) sets $\mathbf{A}_s$ to share the variogram's rotation but with its own SCALES, typically

The factor $\kappa$ is the SEARCH-RADIUS MULTIPLIER. Why 1.5–3? Two competing pressures:

• Too small ($\kappa < 1$): samples just past the variogram range are excluded, but their CORRELATION with the target is near zero — they would have got near-zero kriging weight anyway. The cost of including them in the matrix is small. Setting $\kappa < 1$ wastes statistical information by excluding nearby samples whose contributions matter. Worse, in sparse-data regions the search ellipse may capture FEWER than the minimum number of samples needed for a well-conditioned kriging system — leading to either a degraded local estimate or an outright singular matrix.
• Too large ($\kappa > 4$): samples at distance $> a_{\mathrm{maj}}$ have variogram values at the sill, equivalent to no spatial correlation with the target. Their kriging weights are near zero. Including them costs computation (the $n \times n$ matrix grows) but contributes negligibly to the estimate. In sparse-data regions a very large search ellipse occasionally captures useful far samples, but more typically it just slows the kriging run without improving accuracy.

The 1.5–3× window balances both. $\kappa = 2$ is a common default — large enough to catch a few correlated samples just past the range, small enough that the matrix stays manageable. Pyrcz & Deutsch 2014 §6 reports that in production reservoir-characterisation workflows, the search-multiplier $\kappa$ is one of the few parameters that practitioners almost never tune by formal optimisation; the 1.5–3× heuristic captures essentially all the practical accuracy gain.

The search ellipse can also be capped by SAMPLE-COUNT LIMITS in addition to spatial extent. A typical specification is "search ellipse with $\kappa = 2$, minimum 4 samples, maximum 16 samples; if the ellipse captures fewer than 4 samples, abort the kriging at this target (mark as unestimated)". The maximum cap prevents the matrix from growing unboundedly in dense-data regions; the minimum cap prevents the matrix from becoming under-determined in sparse regions. Both caps are practitioner conventions, not deep theorems — most software exposes them as user-configurable parameters.

Search-ellipse coverage — second widget

The second widget for §4.3 shows the search ellipse in action. A scatter of 80 samples on a unit square; a target at $(0.5, 0.5)$; a fixed reference VARIOGRAM ellipse (5 : 1 anisotropic, $\theta = 30^{circ}$, drawn dashed in pink); and a SEARCH ellipse whose angle, major radius and minor radius you set. Samples inside the search ellipse are highlighted green; samples outside are grey. A toggle forces an ISOTROPIC (circular) search at the same $r_{\mathrm{maj}}$ for direct comparison.

Three things to do with this widget. First, with the search in ANISOTROPIC mode, set the search angle $\theta_s = 30^{circ}$ (matching the variogram), the major radius $r_{\mathrm{maj}} = 0.5$, and the minor radius $r_{\mathrm{min}} = 0.10$ (giving a 5 : 1 search shape matching the 5 : 1 variogram, sized at $\kappa = 1.67\times$ the variogram range). Read off the number of samples inside the search — this is the count that will enter the kriging system at this target. Slide $r_{\mathrm{maj}}$ up to 0.6 and 0.8; the count grows; at $r_{\mathrm{maj}} = 0.8$ the search ellipse covers most of the square and most samples are eligible. This is the $\kappa$ slider sweep.

Second, toggle to ISOTROPIC search at $r_{\mathrm{maj}} = 0.5$. The ellipse becomes a circle of radius 0.5. Look at what changed: the samples that were INSIDE the long horizontal part of the anisotropic ellipse but OUTSIDE the circle are now grey (lost), and the samples that were INSIDE the perpendicular short minor axis of the anisotropic ellipse are now ALSO inside the circle (correctly retained). But — and this is the key — the circle ALSO captures samples in the direction perpendicular to the variogram's minor axis that are FAR beyond the variogram's minor range. Those samples have zero spatial correlation with the target (the variogram has reached the sill at lag $\approx a_{\mathrm{min}} = 0.06$), so they contribute essentially-zero kriging weight, but they cost a matrix row. The isotropic search at the same major radius is WORSE on both counts: fewer correlated major-direction samples, more uncorrelated minor-direction samples.

Third, slide the search angle $\theta_s$ away from $30^{circ}$ to 90°. The search ellipse rotates 60° away from the variogram's orientation. Now even fewer of the correlated major-direction samples fall inside the search, and the search is misaligned with the field's correlation structure. Misorienting the search is the discrete-case version of misorienting the variogram in §3.4 — both produce the same downstream failure mode: kriging with a search ellipse rotated 30° or 45° off the variogram's axis collects systematically WORSE samples for the local estimate. The defensible practice is to ALWAYS YOKE the search-ellipse orientation to the variogram orientation, even if you tune their sizes independently.

Practical specification — what a defensible variogram report contains

The §4.3 mathematical machinery determines what a defensible variogram report MUST document. Six numbers are not enough; you also need the conventions. The standard checklist (synthesising Goovaerts 1997 §4.3, Deutsch & Journel 1998 §III.4, Chilès & Delfiner 2012 §2.5):

• Family choice and parameters. Spherical / Exponential / Gaussian / nested combinations; sill $c$, nugget $c_0$; for nested structures, the per-structure sill and range.
• 2-D: three numbers + convention. $\theta, a_{\mathrm{maj}}, a_{\mathrm{min}}$; whether $\theta$ is measured CW from north or CCW from east; the units of $a_{\mathrm{maj}}, a_{\mathrm{min}}$ (metres, kilometres, feet); whether the ranges are TRUE (Spherical) or PRACTICAL 95% (Exponential, Gaussian).
• 3-D: six numbers + convention. Three Euler angles ($\theta, \phi, \psi$) plus three ranges ($a_{\mathrm{maj}}, a_{\mathrm{med}}, a_{\mathrm{min}}$); the Euler-angle convention (GSLIB / Petrel / Leapfrog / academic); the rotation matrix written out explicitly; the orientation of the medium axis (in-bedding vs cross-bedding).
• Search-ellipse specification. Either the same shape as the variogram with a stated multiplier $\kappa$ (typical $\kappa = 2$), or a separate search-ellipse spec with its own angle and radii; minimum and maximum sample counts per local kriging system; octant or quadrant search subdivisions if used.
• Diagnostic of fit quality. Per-direction sample counts (especially for the minor direction in strong-anisotropy fields, where pair counts can be too low to support the claimed ratio); the directional misfit $|\theta_{\mathrm{fit}} - \theta_{\mathrm{latent}}|$ from §3.4 if a latent direction is known; weighted SSE of the empirical-to-modelled variogram across all bins.

A variogram report missing any of these items is partially uninterpretable. A report that says "5 : 1 anisotropic Spherical, range 200 m" is missing direction, convention, sill, nugget, search-ellipse spec, and fit diagnostics — six items out of six. The defensible report explicitly lists every number, every convention, and every choice, so a downstream geomodeler can implement the same model exactly. This matters because the same six-parameter spec can describe physically opposite orientations depending on the convention.

Geometric vs zonal anisotropy — what §4.3 does and does not handle

The construction in §4.3 — wrap an isotropic family in an anisotropic-ellipse transform — handles geometric anisotropy: ranges differ across directions, but the SILL is the same. The model is one continuous surface $\gamma(\mathbf{h})$ that reaches a common sill $c$ along every direction (eventually). For most earth-science fields with directional structure, geometric anisotropy is a defensible first model.

Zonal anisotropy is different: a directional component adds EXTRA variance in one direction only, producing a different effective sill along different axes. The empirical signature is a directional variogram that plateaus at a different value along the major axis than along the minor axis. Goovaerts 1997 §4.3 and Wackernagel 2003 §7.5 develop this case: zonal anisotropy is modelled as a NESTED STRUCTURE (§4.4) — a sum of a geometric-anisotropic component (shared sill) plus a one-direction-only component (extra sill in the zonal direction). The zonal piece has $a_{\mathrm{maj}}$ infinite or near-infinite along its non-zonal directions, so it contributes only along the zonal axis.

§4.3 deliberately stays with the geometric case. §4.4 develops zonal anisotropy as one species of the more general nested-structures machinery. The two are not exclusive: many real fields show geometric anisotropy in plan view and zonal anisotropy across the vertical — a horizontal sedimentary fabric (geometric) layered on top of bedding-controlled vertical decorrelation (zonal). The compound model has three or four nested structures with carefully tuned per-direction anisotropy ranges and sills.

Try it

• In aniso-ellipsoid-builder, set $\theta = 0^{circ}$, $a_{\mathrm{maj}} = 0.50$, $a_{\mathrm{min}} = 0.10$, $c = 1.0$. Read off $\gamma$ along the major direction at $h = 0.10, 0.25, 0.50$ — you should see roughly $\gamma_{\mathrm{maj}}(0.10) = 1.5(0.10/0.50) - 0.5(0.10/0.50)^3 \approx 0.296$, $\gamma_{\mathrm{maj}}(0.25) = 0.687$, $\gamma_{\mathrm{maj}}(0.50) = 1.000$. Now read off $\gamma$ along the minor direction at the same three lags: $\gamma_{\mathrm{min}}(0.10) = 1.000$ (already at the sill — the minor range is 0.10), and the same for 0.25 and 0.50. The same lag distance gives very different γ values along the two directions; THAT is anisotropy.
• Slide $\theta$ from $0^{circ}$ to $60^{circ}$ in the aniso-ellipsoid-builder. The ellipse rotates rigidly; the right-panel γ curves are unchanged. The orientation $\theta$ and the shape $(a_{\mathrm{maj}}, a_{\mathrm{min}})$ are INDEPENDENT parameters. In the empirical-variogram workflow of §3.4, the rose plot identifies $\theta$ from the angular distribution of the directional ranges; here the slider shows what changing $\theta$ does geometrically.
• In aniso-ellipsoid-builder, set $a_{\mathrm{maj}} = a_{\mathrm{min}} = 0.30$ (ratio 1 : 1). The two γ curves overlay and the ellipse becomes a circle. Now slide $a_{\mathrm{min}}$ down through 0.20, 0.10, 0.05. Watch the ratio readout climb (1.5, 3, 6 : 1) and the two curves separate. Categorise each setting using the on-screen message: which crosses from MILD to MODERATE to STRONG?
• In search-ellipse-coverage, with the default settings ($\theta_s = 30^{circ}$, $r_{\mathrm{maj}} = 0.50$, $r_{\mathrm{min}} = 0.10$), read the "Samples inside search" count. Toggle to ISOTROPIC search — the $r_{\mathrm{min}}$ slider locks to match $r_{\mathrm{maj}}$. Read the new count. How many samples did the isotropic search ADD or LOSE relative to the matched-shape anisotropic search? Are the added samples in the major or minor direction relative to the variogram?
• In search-ellipse-coverage, set the search shape to anisotropic, match the variogram angle ($\theta_s = 30^{circ}$), and sweep $r_{\mathrm{maj}}$ from 0.10 (very small) through 0.30 (around the variogram range) up to 0.70 (much larger). Watch how the count of eligible samples grows. Match the practitioner guideline $\kappa \in [1.5, 3]$ to the slider position: which range of $r_{\mathrm{maj}}$ lies in that window for this $a_{\mathrm{maj}} = 0.30$ variogram?
• In search-ellipse-coverage, deliberately MISORIENT the search: set $\theta_s = 90^{circ}$ while the variogram remains at $30^{circ}$. The search ellipse rotates 60° off the variogram's axis. Compare the count to the matched-orientation case at $\theta_s = 30^{circ}$. The misoriented search picks up DIFFERENT samples; which are now mostly minor-direction (uncorrelated)? Think about what this means for kriging: misoriented search is the discrete analogue of fitting the wrong variogram orientation.
• Without coding: a reservoir-engineering team reports a Spherical variogram fit with $\theta = 60^{circ}$, $a_{\mathrm{maj}} = 800$ m, $a_{\mathrm{min}} = 120$ m, sill 0.04, nugget 0.005. Compute the anisotropy ratio and classify it (mild / moderate / strong / extreme). What search-ellipse size and shape would you recommend? Why does the $\theta$-convention (CW from north vs CCW from east) matter for the production geomodeller?
• Without coding: a 3-D geomodel for a fluvial reservoir reports variogram parameters "azimuth 110°, dip −5°, plunge 0°, a_maj = 1500 m, a_med = 400 m, a_min = 8 m". Compute the two anisotropy ratios. Is the horizontal-to-vertical ratio realistic for a fluvial deposit? What physical processes produce a horizontal range of 1500 m and a vertical range of only 8 m? What does the negative dip mean here?

Pause and reflect: §3.4 fit a 2-D anisotropy ellipse from observed directional ranges; §4.3 wraps any permissible isotropic variogram inside an anisotropic-ellipsoid transform, and then introduces a SEPARATE search ellipse for kriging's neighbourhood selection. The two ellipses share orientation and aspect but are sized independently. What information would you most want to document about the variogram-ellipse / search-ellipse pair before handing the model to a downstream geomodeller, so the geomodel is reproducible without ambiguity?

What you now know — and what Part 4 will build on it

You can build the anisotropic variogram model from any permissible isotropic family via $\gamma(\mathbf{h}) = \gamma_{\mathrm{iso}}(|\mathbf{A},\mathbf{h}|)$, with $\mathbf{A}$ a rotation-then-scaling matrix that maps the anisotropy ellipsoid to a unit sphere. You can write the 2-D form in terms of three numbers $(\theta, a_{\mathrm{maj}}, a_{\mathrm{min}})$ and the 3-D form in terms of six (three Euler angles + three ranges). You understand the practitioner-convention zoo for the Euler angles and the defensible-report obligation to document the convention and write the rotation matrix out explicitly.

You can distinguish GEOMETRIC anisotropy (different ranges, shared sill — the §4.3 case) from ZONAL anisotropy (different ranges AND different sills — handled in §4.4 as a nested structure). You can classify a geometric-anisotropy ratio as mild (1–2 : 1), moderate (2–5 : 1), strong (5–10 : 1), or extreme (> 10 : 1), and you know roughly which fields produce which ratios — sedimentary fabric tends to moderate, channelised reservoirs to strong, fault-zone anisotropy to extreme, isotropic noise to mild.

You can describe the SEARCH ELLIPSE — kriging's neighbourhood-selection geometry — and its standard 1.5–3× sizing rule relative to the variogram range. You understand the two competing pressures: too small ⇒ insufficient samples, ill-conditioned local kriging; too large ⇒ uninformative far samples, bloated matrix. You can identify the consequences of an anisotropy–search mismatch: isotropic search on an anisotropic field wastes major-direction samples and includes uninformative minor-direction ones; misoriented search rotates the eligible-sample set off the field's correlation structure.

Part 4 continues here. §4.4 develops NESTED STRUCTURES: sums of permissible variograms (possibly each with its own anisotropy ellipse, possibly with zonal components for shared-orientation but different-sill behaviour), and the rules that govern how the per-structure sills add up. §4.5 covers FITTING — by eye, by weighted least squares, by maximum likelihood — and how to choose the parameters jointly from data in a way that respects sample counts, lag-bin tolerances, and the permissibility constraint. Part 5 plugs the (possibly anisotropic, possibly nested) model into the kriging system, USING the search ellipse from §4.3 to subset the samples that enter each local linear system.

Variogram and search live together. A defensible kriging workflow specifies both, documents both, and keeps their orientations yoked. Anisotropy is not a cosmetic refinement — it is the lever that captures most of the directional information in a real spatial dataset. Skip it on a clearly anisotropic field and the kriged map is systematically wrong along the major direction (over-smoothed) and along the minor direction (under-smoothed). Get it right and the map preserves the directional structure the data already showed you in Part 3.

Mark section complete →

References

• Matheron, G. (1971). The Theory of Regionalized Variables and Its Applications. Les Cahiers du Centre de Morphologie Mathématique, No. 5. Fontainebleau, France: École Nationale Supérieure des Mines de Paris. (The foundational reference. Introduces the anisotropic variogram via a coordinate transform and establishes the permissibility-preserving nature of the construction. The geometric / zonal distinction is laid out in the modern form here.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford University Press. (§4.3 develops the anisotropic variogram in a practitioner-oriented format. §5.6 covers the search-ellipse-sizing heuristic (1.5–3× the variogram range) and the minimum / maximum sample-count caps for the local kriging neighbourhood. The standard practitioner reference for the §4.3 material.)
• Deutsch, C.V., Journel, A.G. (1998). GSLIB: Geostatistical Software Library and User's Guide (2nd ed.). Oxford University Press. (§II.3 and §III.4 document the GSLIB Euler-angle convention (azimuth, dip, plunge) and the rotation-matrix form used inside the variogram-modelling and kriging routines. The canonical software-side reference; reading it is the only way to verify that a GSLIB-format variogram report has been re-implemented correctly in another package.)
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling (2nd ed.). Oxford University Press. (§5 surveys representative 3-D anisotropy parameters across many reservoir analogue datasets — sedimentary fabric, channelised reservoirs, fractured carbonates — and reports the typical horizontal anisotropy ratios (2 : 1 to 5 : 1) and horizontal-to-vertical anisotropy ratios (10 : 1 to 100 : 1) that practitioners encounter in routine work. §6 covers search-ellipse practice in production reservoir-characterisation workflows.)
• Chilès, J.-P., Delfiner, P. (2012). Geostatistics: Modeling Spatial Uncertainty (2nd ed.). Wiley. (§2.5 covers the anisotropic variogram in the modern comprehensive form. The geometric / zonal distinction, the permissibility-preserving coordinate transform, and the 3-D Euler-angle formulation are all developed at a mathematical-statistics level of rigour suitable for the reader who wants to verify GSLIB-style implementations from first principles.)
• Isaaks, E.H., Srivastava, R.M. (1989). An Introduction to Applied Geostatistics. Oxford University Press. (Chapter 16 covers the variogram model including its anisotropic generalisation at an entry level. The book's pedagogy informs the choice of the 5 : 1 anisotropy ratio in the §4.3 search-coverage widget — common enough to be realistic, large enough to make the isotropic-vs-anisotropic comparison visually obvious.)
• Wackernagel, H. (2003). Multivariate Geostatistics: An Introduction with Applications (3rd ed.). Springer. (§7.5 develops the zonal-anisotropy case as a nested-structure construction — preview of §4.4. The book's treatment of how zonal anisotropy interacts with the kriging system gives the connection between §4.3 modelling and Part 5 kriging that motivates the search-ellipse-coverage widget.)
• Cressie, N. (1993). Statistics for Spatial Data (revised ed.). Wiley. (§2.5 develops the permissibility-preserving nature of the anisotropic coordinate transform formally; §3.4 covers the search-neighbourhood specification from a mathematical-statistics perspective. The cleanest textbook derivation of why a permissible isotropic variogram wrapped in any positive-definite scaling matrix is still permissible.)