Comparing LLM-Generated DEMs to Traditional Fractal and Displacement Methods

AA Digital Elevation Model (DEM) is a raster grid that assigns an elevation value to each cell of a geographic area, and it underpins most terrain work in GIS, hydrology, disaster planning and computer graphics alike [1]. Two distinct traditions produce them for synthetic landscapes — the fractal/procedural-noise family used in games and simulation, and the deep-learning family that has more recently absorbed diffusion models and, tentatively, large language models. This report compares the two traditions on their own terms: what each actually optimises for, where each breaks, and what a hybrid pipeline would need to reconcile.

What a DEM Is and How It Is Normally Made

DEMsDEMs come in two flavours that are frequently conflated. A Digital Terrain Model (DTM) represents the bare-earth surface with vegetation and structures stripped out; a Digital Surface Model (DSM) keeps everything on top of the ground — canopy, rooftops, powerlines. Confusingly, the USGS's own "DEM" terminology historically means bare-earth (i.e. closer to a DTM), which is a genuine source of cross-dataset confusion worth knowing about before comparing two "DEM" datasets from different providers [2].

Real-world DEMs are built from LiDAR, photogrammetry, synthetic-aperture radar, ground survey, or digitised contour lines, each trading resolution, cost and vegetation penetration against the others. None of that machinery is relevant to synthetic terrain — the subject of this report — where there is no ground to survey and the entire elevation field has to be invented by an algorithm or a model. That invented-terrain problem, not real-world DEM acquisition, is where fractal methods and LLMs actually compete.

The Fractal and Displacement Tradition

ProceduralProcedural terrain has relied on the same handful of ideas since the early 1980s. Fournier, Fussell and Carpenter's 1982 paper modelled terrain as a sample path of fractional Brownian motion, giving a way to compute a surface to arbitrary detail without storing the whole thing in advance [3]. Perlin's 1985 "Image Synthesizer" introduced gradient noise as a cheaper, more controllable substitute for exact fractal computation, and Perlin noise (and its later Simplex variant) has been the default heightfield primitive in games ever since [4].

Two recursive-subdivision algorithms sit alongside noise-based methods:

• Midpoint Displacement starts from a line or square with random corner values, recursively subdivides, and displaces each new midpoint by a random offset that shrinks with each iteration.
• Diamond-Square extends the same idea to 2D heightmaps via alternating "diamond" and "square" averaging-plus-displacement steps, and is the more common choice for heightmap generation in practice.

All of these methods are deterministic given a seed, cheap enough to run in real time, and easy to reason about — the roughness parameter (the Hurst exponent, informally "H") has a direct, predictable effect on the output. Their well-documented failure mode is artefactual regularity: naive Midpoint Displacement produces visible grid-aligned creases, and both methods struggle to produce anything that doesn't look "fractal" — river deltas, sedimentary terracing, or the low-frequency asymmetry of real mountain ranges are outside what self-similar noise can represent without substantial hand-authored post-processing.

More recent work narrows that gap without abandoning the noise-based family. Thorimbert and Chopard's polynomial method generates terrain from coherent noise at speeds competitive with fractional Brownian motion while producing comparable fractal-dimension statistics to real terrain data, generalising to arbitrary dimensions and gradient smoothness [5]. Jain, Sharma and Rajan combined a diffusion-based generative network with Perlin-noise-based patch blending to get infinite, multi-resolution terrain that keeps the controllability of noise while borrowing structure from a learned model [6] — itself evidence that the fractal tradition and the learned-model tradition are already converging rather than competing head-on.

Where Generative and Language Models Enter

DeepDeep generative models arrived in this space well before LLMs did, and it is worth keeping the two apart. Beckham and Pal trained a DCGAN on NASA satellite-derived heightmaps and showed it could generate plausible 512px heightfields from a latent vector, without any hand-written noise function [7] — evidence that a purely learned model can substitute for procedural noise on realism grounds, at the cost of losing the direct parametric control noise-based methods offer. Gavriil, Muntingh and Barrowclough applied a Wasserstein GAN with contextual attention to a genuinely different problem — filling voids in real DEMs captured over Norway — and found it produced semantically plausible fill data where classical interpolation could not [8]. Howard et al. compared Perlin noise, Diamond-Square and Worley noise against each other specifically as terrain generators for curriculum reinforcement learning, and found real, usable differences between them in the diversity and difficulty of the terrains each one produces — a reminder that "which noise function" is not a cosmetic choice even within the purely fractal family [9].

LLMs proper are a much newer and thinner strand of this work, and the claims made for them should be read narrowly. The closest documented system, ChatGeoAI, is built on Llama 2 and translates natural-language requests into executable PyQGIS scripts for geospatial analysis [10] — a genuinely useful capability, but one that operates on existing geospatial data and workflows rather than generating a novel elevation field from nothing. A 2024 systematic review of GPT/LLM/GAI use in geospatial science screened close to 300 papers down to 26 that actually applied these models to geospatial tasks [11] — a small and recent literature, concentrated on query interpretation, code generation and data-format translation rather than direct terrain synthesis. No published, reproducible pipeline in this literature has an LLM emit an elevation grid the way a GAN or a noise function does; what exists is LLMs orchestrating other tools (GIS libraries, GAN pipelines) around the actual terrain-generation step, not replacing it.

Comparison

Aspect	Fractal / noise-based	Learned (GAN/diffusion)	LLM-orchestrated
What it actually produces	An elevation grid, directly, from a seed and parameters	An elevation grid, sampled or reconstructed from a trained model	Code or a plan that invokes another tool; not the grid itself
Data requirement	None — generated from scratch	A training set of real or synthetic DEMs	Pretraining corpus plus, typically, a downstream generative or GIS tool
Determinism / reproducibility	Fully deterministic given a seed	Stochastic sampling; reproducible only with a fixed seed and checkpoint	Stochastic at the language layer, on top of whichever tool it drives
Realism ceiling	Bounded by self-similar statistics; struggles with non-fractal features	Can learn non-fractal structure present in training data	Inherits whichever downstream method it orchestrates
Compute cost	Cheap; real-time capable	Training is expensive; inference is usually fast	Adds inference cost on top of the downstream tool's own cost
Maturity for this specific task	Forty-plus years of production use	Active research since roughly 2017 (GANs), extending to diffusion since 2022	A handful of papers since 2023, mostly on query/workflow tasks, not synthesis

Synthesis

TheThe honest comparison is not "LLMs versus fractals" but three tiers of maturity answering different questions. Fractal and noise-based methods remain the right default when the requirement is real-time, parametrically controllable, artefact-free terrain, and forty years of production use back that up. Learned generative models (GANs, and now diffusion) are the right tool when the requirement is matching the statistical texture of real terrain, or filling gaps in real captured data, and the recent Perlin-plus-diffusion hybrids suggest the two traditions are already merging rather than one replacing the other. LLMs, on the current evidence, are not yet a terrain-generation method in the same sense as the other two — they are a natural-language front end that can drive a GIS pipeline or a generative model, which is valuable, but claims that an LLM itself synthesises a novel elevation field belong to a different and much thinner part of the literature than claims about GANs or diffusion doing so.

The clearest opportunity is not asking an LLM to replace Diamond-Square, but using it where its actual demonstrated strength lies — as the natural-language interface that selects, parameterises and chains the fractal, GAN and diffusion methods above, the way ChatGeoAI already does for existing GIS workflows. That is a smaller and more defensible claim than "LLMs generate terrain," but it is the one the evidence in this literature actually supports.

References

1. Elevation Modeling — DEM, DTM and DSM: Definitions, Differences, and Uses. GIS Geography. https://gisgeography.com/dem-dsm-dtm-differences/
2. Digital elevation model. Wikipedia. https://en.wikipedia.org/wiki/Digital_elevation_model
3. Fournier, A., Fussell, D., & Carpenter, L. (1982). Computer Rendering of Stochastic Models. Communications of the ACM, 25(6), 371–384. https://dl.acm.org/doi/10.1145/358523.358553
4. Perlin, K. (1985). An Image Synthesizer. ACM SIGGRAPH Computer Graphics, 19(3), 287–296. https://dl.acm.org/doi/10.1145/325334.325247
5. Thorimbert, Y., & Chopard, B. (2016). Polynomial methods for Procedural Terrain Generation. arXiv:1610.03525. https://arxiv.org/abs/1610.03525
6. Jain, A., Sharma, A., & Rajan, K. S. (2022). Adaptive & Multi-Resolution Procedural Infinite Terrain Generation with Diffusion Models and Perlin Noise. Proceedings of the Thirteenth Indian Conference on Computer Vision, Graphics and Image Processing (ICVGIP '22). https://dl.acm.org/doi/abs/10.1145/3571600.3571657
7. Beckham, C., & Pal, C. (2017). A step towards procedural terrain generation with GANs. arXiv:1707.03383. https://arxiv.org/abs/1707.03383
8. Gavriil, K., Muntingh, G., & Barrowclough, O. J. D. (2019). Void Filling of Digital Elevation Models with Deep Generative Models. IEEE Geoscience and Remote Sensing Letters, 16(10), 1645–1649. https://arxiv.org/abs/1811.12693
9. Howard, D., Kannemeyer, J., Dolcetti, D., Munn, H., & Robinson, N. (2022). Assessing Evolutionary Terrain Generation Methods for Curriculum Reinforcement Learning. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO '22). https://arxiv.org/abs/2203.15172
10. ChatGeoAI: Enabling Geospatial Analysis for Public through Natural Language, with Large Language Models. ISPRS International Journal of Geo-Information, 13(10), 348. https://www.mdpi.com/2220-9964/13/10/348
11. GPT, large language models (LLMs) and generative artificial intelligence (GAI) models in geospatial science: a systematic review. International Journal of Digital Earth. https://www.tandfonline.com/doi/full/10.1080/17538947.2024.2353122