Sketch-based modelling and visualization of geological deposition

Mattia Natali a,n, Tore Grane Klausen b, Daniel Patel a,c

a Department of Informatics, University of Bergen, P.O. Box 7803, N-5020 Bergen, Norwayb Department of Earth Science, University of Bergen, P.O. Box 7803, N-5020 Bergen, Norwayc Christian Michelsen Research AS, P.O. Box 6031, N-5892 Bergen, Norway

a r t i c l e i n f o

Article history:Received 21 August 2013Received in revised form26 February 2014Accepted 27 February 2014Available online 13 March 2014

Keywords:Illustrative visualizationFluvial depositionStratigraphic evolution

a b s t r a c t

We propose a method for sketching and visualizing geological models by sequentially defining strati-graphic layers, where each layer represents a unique erosion or deposition event. Evolution of rivers anddeltas is important for geologists when interpreting the stratigraphy of the subsurface, in particular forhydrocarbon exploration. We illustratively visualize mountains, basins, lakes, rivers and deltas, and howthey change the morphology of a terrain during their evolution. We present a compact representation ofthe model and a novel rendering algorithm that allows us to obtain an interactive and illustrative layer-cake visualization. A user study has been performed to evaluate our method.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Geologists are interested in better tools for externalizing theirideas on the earth's behaviour. They want to do this in anexpressive and simple way, which is particularly important forcommunicative purposes. Current modelling tools in geology havea high learning curve and are tedious to use (Caumon et al., 2009).We present a simple sketching interface and a data structure forcompact representation and flexible rendering of subsurface layerstructures. This is useful for representing mountains, basins, lakes,rivers and deltas. Rivers and deltas change the morphology of aterrain through erosional and depositional processes. This is left asimprints in layers of the terrain. Our technique provides a way tosketch and visualize such layers (Fig. 1). Our representation is wellsuited to obtain interactive layer-cake visualizations, resulting ingeological illustrations that are helpful for education and commu-nication. Conventional hydrocarbon reservoirs (and aquifers) arefound in porous bodies of rock. Examples of such rock bodiesinclude sandstone, which is found in sedimentary basins and has ahigh preservation potential (Hinderer, 2012). The sandstone might,under favourable circumstances and the right basin development(where hydrocarbon source rock and reservoir seal is present),become a reservoir for hydrocarbons. This is, in a crude sense, thereason why these rock bodies receive a lot of focus in geology, anda motivation to try to understand them to the full extent that dataallows.

Available data (e.g. seismic, well logs or core) can often be oflimited resolution and extent. In these cases, geologists have todevelop conceptual ideas to describe the shape of rock bodies, and,from that, which processes were involved in their deposition. Theprocesses involved in the deposition of sandstone bodies areknown to vary between different depositional environments, andcan thus be differentiated based on observations and interpreta-tion (Reading, 1996). To develop good conceptual models for thereservoir sandstone, their horizontal and cross-sectional charac-teristics are often highlighted by schematic block diagrams (e.g.Gani and Bhattacharya, 2007, Pore¸bski and Steel, 2003). In Fig. 2(left), an example of a hand-drawn block diagram depicting ameandering river channel is shown. It illustrates the aerial and thecross sectional expression of sandstone point bars (sedimentsdeposited along the inner bank of a meandering stream) andhow one sandstone body is overlaid by another. Internal architec-ture is important in the sense that it tells how the depositionalelement (e.g. channel or delta) evolved, and subsequently howsmall-scale heterogeneities such as mudstone might be distributedwithin the overall sandy body. This can have direct implicationfor hydrocarbon fluid extraction. Our approach offers a new wayof producing illustrations by performing interactive erosion anddeposition that lets the illustrator mimic processes that sheinterprets to have been the cause for the sandstone deposition.An additional consequence of our 3D sketched models is theirmanoeuvrable cutting planes that enable multiple cross-sectionvisualizations. This helps in understanding complex internal layer-ing within the sandstone, otherwise not intuitively apprehensible(Bridge, 1993). Among the depositional systems that may result inhydrocarbon accumulation, rivers and deltas are central.

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journal homepage: www.elsevier.com/locate/cageo

Computers & Geosciences

http://dx.doi.org/10.1016/j.cageo.2014.02.0100098-3004/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ47 555 84513.E-mail addresses: mattia.natali@uib.no (M. Natali),

Tore.Klausen@geo.uib.no (T.G. Klausen), daniel@cmr.no (D. Patel).

Computers & Geosciences 67 (2014) 40–48

In summary, our overall contribution is a sketch-based systemthat makes it possible to quickly build 3D interactive geologicalillustrations from scratch. Sequentially defining alterations on amodel is less laborious than drawing 2D illustrations; this opens upfor discussions, fast hypothesis testing and creation of time-seriesillustrations. A novel central aspect lies in the proposed datastructure (each stratigraphic layer corresponds to a composition ofone or more heightmaps) and the way it is processed to rendervolumetric models. The main features that characterize our toolconsist of operators that interpret each sketch to generate depositionand erosion processes; in particular, rivers and channels, mountains,basins, deltas and intermediate stages of their evolution.

2. Related work

Little work exists on modelling and visualizing deltas in computergraphics. For modelling rivers and erosion in a geological setting,

most of the methods are based on fractal noise generation andphysical based processes (for instance the works of Benes et al., 2006and Stava et al., 2008). Such procedural approaches reduce thedegree of control over the landscape development. Other sketch-based techniques have been introduced to model landforms on aterrain (Hnaidi et al., 2010; Gain et al., 2009), but they only considerthe top surface and not subsurface structures, as we do. Some workin geology focuses on producing layered representations of the earthby modelling the mesh of each separating surface individually, likeBaojun et al. (2009), but are not sketch-based and therefore timeconsuming. Amorim et al. (2012) address 3D seismic interpretationby modelling one surface at a time, while Cutler et al. (2002) proposea procedural method to obtain layered solid models. None of thesetwo last works consider fluvial systems consisting of lakes, rivers ordeltas.

A state of the art report on modelling of terrains and subsurfacegeology was presented by Natali et al. (2013). There two works arereviewed that describe volumetric representations applicable ingeology, although they do not discuss how to sketch them fastas we do. Takayama et al. (2010) obtain layered models usingdiffusion surfaces. Wang et al. (2011) represent objects using signeddistance functions. Composite objects are created by combiningimplicit functions in a tree structure which makes it possible toproduce volumes made of many smaller inner components.

Geostatistics: The Surface- and the event-based modelling haverecently been developed as a branch of geostatistics. Xie et al. (2000)suggest depositional and erosional surface-based models as collec-tions of separate sediment units, each of them stochastically gener-ated. Lopez et al. (2001) create their 3D models, in the contextof meandering channelized reservoirs, combining a process-basedapproach to model river motion and deposition, and a stochasticapproach to respect vertical proportion curves that reflect naturalbehaviour. Pyrcz et al. (2005), Pyrcz (2004), Michaels et al. (2010),and Abrahamsen et al. (2012) use statistical analysis on process-based models to create surface-based distributary lobe models withthe addition of available geologic information. Pyrcz et al. (2012)present new applications for event-based models and methods to

Fig. 1. A 3D model created using our sketch-based approach to shape surface andsubsurface geological features. Bottom right inset shows a map view of thesketched strokes in blue overlaid on the model. (For interpretation of the referencesto colour in this figure caption, the reader is referred to the web version of thispaper.)

Fig. 2. Comparison between a 2D hand made illustration of a river sedimentation process (left) which took one working day to produce and the result achieved with ourapproach (right) which took 1 h to produce. The example is a real case analysis built from field observations.

M. Natali et al. / Computers & Geosciences 67 (2014) 40–48 41

generate them with hand editing. Bertoncello et al. (2013) constrainsedimentary surface-based models to field data (wells and thicknessdata interpreted from geophysics, such as seismic data).

Geological illustrations: Geological illustrations let geologistsexpress and explain subsurface processes of how the earth hasevolved and can be used to predict and discuss future scenarios. Inaddition to enabling communication between experts, geologicalillustrations are heavily employed for teaching purposes as theyare expressive and easy to comprehend. This process of externa-lizing ideas is a crucial aspect in geology (as highlighted by Lidalet al., 2013).The added value of our 3D models with internalstructures compared to static 2D illustrations made on paper orwith 2D raster or vector drawing software is the support ofinteractive exploration. Creating a volumetric 3D model allowsfor visualizing arbitrary cross-sections of the model and ensuresspatial consistency of internal structures. For hand-drawn 2Dillustrations, spatial consistency between cross-sections is notensured as features can be drawn independently on each cross-section without honouring the underlying 3D structures. We arenot aware of any work similar to ours for fast sketching of 3Dgeological illustrations containing internal structures. Related isthe work by Natali et al. (2012) which is able to imitate 3Dgeological illustrations by supporting 2D drawing on the surfacesof a hollow 3D boxlike object.

Fluvial landscape evolution: When the top surface of a geologi-cal model contains rivers and streams, or has been shaped bythem, it is called a fluvial landscape. A clear classification ofdifferent methods that have been used to model or simulate afluvial system and the surrounding landscape can be found in thework by Coulthard and Van De Wiel (2012). Models concerningfluvial geomorphology are made that attempt to retrieve thehistory or to predict the evolution of rivers (Fig. 2 is an exampleof a fluvial geomorphology illustration). Such models are alsonecessary when a large temporal scale process is considered andcannot be studied with just observations of the real world, whichare in a small temporal scale compared to the river history. Thework on river geomorphology concerned with simulation oferosion and sediment transport usually utilizes physically basedconstraints such as the conservation of mass and momentumof the river flow, formally expressible with the Navier–Stokesformula.

Layered representations: A layered heightmap representationhas been proposed earlier (Benes and Forsbach, 2001), but only inthe context of terrain visualization and erosion, and not forrepresenting subsurface structures as we propose. A representa-tion related to ours is used by Lemon and Jones (2003), employingmeshes instead of heightmaps. They present an approach forgenerating solid models from borehole data. Their model con-struction is simplified by representing horizons as triangulatedsurfaces, where vertices maintain the same ðx; yÞ position for everysurface, and only the z-value varies. This simplifies intersectiontesting and topological relations between surfaces. Inspired by this

work, we use heightmaps to represent layers and replace inter-section tests with simple and fast arithmetic operations. Inaddition, correct topology between layers is implicitly maintained.Invalid layers, layers with holes, or empty spaces inside models areautomatically avoided. Peytavie et al. (2009) also use a layeredrepresentation, where subsurface layers are discussed, but only forthe purpose of realistic modelling of the top terrain surface. Thelayers represent materials that interact at the top surface such asair, water, boulders, sand and ground. We model the top surface,but we also include internal structures and landscape evolution(Figs. 2 and 3) which can be rendered volumetrically and witharbitrary cuts.

3. Description of our approach

The entire approach is based on two synchronized datastructures; the relative layers and the absolute layers. The relativelayers let us keep geological processes independent of each otherwith respect to time, allowing us to rearrange layers in any orderwithout further computations. The absolute layers is derived fromrelative layers and is used for fast rendering. The i-th absolute layerhabsi ðx; yÞ is defined in each of its points ðx; yÞAG by the height ofthe top surface of the layer in the reference coordinate system ofthe model, 8 i¼ 1;…;n. Where n is the number of layers, G is auniform grid and habs0 ðx; yÞ is defined to be zero in every point.

The i-th relative layer hreli ðx; yÞ is defined in each of its pointsðx; yÞAG by the displacement

Δhabsi ðx; yÞ ¼ habsi ðx; yÞ�habsi�1ðx; yÞof two consecutive absolute layers, 8 i¼ 1;…;n.

The relation between relative and absolute layers lets us expressthe i-th absolute layer as

habsi ðx; yÞ ¼ ∑i

k ¼ 1hrelk ðx; yÞ; 8 i¼ 1;…;n:

Fig. 4 illustrates the relation between relative and absolute layersand shows how the final model is reached at each step i. Thenumber of layers of the final composition is not known a priori,therefore we will only give a number to the last l layers, i.e. L1i ;…; Lli,with corresponding thickness values T1

i ðx; yÞ;…; Tliðx; yÞ. In the third

column of Fig. 4, for instance, we have one layer each at steps 1 and2, namely L11 and L12 , but with different shapes due to differentthicknesses; two layers at steps 3 and 4, that is ðL13; L23Þ and ðL14; L24Þrespectively. Unique properties can be associated to each layer sothat they can be represented differently in the final rendering.

3.1. Data structure description

A final composition of the model is made of a sequence ofchronologically ordered relative layers (both positive and negativevalues are feasible). When, at step i and in a specific point ðx; yÞAG,

Fig. 3. River evolution example. Left: first and last configurations of the river are sketched and imprinted. Right: imprint of additional five intermediate stages of thedepositional history.

M. Natali et al. / Computers & Geosciences 67 (2014) 40–4842

hreli ðx; yÞZ0, last layers of the final composition are not modifiedand a new one changes in ðx; yÞ to have thickness hreli ðx; yÞ. Whenhreli ðx; yÞo0, this affects previous layers and they must be changedaccordingly. Fig. 5 gives us an example of this occurrence. Theprocess develops as follows: the erosion expressed by hrel

i ðx; yÞdecreases the thickness Tl�1

i ðx; yÞ of the previous layer; ifhreli ðx; yÞ4Tl�1

i ðx; yÞ, then Ll�2i is also involved and Tl�2

i ðx; yÞdecreased; the iteration analogously continues until layer Lik, suchthat hreli ðx; yÞr∑k

j ¼ l�1Tjiðx; yÞ. In Fig. 5, Li2 and Li

3, iAf4;5g, are setto zero thickness in our data structure and will not be visible in therendered model.

Our model is obtained through an accumulation of relativelayers, defined on uniformly sampled grids represented as 2Dheight maps. Each relative layer indicates the amount of deposition(positive values) or erosion (negative values) that generates achange in height with respect to the shape of the previous layer.For instance, to achieve a simple model made of a starting land-scape and a river, the first relative layer (always also the firstabsolute layer, see top row Fig. 4) is used as an initial ground layer ofconstant thickness. Afterwards, the bed of the river is carved out bythe second relative layer (second row in Fig. 4). This relative layercontains negative values that indicate the erosion depth for eachpoint of the grid. Compared to using a boundary representation(Natali et al., 2012), we avoid time consuming triangle-intersectiontesting and vertex alignment between the boundaries of individual

layers. The process of erosion and deposition is straightforwardto realize and can be performed in parallel compared toconstructive solid geometry algorithms performed on a boundaryrepresentation.

3.2. Rendering the model

We visualize the model by applying ray-casting based volu-metric rendering. Volumetric rendering inherently supports volu-metric variations and translucency. Therefore our method canhandle procedural 3D textures for defining different rock appear-ances, co-rendering of the model together with volumetric datasuch as seismic data of the area being modeled, or varying layerproperties such as grain size in a delta deposit. As opposed tostandard volumetric rendering which accesses a 3D array repre-senting a discretization of the model space, we perform ray-casting directly on our compact absolute layer structure. Thecolour of each pixel on the screen is found by sending a ray fromthe pixel into the scene and accumulating the colour along the ray.For each sample position on the ray, we identify in which layer weare which decides which colour and opacity the sample has.We perform the ray casting on the GPU to enable fast parallelprocessing of rays.

Each layer can be considered as a 2D height field representingthe thickness for each (x,y) position (as shown in Fig. 5). We storen layers, each with dimension l�m, as a 3D texture with dimen-sion l�m� n. For interactive rendering, we need a fast way toidentify which layer a sample at position ðx; y; zÞ is. Each (x,y)position in an absolute layer stores four components: the startingand the ending position of the layer and the index of the next andthe previous nonempty layer. See Fig. 6 and the accompanyingTable 1 describing the four values for each of the layers for positionx¼ x3. In Fig. 6, the layer-index of the first sample at x¼ x1,representing the first sample on each ray, is found by iteratingover all layers until the interval containing the sample's z positionis found. This search can be accelerated by performing a binarysearch. The search returns layer 2. For the next sample havingx¼ x2, we first look up in the same layer as the previous sample(layer 2), and retrieve the start and the end of the interval. Thesample's z value is within this interval and we are therefore inthe same layer. This lookup requires only one texture access. Thenext sample at x¼ x3 has crossed a layer boundary. The content ofall layers for position ðx3Þ is shown in Table 1. The layer interval forL2 is below the z value for the sample, therefore we search inlayers above to find the correct interval. We look up in the firstnonempty interval L6 which the sample is inside.

With volume rendering we are able to change optical proper-ties along the ray inside a layer as, for instance, a function of thedistance to the source of deposition for communicating propertiessuch as grain size (see Fig. 7).

Fig. 4. Relation between relative layer, left column and absolute layer, middlecolumn. The right column shows the accumulated final model.

Fig. 5. Interaction between the i-th relative layer hreli and the previous layersL1i ;…; Lli . This example refers to a single point ðx ; yÞ of the grid G. Only hrel

4 ðx ; yÞo0,therefore from step 3 to step 4 we observe an erosion.

Fig. 6. Using the absolute data structure for fast access along a ray cast from leftto right.

M. Natali et al. / Computers & Geosciences 67 (2014) 40–48 43

On cutting planes, a black separator line is drawn betweenadjacent layers. This makes it easier to distinguish between differentdepositions. However it is possible to assign different layers into thesame layer group to avoid separator lines between them. This isnecessary for giving the brown soil in Fig. 2 one homogeneousappearance even though it consists of several smaller layers inter-mixed with river layers.

3.3. Geological concepts through sketches

Sketching geological features on the model is based on twotypes of strokes: the open and the closed curve. They are the basisfor all geological features. In addition, there is the possibility toadd a layer of constant thickness for expressing a sedimentationprocess equally distributed in the landscape. Every sketched curveconsists of an array of points that has been subject to a Chaikinsubsampling (Chaikin, 1974) to reduce the number of sampleswhile still maintaining its shape. In the following paragraphs, wedescribe how the sketches are interpreted, classified and con-verted to relative layers.

Modelling a river: When the first and the last point of a curveare not close to each other (according to a predefined threshold),we consider the curve as a definition of the centreline of a river asseen from above (map-view). To allow branches in a river, the lastdrawn curve is merged with the previous one if they intersect.

Meandering rivers can be sketched by defining only the startand the end configuration and specifying the number of inter-mediate rivers. Intermediate rivers are then calculated as linearinterpolations of the first and the last configuration. The inter-polation between two curves is done by resampling both curvesuniformly, so that they have the same number of points andat the same curve-length distance. Instead of a physically basedinterpolation, we use an interpolation based on curve-lengthparameterization. The simplicity results in predictable behaviour.

This enables the user to express any kind of river evolution bymanually sketching interpolating rivers. For simple cases, a startand end river is enough. For more complex cases, the user mustsketch several key rivers which the system will then interpolatebetween.

Given two rivers, defined by two curves Cs and Ce, thatrespectively describe the starting and the ending configuration,we proceed as follow. Let us assume the behaviour of the riverduring its evolution is expressed by r intermediate steps and eachcurve is resampled into p points. Every intermediate curve Ci isdefined as a set of vertices Vs

i , where s¼ 1;…; p. The interpolationbetween a vertex Vs

s on Cs and a vertex Vse on Ce at intermediate

step i is given by

Vsi ¼ ð1�tÞVs

s þtVse ;

where i¼ 1;…; r and tA ½0;1�.We now introduce a mathematical description of our river

sections. The flow of a river has decreasing velocity from themiddle of its bed to its banks, and erosion is proportional tovelocity. We model this with a function that smoothly goestowards 0 with the distance to the centreline (see Fig. 8(top)),although our system can easily be extended to support anyanalytic or user-sketched definition of the river section. Whenthe user draws the centreline of a river, a scaling factor for riverdepth and a scaling factor for river width can be set. Each erodinglayer associated with rivers is always coupled with complemen-tary depositional layers, i.e. defined by grids with opposite values.

Fig. 8, top left, shows the graph describing the river bedsection, given by hαðzÞ ¼ αhðzÞ, where αAR and h(z) is defined asð sin ðπz�π=2Þ�1Þ=2 8zA ½0;1�0 otherwise. Here z¼ d=r, where dis the distance to the centreline and r is half of the width ofthe river. d is defined to be the minimum of the distances to thesegments of line defining the curve. Since dZ0 and r40, zA ½0;1�is equivalent to the more intuitive relation drr; i.e. the height h isinversely proportional to the distance dwhen the considered pointof the grid G (relative layer) is inside the neighbourhood definedby r. Outside this neighbourhood, the height is set to zero.

During the erosion process along the centreline, there is noneed to evaluate the distance for every point of the grid thatdefines the relative layer. We consider a subset Si of the grid G, inwhich to check the amount of erosion to apply, that is the union oftwo squared subset of G, SiA and SBi ; centred respectively at the twolimit points A and B of the i-th segment of the centreline, and withhalf-side length L¼

ffiffiffi

2p

T ; T ¼ JA�BJ=2.Modelling a delta: When the first and the last point of a curve are

close enough (according to a predefined threshold), we consider

Table 1The absolute layer structure used for volume rendering. The table shows layerproperties for position x3 of the example in Fig. 6.

Layer Start End Prev layer Next layer

L1 0 2 – L2L2 2 6 L1 L6L3 6 6 L2 L6L4 6 6 L2 L6L5 6 6 L2 L6L6 6 9 L2 –

Fig. 7. Cross-sectional view showing the internal structure of a delta deposit,where different grain sizes are conveyed by colour transition. Water is set to behighly translucent. (For interpretation of the references to colour in this figurecaption, the reader is referred to the web version of this paper.)

Fig. 8. Top: the function hα representing a river section. CR is a arbitrary point alongthe river centreline. Bottom: the function hα defining delta deposition. CD is aarbitrary point along the delta contour.

M. Natali et al. / Computers & Geosciences 67 (2014) 40–4844

the curve as closed. A closed curve is interpreted as the outerboundary of a delta as seen from above (map-view). The user canalso specify a scaling factor to control the height of the delta. In thefollowing, we introduce a mathematical description of our deltasections (shown in Fig. 8(bottom)).

This time, the image of function h : ½0;1�⟶½0;1� has positivevalues, since it implies a deposition, resulting in a formation of adune in the delta region. Cross-sections of deltas are describedby the function hαðzÞ ¼ αhðzÞ, where αAR and h(z) is definedas ð sin ðπz�π=2Þþ1Þ=2 8zA ½0;1�0, otherwise. Here z¼ d=dmax,where d is the distance to the contour and, different from theriver case, dmax is the maximum distance to the contour of all thepoints of the grid G inside the contour. Once more, since dZ0 anddmax40, zA ½0;1� is equivalent to drdmax, but now the height h isdirectly proportional to the distance d to the contour of the deltawhen the considered point of the grid is inside the contour.Outside the delta boundary, the height is set to zero.

During the deposition process inside a delta contour, weconsider a subset Si of the grid, in which to check if a point isinside the boundary of the delta or not, that is centred in the i-thpoint Vi of the contour C and has half-side length

Li ¼maxWAC

JVi�W J :

Modelling mountains, lakes and constant layers: There is thepossibility to create geological shapes other than rivers and deltas,by building on the previous definitions. Using the delta definitionwith negative values, we can erode the shape of a lake or basin.The user can also specify if an additional deposition of the sameamount as the erosion should be applied to effectively fill up thehole again. The fill will be a unique layer that can be assignedmaterial properties. For instance, one can use transparency to givethe effect of a water-filled lake. In such a way, lakes or valleys can beadded to the model using a negative scaling factor (as in Fig. 9), ormountains with a larger amount of deposition (as in Fig. 10). Deposi-tion taking place over water-filled rivers or lakes will act naturally onthe water by sinking to the bottom. In practice, this is realized by re-arranging relative layers so that deposit occurs before the relativelayer consisting of water. Then the water layer is reduced in height bythe amount of sunk material, but never more than its depth. Thisenables us to place the small islands appearing in the river in Fig. 2(right). In addition, layers of constant, user defined thickness may beintroduced in the layer-cake.

4. Results

The examples used in this paper are generated through a sketch-based technique to acquire a shape definition of the different relative

layers, as in Fig. 9, where the shape of each lake is defined by a singleuser stroke.

Several geological features can be generated with our method. Ariver with its sedimentary history can be achieved by drawing theinitial and the final configuration of its shape evolution, as illustratedin Fig. 3. A delta representation is obtained by sketching a closedcurve. Fig. 1 contains an example of a delta. The grain size of deltadepositions is dependent on the distance to the mouth of the riverwhich deposits the material. Heavy particles fall down first, whilesmaller particles are suspended in the flow for longer distances. Oursystem allows the user to describe this feature as shown in Fig. 7. Acolour transfer function is used to map distances to colours. Lakes andempty basins (as shown in Fig. 9) are simple shapes that are notalways relevant for exploration purposes, but important to givea context to an illustration. Mountains can be made to improvethe context of the illustration (Fig. 10) as they are often the sourceof depositional material. A composition of all the previous featuresproduces a global overview of a geological scenario, as depicted inFig. 1. To show that our method can create results comparable withillustrations, we reproduced a detailed 3D version of an illustrationdescribing fluvial evolution as demonstrated in Fig. 2.

An important characteristic of our model representation andvisualization is the ability to incorporate intermediate stages oferosion and deposition that define the sedimentary evolution of ageological scenario. This is important for fluvial systems which arecentral to understand for geologists. The illustration on the left sideof Fig. 2 was made to capture the different depositional character-istics of a river system from a proximal to distal position. It has beenmade in conventional 2D vector graphics software based on drilledwells and interpreted river footprints found in 3D seismic datasets inthe Barents Sea (Klausen et al., 2012). Such drawing processes aretime-consuming, and in this particular case the illustration took a fullday's work to complete. The resulting block diagram is also implicitlystatic, which entails that upon finalizing, it cannot be easily altered. Ittook 1 h to reproduce the illustration with our approach giving a fast,automated and interactive way of drawing the diagram makingsubsequent alterations less laborious. Cutting into the model addi-tionally gives the illustrator a unique opportunity to cross-referenceher illustration and quickly show the internal implications of externalshapes (e.g. the internal shape of a point bar drawn from above canbe examined at various angles and cuts, as shown in Figs. 3 and 11).Drawing schematic block diagrams in conventional illustration soft-ware is today a natural approach when investigating 3D seismic data,consequently our technique could be highly relevant for illustrationsbased on such data. Our geologist co-author pointed out thatwith our approach, the user benefits from an interactive, on-the-flymethod of drawing, instead of having to run through differentprocessing steps (in 3D software) or meticulous block-diagramdrawings (in 2D drawing software). Fig. 11 demonstrates the ease

Fig. 9. Two areas eroded into three layers of constant thickness. The rightmost areawas filled with material, which has been assigned a translucent water appearance.

Fig. 10. Layer-cake model with few elements. The distance-based height calcula-tion results in mountain-like features with a varying mountain ridge height.

M. Natali et al. / Computers & Geosciences 67 (2014) 40–48 45

of drawing time-series illustrations with our approach by addingadditional strokes to an initial model. Images on the left side appearin a physical geology book by Thompson and Turk (1998). We createdthe first time step by drawing two rivers using two strokes andthen additionally one stroke each for the following three images. Thebottom-right corner is a magnification of a cut-view of the lastillustration showing each deposition with individual colouring.

A model consisting of 20 layers is rendered at 40 framesper second in a window of size 1200�600. Procedurally generat-ing the 20 relative layers and transforming them to absolute layerstakes approximately 3 s with our unoptimized code. Timings havebeen performed on an Intel Xeon E5620 CPU (12 GB of RAM) withan NVIDIA GeForce GTX 580 GPU (1.5 GB of dedicated memory).

5. User evaluation

To evaluate the usability and the generality of our tool, weperformed two separate user studies on domain experts. One

study consisted of 6 geological domain experts (users A–F), whoindividually watch and give feedback on a tutorial video (acces-sible here Natali, 2013). The video demonstrates how to createeach of the geological features supported in our approach followedby a more complete example in the end. Feedback was given bymeans of grading a list of statements (Table 2) about our approachand by giving additional comments. Results from the gradings areshown in Fig. 12(a). All participants have or are taking a degreein geology. Users A, B, C, D, E, and F are respectively a Master'sstudent, PhD student, PhD student, PhD student, universityresearcher with PhD degree, geologist at Norwegian Petroleumdirectorate with Master's degree.

In our other study, an expert in the tool individually gave adifferent group of 4 users a tutorial and then let them use the toolfor about 45 min. During the session, the tool expert was presentto provide assistance, for discussions and answering questions.After each session, the participants rated the same statements asin the first study. Results are shown in Fig. 12(b). In addition, thetool expert noted comments that arose while using the tool. Users

Fig. 11. Left: sequence of illustrations showing the evolution of a meandering river with oxbow shape by Thompson and Turk (1998). Right: Reproduction with our approachrequiring only five strokes.

M. Natali et al. / Computers & Geosciences 67 (2014) 40–4846

G, H, I and J have respectively the background of professionalillustrator, post-doc in mining geology, post-doc in sedimentarygeology, professor in sedimentary geology.

Some general observations are that all average scores withineach question were higher than 0, indicating an overall positivefeedback, and all individual scores were 0 or higher, except for ascore of �1 given to question 10 by user A and question 12 by userJ which we will elaborate on. Also, participants who were able totest the tool scored each question equally or more highly thanparticipants who only watched the video, except for question 12which we will also elaborate on.

Questions 1–7 attempted to quantify the user-friendliness ofour approach. Scores were mostly positive with a few neutralindicating that the tool is easy to use. (Survey a had minimum 1 onquestion 1 and 6 and minimum 0 on the others, however all withaverage closer to 1 than to 0. Survey b had minimum 1 on all thesequestions.). Questions 8–12 attempted to quantify the usefulnessof the approach in terms of the ability to rapidly create geologicillustrations (question 8); advantages of 3D volumetric illustra-tions over 2D static illustrations (question 9); the expressivenessof the final illustrations (question 10); the ability to use the resultsfor communicating reasons (question 11) and for teaching pur-poses (question 12). Question 8 had 8 positive scores and twoneutral which can be considered as good, while question 9 had allpositive scores. Question 10 had positive scores from all of theutilizers of our method, worse scores were given by the ones who

only watched the explanatory video, although the average scorefor them was positive. User A, a Master's student, gave a negativescore based on the argument that if someone is very good inmaking 2D illustrations, then our 3D modelling approach willbe less expressive and might even be slower. Question 11 had7 positive and 3 neutral scores which could be considered a goodevaluation. User H gave score 0 based on that the approachseemed powerful for sedimentary systems, but does not go beyondthat in terms of modelling features of structural geology such asfaulting. Finally, question 12 had 9 positive and one negative score.User J, a Professor, gave the negative score because of therestrictions with our approach focusing on specific features andrequiring a computer compared to a classical classroom setting fora teacher that is trained at drawing geological illustrations ona blackboard. We also received several positive comments withrespect to the usefulness of our approach. One commented that“The system seems really useful if you study a fluvial systembehaving in a regular way”. Several commented that there is a lotof potential in our approach both for visualizing systems in 3D andfor speeding up the process of creating illustrations. Finally,volumetric illustration allowing cuts through the volume wasappreciated. The user study indicates that our approach is userfriendly and that it covers most of the structures relevant forfluvial systems and depositions but is too limited for more generalgeology such as structural geology which however was notour focus.

Suggestions for further improvements were the possibility toexport models to vector-based illustration software to be able toadd finishing touches. Two users expressed the wish of having acoordinate system that would allow them to use measurementsas input.

6. Conclusion

We have identified sketching needs by geologists who modelfluvial systems based on interpretations. To address these needswe have constructed a sketch-based interface, a layered datarepresentation and a rendering approach. The data structure leadsto an intuitive definition of the geological process of depositionand erosion and requires only basic arithmetic calculations forrealizing the model. This results in a volumetric model whichrelieves us from topology and intersection testing required for

Table 2Statements graded in the user study.

Scale is �2¼very bad, �1¼bad, 0¼neutral, 1¼good, 2¼very good.1. The approach is fast to learn how to use2. The approach is easy to use3. It is simple to sketch layers of constant thickness4. It is simple to sketch deltas/mountains5. It is simple to sketch lakes6. It is simple to sketch rivers7. It is simple to sketch sequences of intermediate rivers8. The approach allows for rapid sketching of geological illustrations9. 3D rotation and volumetric cutting into the model improvesunderstanding10. The geological features that the approach attempts at expressingare satisfactorily reproduced in the illustrations it creates11. The approach provides support for communication12. The approach is suitable for teaching purposes

Fig. 12. User feedback on our approach obtained from domain experts.

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boundary representations. This work has been performed in tightcollaboration with a geologist defining the problem domain, andwho is a co-author of the paper.

Limitations: Due to our compact representation of the modelusing 2D grids, a layer cannot fold back onto itself. Hence, we arenot able to create geological scenarios with overhangs. Withrespect to depositions in general and specifically rivers and deltas,this is not a major problem as they originate from erosional anddepositional processes that mostly work in the vertical direction.

Our approach focuses on fast creation of illustrations for com-munication purposes and unconstrained hypothesis testing. There-fore we did not build in constraints and elements of physicssimulation as this can burden the user with complex input para-meters and long processing times. It is up to the user to honourgeological and physical constraints such as unrealistic terraingradients and slope stability.

Future work: As tectonics, subsidence, slope failures and faultscan be handled by our data structure with small modifications, weplan to include these geological processes in a next prototype ofour sketch-based system. The calculation of the relative and theabsolute layers is parallelizable and could in a future version beperformed on the GPU.

Another improvement to our models would be given by includ-ing the technique of Zhu et al. (2011) to enrich channels withanimated visualization of flow.

Finally, 3D models obtained with our approach that includesome level of heterogeneity, as suggested with grain size trends indeltas (Fig. 7), could serve as training images for Multiple-PointStatistics (MPS) (Caers, 2001; Strebelle, 2002; Arpat and Caers,2005).

Acknowledgments

The authors would like to thank all reviewers who gavecomments and suggestions for improvements, and Ivan Viola forfruitful discussions. This work is funded by the Petromaks programof the Norwegian Research Council through the Geoillustratorproject (#200512).

References

Abrahamsen, P., Kolbjoernsen, O., Hauge, R., 2012. Geostatistics Oslo, Reading(1996), Thompson and Turk (1998). Springer

Amorim, R., Brazil, E.V., Patel, D., Sousa, M.C., 2012. Sketch modeling of seismichorizons from uncertainty. In: Proceedings Symposium on Sketch-Based Inter-faces and Modeling, pp. 1–10.

Arpat, G., Caers, J., 2005. A multiple-scale, pattern-based approach to sequentialsimulation. In: Quantitative Geology and Geostatistics, vol. 14. pp. 255–264.

Baojun, W., Bin, S., Zhen, S., 2009. A simple approach to 3D geological modellingand visualization. Bull. Eng. Geol. Environ. 68 (4), 559–565.

Benes, B., Forsbach, R., 2001. Layered data representation for visual simulationof terrain erosion. In: Proceedings Spring Conference on Computer Graphics,pp. 80–85.

Benes, B., Tesinsky, V., Hornys, J., Bhatia, S.K., 2006. Hydraulic erosion. Comput.Anim. Virtual Worlds 17 (2), 99–108.

Bertoncello, A., Sun, T., Li, H., Mariethoz, G., Caers, J., 2013. Conditioning surface-based geological models to well and thickness data. Math. Geosci. 45 (7),873–893.

Bridge, J., 1993. The interaction between channel geometry, water flow, sedimenttransport and deposition in braided rivers. In: Braided Rivers, vol. 75. Geolo-gical Society Special Publication, pp. 13–71.

Caers, J., 2001. Geostatistical reservoir modelling using statistical pattern recogni-tion. J. Pet. Sci. Eng. 29 (3), 177–188.

Caumon, G., Collon-Drouaillet, P., Le Carlier de Veslud, C., Viseur, S., Sausse, J., 2009.Surface-based 3D modeling of geological structures. Math. Geosci. 41 (8),927–945.

Chaikin, G., 1974. An algorithm for high speed curve generation. Comput. Graph.Image Process. 3, 346–349.

Coulthard, T.J., Van De Wiel, M.J., 2012. Modelling river history and evolution.Philos. Trans. A Math. Phys. Eng. Sci. 370 (1966), 2123–2142.

Cutler, B., Dorsey, J., McMillan, L., Müller, M., Jagnow, R., 2002. A proceduralapproach to authoring solid models. ACM Trans. Graph. 21 (July (3)), 302–311.

Gain, J., Marais, P., Straßer, W., 2009. Terrain sketching. In: Proceedings Symposiumon Interactive 3D Graphics and Games, pp. 31–38.

Gani, M.R., Bhattacharya, J.P., 2007. Basic building blocks and process variability of acretaceous delta: internal facies architecture reveals a more dynamic interac-tion of river, wave, and tidal processes than is indicated by external shape.J. Sediment. Res. 77, 284–302.

Hinderer, M., 2012. From gullies to mountain belts: a review of sediment budgets atvarious scales. Sediment. Geol. 280, 21–59.

Hnaidi, H., Guérin, E., Akkouche, S., Peytavie, A., Galin, E., 2010. Feature basedterrain generation using diffusion equation. Comput. Graph. Forum 29 (7),2179–2186.

Klausen, T., Helland-Hansen, W., Laursen, I., Gawthorpe, R., 2012. Fluvial Geome-tries and Reservoir Characteristics of a Triassic Coastal Plain System in theNorwegian Barents Sea. AAPG Search and Discovery.

Lemon, A., Jones, N., 2003. Building solid models from boreholes and user-definedcross-sections. Comput. Geosci. 29 (5), 547–555.

Lidal, E.M., Natali, M., Patel, D., Hauser, H., Viola, I., 2013. Geological storytelling.Comput. Graph. 37, 445–459.

Lopez, S., Galli, A., Cojan, I., 2001. Fluvial meandering channelized reservoirs: astochastic and process-based approach. In: Proceedings Annual Conference ofthe International Association of Mathematical Geologists.

Michaels, H., Gorelick, S., Sun, T., Li, H., Boucher, A., Caers, J., 2010. Combininggeologic-process models and geostatistics for conditional simulation of 3-dsubsurface heterogeneity. Water Resour. Res. 46.

Natali, M., 2013. Illustrative Modelling of Fluvial Systems. ⟨http://folk.uib.no/mna024/public/tutorial.mp4⟩.

Natali, M., Lidal, E. M., Parulek, J., Viola, I., Patel, D., 2013. Modeling terrains andsubsurface geology. In: Eurographics 2013. Girona, Spain.

Natali, M., Viola, I., Patel, D., 2012. Rapid visualization of geological concepts. In:Proceedings 25th SIBGRAPI Conference on Graphics, Patterns and Images.

Peytavie, A., Galin, E., Merillou, S., Grosjean, J., 2009. Arches: a framework formodeling complex terrains. Comput. Graph. Forum (Proc. Eurographics) 28 (2),457–467.

Pore¸bski, S.J., Steel, R.J., 2003. Shelf-margin deltas: their stratigraphic significanceand relation to deepwater sands. Earth-Sci. Rev. 62, 283–326.

Pyrcz, M., 2004. The Integration of Geologic Information into Geostatistical Models(Ph.D. thesis). University of Alberta, Edmonton, Alberta.

Pyrcz, M., Catuneanu, O., Deutsch, C., 2005. Stochastic surface-based modeling ofturbidite lobes. Am. Assoc. Pet. Geol. Bull. 89 (2), 177–191.

Pyrcz, M., McHargue, T., Clark, J., Sullivan, M., Strebelle, S., 2012. Event-basedgeostatistical modeling: description and applications. In: Geostatistical Con-gress, Oslo, Norway.

Reading, H., 1996. Sedimentary Environments: Processes, Facies and Stratigraphy.John Wiley & Sons

Stava, O., Benes, B., Brisbin, M., Krivanek, J., 2008. Interactive terrain modeling usinghydraulic erosion. In: Proceedings ACM SIGGRAPH/Eurographics Symposiumon Computer Animation, pp. 201–210.

Strebelle, S., 2002. Conditional simulation of complex geological structures usingmultiple-point statistics. Math. Geol. 34 (1).

Takayama, K., Sorkine, O., Nealen, A., Igarashi, T., 2010. Volumetric modeling withdiffusion surfaces. ACM Trans. Graph. (Proc. ACM SIGGRAPH) 29 (6).

Thompson, G., Turk, J., 1998. Introduction to Physical Geology. Saunders GoldenSunburst Series. Saunders College Pub.

Wang, L., Yu, Y., Zhou, K., Guo, B., 2011. Multiscale vector volumes. ACM Trans.Graph. 30 (December (6)) 167:1–167:8.

Xie, Y., Deutsch, C.V., Cullick, A.S., April 2000. Surface-geometry and trend modelingfor integration of stratigraphic data in reservoir models. In: GEOSTATS 2000,Geostatistical Congress.

Zhu, B., Iwata, M., Haraguchi, R., Ashihara, T., Umetani, N., Igarashi, T., Nakazawa, K.,2011. Sketch-based dynamic illustration of fluid systems. In: ProceedingsSIGGRAPH Asia. SA '11. pp. 134:1–134:8.

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