# Assistant conversation T839GX - Canonical URL: - Resource IRI: - Type: assistant conversation - Mode: chat ## Transcript ### 1. User can you explain the chi transform and how it's used for the stream power erosion stuff? ### 2. Assistant tool calls #### Tool call: `search_text` ```json { "query": "chi transform stream power erosion χ ksn concavity" } ``` ### 3. Tool result: search_text Exact matches 1. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 8 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #A8NK92 6. Results #AWRWEN 6.4. Stream power erosion Matching excerpt #76252N: Stream power erosion without the uplift. The stream power erosion can be used without the uplift and it adds a global hydrological realism to an existing scene as shown on an example of a fractal terrain enhanced with erosion in Figure 14. As the erosion converges toward a flat terrain, it is necessary to use small time steps and to stop the simulation after only a few iterations. 2. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 0 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #LEPLXD Abstract Matching excerpt #EU9PCE: At large scale, landscapes result from the combination of two major processes: tectonics which generate the main relief through crust uplift, and weather which accounts for erosion. This paper presents the first method in computer graphics that combines uplift and hydraulic erosion to generate visually plausible terrains. Given a user-painted uplift map, we generate a stream graph over the entire domain embedding elevation information and stream flow. Our approach relies on the stream power equation introduced in geology for hydraulic erosion. By combining crust uplift and stream power erosion we generate large realistic terrains at a low computational cost. Finally, we convert this graph into a digital elevation model by blending landscape feature kernels whose parameters are derived from the information in the graph. Our method gives high-level control over the large scale dendritic structures of the resulting river networks, watersheds, and mountains ridges. 3. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 5 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #7E86TB 5. Erosion Matching excerpt #ECXQM8: Correction based on thermal erosion. While the simulation of the stream power equation works efficiently for carving the bottom of the rivers, other phenomena may be predominant in some cases, in particular for low drainage areas. In such cases, the stream erosion equation produces unrealistic sharp and high peaks. 4. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 2 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #ZRSZUM 3. Background and Overview #JT9864 3.1. Geological Background Matching excerpt #7GS7F8: Fluvial erosion is the erosion of the bedrock material and its transportation downhill by streams. It is caused by the shear stress exerted by running water and the sediment it contains onto the bed of a stream. The interaction between the fluvial erosion and the tectonic uplift has been studied for many years in geology and is usually modeled by the stream power equation [WT99]: 5. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 10 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #PZFF56 7. Results #BK5LST 7.2. Ablation study Matching excerpt #3WX3ZE: Hillslope erosion stabilizes the slopes of the mountains and acts as the main erosion process at low drainage where the stream power law becomes negligible [LD03]. We show the impact of hillslope in Figure 9, where we show the analytical solutions of the stream power law only at \delta x = 50 m (left), compared with our modified formulation that includes the hillslope erosion (Section 6.1, right of Figure 9). Without hillslope, we observe the emergence of unrealistically sharp ridges and peaks. 6. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 7 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #97YQQD 6. Other erosion factors Matching excerpt #WDG5SJ: The stream power law presents a singularity when the drainage area reaches zeros: when time increases, the solution converges to infinitely steep ridges. This is documented by geologists [LD03], who suggest that the stream power is not applicable in these locations, or at least dominated by other processes. We explore two possible predominant processes: hillslope erosion and landslides, and propose approximations to easily integrate them into our algorithm. 7. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 2 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #QW6CT7 3. Background and overview #MYA2EJ 3.1. Landscape dynamics in geology Matching excerpt #ESC38J: A first common modeling approximation is to consider surface evolution as a detachment-limited process (as opposed to transport-limited) where the evolution of surface elevation z is directly related to the competition between rock uplift and erosion rate, and not to the capacity of the rivers to transport or deposit sediments [How94]. In this setting, erosion is generally described using the stream power incision model [HK83, How94, WT99, Lag14], where erosion rate is a power law of the surface slope \|\nabla z\| , and drainage area A that acts as a proxy for the river discharge. The drainage area A(\mathbf{x}) is defined at a position \mathbf{x} as the area of the drainage basin - or catchment - upstream of \mathbf{x} . Coupled with the uplift u , the Stream Power Law expresses the rate of change of surface elevation: 8. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 12 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #PZFF56 7. Results #VAST65 7.4. Limitations Matching excerpt #9NTKS2: Another limitation is inherent to analytical formulation, which restricts the possible combination with other erosion laws. We proposed new models for hillslope and thermal erosion which have a critical impact when fluvial erosion is low but including other processes, sediment deposition, or simply a non-linear dependency to the slope in the stream power law (the exponent n \neq 1 ) might prove challenging. 9. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 7 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #34USBX 6. Application to landscape simulation #BTBPNF Algorithm 6: Extracting a depression-free water surface Matching excerpt #QKYV46: Terrain erosion. Flow routing is a critical component of terrain erosion, as it provides the mechanism by which material is worn away and transported, ultimately over geological time shaping the mountains themselves. Furthermore, even the ordering of recipient cells plays a role in the implicit integration of the stream power law erosion equation. This is expressed as [CBC*16]: 10. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 7 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #34USBX 6. Application to landscape simulation Matching excerpt #V92Q79: Here we show how we apply flow and depression routing to accelerate landscape simulation, with use cases in river and lake modeling, terrain erosion, and ecosystem simulation. In particular, our GPU algorithms together provide the discharge necessary for erosion simulation. Furthermore, they enable a novel GPU-based solution for implicit time-stepping of erosion using the Stream Power Law, which we combine with a simulation of sediment deposition. Approximate matches 1. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 2 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #ZRSZUM 3. Background and Overview #JT9864 3.1. Geological Background Score: 0.027 Related excerpt #PGNGMP: The stream power equation states that the rate of change of surface topography h(\mathbf{p}) at a position \mathbf{p} is controlled by the balance between the surface uplift u(\mathbf{p}) and the fluvial erosion, which is a function of the local slope s(\mathbf{p}) and the drainage area A(\mathbf{p}) . The local slope s(\mathbf{p}) is defined as the surface topographic gradient: 2. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 7 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #A8NK92 6. Results #AWRWEN 6.4. Stream power erosion Score: 0.027 Related excerpt #L66ZXE: In our approach, we allow only the changes to the drainage area exponent m . Indeed, only the ratio between m and n has a meaning that we can deduce from the equations. Let us suppose we have reached an equilibrium state in a region where u and k are constant over the space. The stream power Equation (1) becomes: 3. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 2 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #ZRSZUM 3. Background and Overview #JT9864 3.1. Geological Background Score: 0.024 Related excerpt #EKXATA: The constants m and n depend on rock strength, climate, and the topology of river networks. While the values of those parameters are poorly understood, the ratio m/n is constrained by the shape of the stream profiles and is thought of being m/n \approx 0.5 [WT99]. As in most geomorphological studies, we use n = 1 and m = 0.5 . Moreover, some geological studies attempt to tune these parameters by example [CB14] and a recent survey [Lag14] studies the limit of geological knowledge regarding the parameters of the stream power equation. 4. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 4 Context: #JJE8HN Procedural Riverscapes #TFD8TX 5. River Network Amplification #S7JXXD 5.2. Riverbed Carving Score: 0.023 Related excerpt #B77LY9: C(r) in Figure 3, but also depend on the local curvature of the river trajectory. They are normalized in the sense that their scale assumes unit area for water in the cross section, and when instantiated, they are scaled according to a factor derived from the flow \phi and elevation of the trajectory. 5. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 4 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #UE2TM4 4. Analytical solutions of the stream power law #QR788H 4.1. The method of characteristics for the stream power law Score: 0.025 Related excerpt #MLVFNW: Our first observation is that the spatial component of the characteristic curve follows the direction of the topographic gradient \frac{\nabla z}{\|\nabla z\|} . Intuitively, this means that the elevation at any point x in the terrain will only depend on elevations downstream of x , by following the path of steepest gradients. We call this path a river path as this is the trajectory naturally followed by the water, and parameterize it by s , the distance between the bound ( s = 0 ) and any point on the path (Figure 3, left). 6. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 7 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #97YQQD 6. Other erosion factors Score: 0.025 Related excerpt #WDG5SJ: The stream power law presents a singularity when the drainage area reaches zeros: when time increases, the solution converges to infinitely steep ridges. This is documented by geologists [LD03], who suggest that the stream power is not applicable in these locations, or at least dominated by other processes. We explore two possible predominant processes: hillslope erosion and landslides, and propose approximations to easily integrate them into our algorithm. 7. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 2 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #QW6CT7 3. Background and overview #MYA2EJ 3.1. Landscape dynamics in geology Score: 0.023 Related excerpt #2DEC4W: where k , m and n are erosion coefficients. Throughout the paper, we will use some of the common values: m = 0.4 and n = 1 . The choice of n = 1 , also commonly used in geomorphology, makes the equation linear and therefore simplifies the derivation of the analytical solutions. While this choice barely impacts the result as the valley profiles are mostly directed by the ration m/n , we acknowledge that the actual values of m and n remain an open question in geomorphology [Lag14]. 8. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 8 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #34USBX 6. Application to landscape simulation #53AQ2N Algorithm 7: Implicit fluvial erosion Score: 0.025 Related excerpt #VPGNAT: Fluvial erosion is not the only erosive process responsible for shaping terrain. Hillslope processes model the gradual accumulation of solid material at the base of mountains and hills [BS97] and is usually expressed as a diffusion equation. We follow Tzathas et al. [TGSC24] and approximate it by including additional terms in the Stream Power Equation, changing kQ^m to kQ^m + k_t + k_h A^{-h} , where k_t and h = 0.6 are hillslope erosion parameters, A is the drainage area (obtained via flow routing with precipitation set uniformly to p = \Delta x^2 ). The parameter k_t accounts for slope-dependent effects (landslides, debris-flow), regrouped in computer graphics under the catch-all term thermal erosion [MKM89]. 9. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 8 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #34USBX 6. Application to landscape simulation #53AQ2N Algorithm 7: Implicit fluvial erosion Score: 0.023 Related excerpt #VU6KLE: where k_d is the sediment deposition coefficient and Q_s is sediment flux, obtained by accumulating the negative elevation balance -\Delta x^2 \frac{\partial z}{\partial t} downstream with flow routing. While Yuan et 10. Source: Scalable real‐time animation of rivers (#XDESU9), Eric Bruneton, Fabrice Neyret, Nicolas Holzschuch, Qizhi Yu, p. 8 Context: #B9LYTR Scalable real-time animation of rivers #TJAYJS 7. Discussion and limitations Score: 0.023 Related excerpt #4Y2DNV: 2D flow hypothesis and terrain slope Our 2D flow hypothesis is valid for constant water depth (and homogeneous velocity profile along each water column). To account for depth h(x,y) variations we should simply conserve q(x,y) = v(x,y)h(x,y) instead of v \cdot \nabla \cdot q = 0 , q = \nabla \times \psi , v = q/h . This supposes either to know h or to deduce it from the terrain elevation z(x,y) . In our static case, the Chézy law provides a convenient approximation: v = C\sqrt{Rs} with C the Chézy constant, s the slope, v = Q/S the average velocity in a vertical section of surface S , perimeter P , and hydraulic radius R = S/P . Assuming the section has a known shape, e.g., a rectangle of known length l and height h , this yields h as a function of s, l, Q . ### 4. Assistant tool calls #### Tool call: `get_document` ```json { "id": "96ZMGK" } ``` #### Tool call: `get_document` ```json { "id": "DWXKYQ" } ``` #### Tool call: `search_text` ```json { "query": "channel steepness chi plot integral drainage area reference concavity" } ``` #### Tool call: `search_text` ```json { "query": "steady state uplift erosion stream power slope area relationship" } ``` ### 5. Tool result: get_document DOCUMENT #96ZMGK Title: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion Outline: - 1 #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion - 1.1 #LEPLXD Abstract - 1.2 #T4ZQ7N 1. Introduction - 1.3 #X3SQUZ 2. Related Work - 1.4 #ZRSZUM 3. Background and Overview - 1.4.1 #JT9864 3.1. Geological Background - 1.4.2 #DH5T8U 3.2. Algorithm Overview - 1.4.3 #3YZ889 4.3. Lake Overflow - 1.5 #7E86TB 5. Erosion - 1.6 #A8NK92 6. Results - 1.6.1 #63EVKU 6.1. Visual realism - 1.6.2 #CBX4QV 6.2. Rendering - 1.6.3 #2RYGP7 6.3. Performance - 1.6.4 #AWRWEN 6.4. Stream power erosion - 1.7 #RRBERX 7. Conclusion - 1.8 #D9ZPU5 References ### 6. Tool result: get_document DOCUMENT #DWXKYQ Title: Physically-based analytical erosion for fast terrain generation Outline: - 1 #JDHNVB Physically-based analytical erosion for fast terrain generation - 1.1 #87Z54H Abstract - 1.2 #8S8H8Z CCS Concepts - 1.3 #TY8V58 1. Introduction - 1.4 #2BWKC4 2. Previous Work - 1.5 #QW6CT7 3. Background and overview - 1.5.1 #MYA2EJ 3.1. Landscape dynamics in geology - 1.5.2 #Q878EF 3.2. Challenges and algorithm - 1.6 #UE2TM4 4. Analytical solutions of the stream power law - 1.6.1 #QR788H 4.1. The method of characteristics for the stream power law - 1.6.2 #S7W7HC 4.2. Recursive algorithm for the 1D analytical solutions - 1.6.3 #5CR379 4.3. Extension to the 2D terrain domain - 1.7 #4VW4H2 5. Combining river network and elevations - 1.7.1 #JHZBJ3 5.1. Fixed-point algorithm - 1.7.2 #3F6MC3 5.2. Accelerating the convergence via multigrid - 1.7.3 #USAEE8 5.3. Optimization-based altitude correction - 1.8 #97YQQD 6. Other erosion factors - 1.8.1 #C4WS38 6.1. Hillslope erosion - 1.8.2 #VEK6N9 6.2. Thermal erosion - 1.9 #PZFF56 7. Results - 1.9.1 #4Q9G6F 7.1. Validation and comparison - 1.9.2 #BK5LST 7.2. Ablation study - 1.9.3 #V6AZDH 7.3. Applicability of our method - 1.9.4 #VAST65 7.4. Limitations - 1.10 #US5Z9M 8. Conclusion - 1.11 #99Q6MG Acknowledgements - 1.12 #7NBERZ Appendix A: Gradient Derivation - 1.13 #5A4SE2 References ### 7. Tool result: search_text Exact matches 1. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 2 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #ZRSZUM 3. Background and Overview #JT9864 3.1. Geological Background Matching excerpt #GPZB7Q: The drainage area A(\mathbf{p}) is the upstream area draining through point \mathbf{p} , assuming that water flows along the topographic gradient (Figure 3). In our implementation, the terrain is represented by a geometric graph \mathcal{G} connecting points sampled over the terrain domain. The drainage area A(\mathbf{p}) is the area associated to the set of points \{\mathbf{q} \in \mathcal{G}\} strictly above \mathbf{p} such that there exists one path of strictly increasing height starting from \mathbf{p} and ending at \mathbf{q} . The factor k is an erosion constant that depends on many factors, such as lithology (the composition of the soil/bedrock), vegetation, climate, and climate variability. 2. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 5 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #7E86TB 5. Erosion Matching excerpt #HUHFWU: Drainage and slope. Let \mathcal{N}_k denote a node of the graph-covering stream trees \tilde{\mathcal{T}} and C(\mathcal{N}_k) the set of its children nodes. The drainage area A_k can be computed using the recursive formula: 3. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 2 Context: #JJE8HN Procedural Riverscapes #Z6DVCN 4. River Network Graph Matching excerpt #JGM58Q: We begin by computing the drainage area A_{ij} for every cell C_{ij} of the input terrain \mathcal{T} (using the method of Freeman [Fre91]). The discrete river network \mathcal{D} is then simply the set of cells that have a drainage area greater than a user-controlled threshold value (see 4. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 3 Context: #JJE8HN Procedural Riverscapes #Z6DVCN 4. River Network Graph Matching excerpt #TRPJZX: Figure 5). One complication is that the drainage calculation assumes that there are no depressions (local minima) in the digital elevation model, since these have no outlet to neighboring cells. To prevent disconnected river graphs we apply an optimal depression filling algorithm [BLM14], which leaves the surface slightly proud with an available flow channel. 5. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 4 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #UE2TM4 4. Analytical solutions of the stream power law #QR788H 4.1. The method of characteristics for the stream power law Matching excerpt #SP6MM5: Compared to previous simulation-based methods that required iterating over time, this solution directly expresses the elevation of the terrain from the uplift and drainage area. This is why we follow geology literature [Ste21] to call this an analytical solution (with respect to time), even if we need to resort to a numerical evaluation of the integral over space. 6. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 10 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #PZFF56 7. Results #BK5LST 7.2. Ablation study Matching excerpt #2P93PV: Choice of the receiver. In Section 4.3, we suggested choosing the receiver randomly among the neighbors below each node. In Figure 8, we compare this strategy with the choice of the lowest neighbor (minimum receiver). Choosing the lowest neighbor leads to the formation of uniform axis-aligned slopes. In practice, this choice constrains the rivers to follow a straight line, reducing the possible convergence of small streams. This is the reason for both the axis-aligned appearance of the slopes and their steepness, as non-merging streams reduce the drainage area locally and therefore the erosion. 7. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 10 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #PZFF56 7. Results #BK5LST 7.2. Ablation study Matching excerpt #TMJFHY: in Figure 10 (left), but the stream power law does not let us control the steepness of the cliff, only the speed at which they propagate away from the low boundary node following the drainage patterns. In contrast, adding thermal erosion as explained in Section 6.2 allows us to adjust the critical angle, and hence the shape of the slopes (Figure 10, right). 8. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 2 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #QW6CT7 3. Background and overview #MYA2EJ 3.1. Landscape dynamics in geology Matching excerpt #ESC38J: A first common modeling approximation is to consider surface evolution as a detachment-limited process (as opposed to transport-limited) where the evolution of surface elevation z is directly related to the competition between rock uplift and erosion rate, and not to the capacity of the rivers to transport or deposit sediments [How94]. In this setting, erosion is generally described using the stream power incision model [HK83, How94, WT99, Lag14], where erosion rate is a power law of the surface slope \|\nabla z\| , and drainage area A that acts as a proxy for the river discharge. The drainage area A(\mathbf{x}) is defined at a position \mathbf{x} as the area of the drainage basin - or catchment - upstream of \mathbf{x} . Coupled with the uplift u , the Stream Power Law expresses the rate of change of surface elevation: 9. Source: Recommended water renderer for procedural hydrological terrain (#JVRSKS) Matching note #JVRSKS: For a game with precomputed geological erosion and hydrology, the best fit is a stylized data-driven hybrid rather than runtime CFD. Reuse channel topology, banks, flow direction, discharge/drainage area, slope, depth/width, curvature, drops, junctions, obstacles, and distance-to-shore as shader/control fields. This closely matches the input assumed by scalable river animation (#QXYWAJ) and Procedural Riverscapes, which derives per-cell slope, volume, and velocity (#X3RVN8) and selects calm, turbulent, wave, cascade, vortex, and ripple primitives from terrain and flow conditions (#GU2NEL, #5NJY7V). Recommended architecture: one shared water material; rivers use generated flow maps to advect two offset normal/detail layers as in Portal 2 (#6ELMAT, #XVFV3N); lakes use low-speed wind ripples and shoreline masks; ocean uses a few art-directed Gerstner/spectral bands plus shore foam. Generate masks for turbulence/foam from normalized stream power, slope, curvature, constriction, drops, and obstacles; use depth for color/opacity and shallow-ground blending; use local feature primitives only at visually important events such as waterfalls, rapids, confluences, and rocks. Apply screen- or distance-dependent LOD, retaining flow direction and wind at distance while removing displacement and local effects (#H2E2UR, #EYM9N6). 10. Source: Legacy sediment: Definitions and processes of episodically produced anthropogenic sediment (#K82AS7), L. Allan James, p. 6 Context: #8A96NY Legacy sediment: Definitions and processes of episodically produced anthropogenic sediment #L3T5NJ 5. Processes governing LS deposition and preservation #7QBVBC 5.1. Sediment production, delivery, and transport capacity Matching excerpt #P2C726: Sediment delivery depends not only on sediment production on hill slopes, but also on conditions that govern deposition and recruitment, including transport capacity, sediment characteristics, and valley-bottom conditions. Many of these factors are scale-dependent and vary systematically with drainage area. Sediment characteristics that influence deliveries include grain size, shape, cementation, imbrication, and armoring. Relevant valley-bottom factors include morphology, floodplain width, position relative to channels, geologic structure, valley gradient, base-level, history of sea-level change, previous history of channel aggradation or incision, glacial history, and human alterations (channel-bed mining, dams, levees, etc.) (Belmont, 2011; Blum and Törnqvist, 2000; Nardi et al., 2006). Storage potential also depends on local Approximate matches 1. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 5 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #7E86TB 5. Erosion Score: 0.029 Related excerpt #HUHFWU: Drainage and slope. Let \mathcal{N}_k denote a node of the graph-covering stream trees \tilde{\mathcal{T}} and C(\mathcal{N}_k) the set of its children nodes. The drainage area A_k can be computed using the recursive formula: 2. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 2 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #ZRSZUM 3. Background and Overview #JT9864 3.1. Geological Background Score: 0.027 Related excerpt #GPZB7Q: The drainage area A(\mathbf{p}) is the upstream area draining through point \mathbf{p} , assuming that water flows along the topographic gradient (Figure 3). In our implementation, the terrain is represented by a geometric graph \mathcal{G} connecting points sampled over the terrain domain. The drainage area A(\mathbf{p}) is the area associated to the set of points \{\mathbf{q} \in \mathcal{G}\} strictly above \mathbf{p} such that there exists one path of strictly increasing height starting from \mathbf{p} and ending at \mathbf{q} . The factor k is an erosion constant that depends on many factors, such as lithology (the composition of the soil/bedrock), vegetation, climate, and climate variability. 3. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 3 Context: #JJE8HN Procedural Riverscapes #Z6DVCN 4. River Network Graph Score: 0.023 Related excerpt #HS22AT: Next, graph nodes are labeled with the terrain slope s and river flow \phi values at their cell position. The latter is a measure of the volumetric rate at which water is carried down the river and an accurate estimation is problematic, since it depends on parameters such as rainfall and soil composition. Instead, we apply a simplified model based on an empirical power law observed in geomorphology [Dun78]: from drainage area A_{ij} [ m^2 ], the flow \phi_{ij} of the river [ m^3 s^{-1} ] is approximated by \phi_{ij} = 0.42A_{ij}^{0.69} . This equation takes into account evaporation and infiltration, which is why the volume of flow is not preserved. 4. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 4 Context: #JJE8HN Procedural Riverscapes #TFD8TX 5. River Network Amplification #S7JXXD 5.2. Riverbed Carving Score: 0.022 Related excerpt #C9VQSA: Rosgen type C (see Figure 12) has strong curvature in its trajectory with a low overall slope (less than 2%). A peculiarity is that the cross-sectional profile is asymmetric in sections of high curvature. A combination of symmetric and asymmetric profiles is thus required and they are combined through linear interpolation. 5. Source: Procedural Riverscapes (#AK7NGE), A. Peytavie, B. Benes, E. Galin, E. Guérin, J. Gain, T. Dupont, Y. Cortial, p. 4 Context: #JJE8HN Procedural Riverscapes #TFD8TX 5. River Network Amplification #S7JXXD 5.2. Riverbed Carving Score: 0.021 Related excerpt #B77LY9: C(r) in Figure 3, but also depend on the local curvature of the river trajectory. They are normalized in the sense that their scale assumes unit area for water in the cross section, and when instantiated, they are scaled according to a factor derived from the flow \phi and elevation of the trajectory. 6. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 8 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #97YQQD 6. Other erosion factors #C4WS38 6.1. Hillslope erosion Score: 0.028 Related excerpt #FJ5WJ6: We integrate from s to the ridge r , where the slope vanishes; and use Hack's law [Hac57] to associate the distance to the ridge with the drainage A : r - s = CA(s)^h , where C is a constant in the range [1.4 – 2], and the exponent is usually set to h = 0.6 : 7. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 4 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #UE2TM4 4. Analytical solutions of the stream power law #QR788H 4.1. The method of characteristics for the stream power law Score: 0.027 Related excerpt #MLVFNW: Our first observation is that the spatial component of the characteristic curve follows the direction of the topographic gradient \frac{\nabla z}{\|\nabla z\|} . Intuitively, this means that the elevation at any point x in the terrain will only depend on elevations downstream of x , by following the path of steepest gradients. We call this path a river path as this is the trajectory naturally followed by the water, and parameterize it by s , the distance between the bound ( s = 0 ) and any point on the path (Figure 3, left). 8. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 7 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #97YQQD 6. Other erosion factors Score: 0.024 Related excerpt #WDG5SJ: The stream power law presents a singularity when the drainage area reaches zeros: when time increases, the solution converges to infinitely steep ridges. This is documented by geologists [LD03], who suggest that the stream power is not applicable in these locations, or at least dominated by other processes. We explore two possible predominant processes: hillslope erosion and landslides, and propose approximations to easily integrate them into our algorithm. 9. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 3 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #QW6CT7 3. Background and overview #Q878EF 3.2. Challenges and algorithm Score: 0.022 Related excerpt #S3XW5N: We follow [Ste21] and order the computation along the river network. This network consists of a set of trees that covers the terrain and represents the progressive merging of high-altitude small streams down to the larger rivers. We obtain the 2D analytical solution by solving the 1D problem on each upstream path embedded in the tree structures. Similarly to the previous implicit time stepping schemes for the Stream Power Law [BW13, CBC*16], we separate the computation into two parts: first we accumulate the drainage area by following the river directions from the high elevations (ridges) of the terrain to the bounds, then we evaluate the analytical solution upstream from the bounds to the ridges. 10. Source: Scalable real‐time animation of rivers (#XDESU9), Eric Bruneton, Fabrice Neyret, Nicolas Holzschuch, Qizhi Yu, p. 8 Context: #B9LYTR Scalable real-time animation of rivers #TJAYJS 7. Discussion and limitations Score: 0.023 Related excerpt #4Y2DNV: 2D flow hypothesis and terrain slope Our 2D flow hypothesis is valid for constant water depth (and homogeneous velocity profile along each water column). To account for depth h(x,y) variations we should simply conserve q(x,y) = v(x,y)h(x,y) instead of v \cdot \nabla \cdot q = 0 , q = \nabla \times \psi , v = q/h . This supposes either to know h or to deduce it from the terrain elevation z(x,y) . In our static case, the Chézy law provides a convenient approximation: v = C\sqrt{Rs} with C the Chézy constant, s the slope, v = Q/S the average velocity in a vertical section of surface S , perimeter P , and hydraulic radius R = S/P . Assuming the section has a known shape, e.g., a rectangle of known length l and height h , this yields h as a function of s, l, Q . ### 8. Tool result: search_text Exact matches 1. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 2 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #ZRSZUM 3. Background and Overview #JT9864 3.1. Geological Background Matching excerpt #PGNGMP: The stream power equation states that the rate of change of surface topography h(\mathbf{p}) at a position \mathbf{p} is controlled by the balance between the surface uplift u(\mathbf{p}) and the fluvial erosion, which is a function of the local slope s(\mathbf{p}) and the drainage area A(\mathbf{p}) . The local slope s(\mathbf{p}) is defined as the surface topographic gradient: 2. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 0 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #T4ZQ7N 1. Introduction Matching excerpt #P29KTM: The key observation of our work is that terrains and mountains are formed by various mutually interacting physical processes acting at geological time and spatial scales [BFH92]. In order to capture its action at large spatial and temporal scales, mountain development should be taken into account. The terrains we observe in nature are emergent phenomena resulting from the erosional response to tectonically-driven uplift, potentially leading to a steady-state or equilibrium situation [How82]. The role of the interaction between the uplift and the erosion has been extensively studied in geomorphology and expressed through different models, such as the stream power equation [WT99]. By bringing this theory to computer graphics we can provide an easily controllable mechanism that generates large scale realistic terrains conforming to a global geomorphological process. 3. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 6 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #A8NK92 6. Results #AWRWEN 6.4. Stream power erosion Matching excerpt #MNZNZU: The erosion parameters in the stream power erosion are not intuitive to set. Even in geology, the impact of these coefficients is not well-understood. The erosion coefficient k and the uplift u are both subject to a multiplication by dt , so only their ratio is relevant. However, its value has a strong influence on the mountain height. We made a series of experiments in order to find a relationship between this ratio and the maximum mountain height. It turned out this relation is linear and the height in kilometers follows the rule h_{max} = 2.244 u/k . 4. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 8 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #A8NK92 6. Results #AWRWEN 6.4. Stream power erosion Matching excerpt #76252N: Stream power erosion without the uplift. The stream power erosion can be used without the uplift and it adds a global hydrological realism to an existing scene as shown on an example of a fractal terrain enhanced with erosion in Figure 14. As the erosion converges toward a flat terrain, it is necessary to use small time steps and to stop the simulation after only a few iterations. 5. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 0 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #LEPLXD Abstract Matching excerpt #EU9PCE: At large scale, landscapes result from the combination of two major processes: tectonics which generate the main relief through crust uplift, and weather which accounts for erosion. This paper presents the first method in computer graphics that combines uplift and hydraulic erosion to generate visually plausible terrains. Given a user-painted uplift map, we generate a stream graph over the entire domain embedding elevation information and stream flow. Our approach relies on the stream power equation introduced in geology for hydraulic erosion. By combining crust uplift and stream power erosion we generate large realistic terrains at a low computational cost. Finally, we convert this graph into a digital elevation model by blending landscape feature kernels whose parameters are derived from the information in the graph. Our method gives high-level control over the large scale dendritic structures of the resulting river networks, watersheds, and mountains ridges. 6. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 2 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #ZRSZUM 3. Background and Overview #JT9864 3.1. Geological Background Matching excerpt #7GS7F8: Fluvial erosion is the erosion of the bedrock material and its transportation downhill by streams. It is caused by the shear stress exerted by running water and the sediment it contains onto the bed of a stream. The interaction between the fluvial erosion and the tectonic uplift has been studied for many years in geology and is usually modeled by the stream power equation [WT99]: 7. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 2 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #QW6CT7 3. Background and overview #MYA2EJ 3.1. Landscape dynamics in geology Matching excerpt #ESC38J: A first common modeling approximation is to consider surface evolution as a detachment-limited process (as opposed to transport-limited) where the evolution of surface elevation z is directly related to the competition between rock uplift and erosion rate, and not to the capacity of the rivers to transport or deposit sediments [How94]. In this setting, erosion is generally described using the stream power incision model [HK83, How94, WT99, Lag14], where erosion rate is a power law of the surface slope \|\nabla z\| , and drainage area A that acts as a proxy for the river discharge. The drainage area A(\mathbf{x}) is defined at a position \mathbf{x} as the area of the drainage basin - or catchment - upstream of \mathbf{x} . Coupled with the uplift u , the Stream Power Law expresses the rate of change of surface elevation: 8. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 4 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #UE2TM4 4. Analytical solutions of the stream power law #QR788H 4.1. The method of characteristics for the stream power law Matching excerpt #GNAVAZ: If this happens for all cells in the terrain, this solution corresponds to the steady state of the stream power law. 9. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 1 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #2BWKC4 2. Previous Work Matching excerpt #SR5QZ4: Methods that simulate hydraulic erosion handle the water dynamics explicitly, which, in theory, increases the physical accuracy of the erosion but introduces numerical constraints that limit them to a smaller spatiotemporal extent. To compensate, the results are scaled up, which therefore reduces the overall plausibility. In contrast, fluvial erosion methods implement models developed in geomorphology, for instance, the stream power law [WT99]. These laws abstract water physics under simpler proxies, e.g., the drainage area that represents water flux (or discharge), which yields simulations that can efficiently cover much larger time spans. Therefore, fluvial erosion allows a tight coupling with the growth of the mountain under tectonic uplift [CBC + 16] to model the formation of large-scale mountain ranges. Uplift was also proposed as a guide for the user to shape the landscape [CCB + 17, SPF + 23]. We build our analytical model upon the laws introduced by fluvial erosion methods, but our mathematical treatment removes the need for costly iterations inherent to simulations. 10. Source: FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation (#NV2YRW), Aryamaan Jain, Bernhard Kerbl, Brandon Finley, Guillaume Cordonnier, James Gain, p. 8 Context: #6UY46T FastFlow: GPU Acceleration of Flow and Depression Routing for Landscape Simulation #34USBX 6. Application to landscape simulation #53AQ2N Algorithm 7: Implicit fluvial erosion Matching excerpt #VPGNAT: Fluvial erosion is not the only erosive process responsible for shaping terrain. Hillslope processes model the gradual accumulation of solid material at the base of mountains and hills [BS97] and is usually expressed as a diffusion equation. We follow Tzathas et al. [TGSC24] and approximate it by including additional terms in the Stream Power Equation, changing kQ^m to kQ^m + k_t + k_h A^{-h} , where k_t and h = 0.6 are hillslope erosion parameters, A is the drainage area (obtained via flow routing with precipitation set uniformly to p = \Delta x^2 ). The parameter k_t accounts for slope-dependent effects (landslides, debris-flow), regrouped in computer graphics under the catch-all term thermal erosion [MKM89]. Approximate matches 1. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 2 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #ZRSZUM 3. Background and Overview #JT9864 3.1. Geological Background Score: 0.03 Related excerpt #PGNGMP: The stream power equation states that the rate of change of surface topography h(\mathbf{p}) at a position \mathbf{p} is controlled by the balance between the surface uplift u(\mathbf{p}) and the fluvial erosion, which is a function of the local slope s(\mathbf{p}) and the drainage area A(\mathbf{p}) . The local slope s(\mathbf{p}) is defined as the surface topographic gradient: 2. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 8 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #A8NK92 6. Results #AWRWEN 6.4. Stream power erosion Score: 0.026 Related excerpt #76252N: Stream power erosion without the uplift. The stream power erosion can be used without the uplift and it adds a global hydrological realism to an existing scene as shown on an example of a fractal terrain enhanced with erosion in Figure 14. As the erosion converges toward a flat terrain, it is necessary to use small time steps and to stop the simulation after only a few iterations. 3. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 6 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #A8NK92 6. Results #AWRWEN 6.4. Stream power erosion Score: 0.025 Related excerpt #MNZNZU: The erosion parameters in the stream power erosion are not intuitive to set. Even in geology, the impact of these coefficients is not well-understood. The erosion coefficient k and the uplift u are both subject to a multiplication by dt , so only their ratio is relevant. However, its value has a strong influence on the mountain height. We made a series of experiments in order to find a relationship between this ratio and the maximum mountain height. It turned out this relation is linear and the height in kilometers follows the rule h_{max} = 2.244 u/k . 4. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 2 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #ZRSZUM 3. Background and Overview #JT9864 3.1. Geological Background Score: 0.025 Related excerpt #MSXUGH: Note that, when applied at the right temporal and spatial scales (typically between 10^5 and 10^7 years and a few tens to hundreds of kilometers), the stream power equation does not only model erosion, but also captures the way a complex relief emerges from a supposedly flat part of the continental crust [How94]. 5. Source: Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion (#96ZMGK), Adrien Peytavie, Bedrich Benes, Guillaume Cordonnier, Jean Braun, Marie-Paule Cani, Éric Galin, Éric Guérin, p. 7 Context: #A48PSA Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion #A8NK92 6. Results #AWRWEN 6.4. Stream power erosion Score: 0.024 Related excerpt #L66ZXE: In our approach, we allow only the changes to the drainage area exponent m . Indeed, only the ratio between m and n has a meaning that we can deduce from the equations. Let us suppose we have reached an equilibrium state in a region where u and k are constant over the space. The stream power Equation (1) becomes: 6. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 2 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #QW6CT7 3. Background and overview #MYA2EJ 3.1. Landscape dynamics in geology Score: 0.027 Related excerpt #ESC38J: A first common modeling approximation is to consider surface evolution as a detachment-limited process (as opposed to transport-limited) where the evolution of surface elevation z is directly related to the competition between rock uplift and erosion rate, and not to the capacity of the rivers to transport or deposit sediments [How94]. In this setting, erosion is generally described using the stream power incision model [HK83, How94, WT99, Lag14], where erosion rate is a power law of the surface slope \|\nabla z\| , and drainage area A that acts as a proxy for the river discharge. The drainage area A(\mathbf{x}) is defined at a position \mathbf{x} as the area of the drainage basin - or catchment - upstream of \mathbf{x} . Coupled with the uplift u , the Stream Power Law expresses the rate of change of surface elevation: 7. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 7 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #97YQQD 6. Other erosion factors Score: 0.025 Related excerpt #WDG5SJ: The stream power law presents a singularity when the drainage area reaches zeros: when time increases, the solution converges to infinitely steep ridges. This is documented by geologists [LD03], who suggest that the stream power is not applicable in these locations, or at least dominated by other processes. We explore two possible predominant processes: hillslope erosion and landslides, and propose approximations to easily integrate them into our algorithm. 8. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 10 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #PZFF56 7. Results #BK5LST 7.2. Ablation study Score: 0.024 Related excerpt #3WX3ZE: Hillslope erosion stabilizes the slopes of the mountains and acts as the main erosion process at low drainage where the stream power law becomes negligible [LD03]. We show the impact of hillslope in Figure 9, where we show the analytical solutions of the stream power law only at \delta x = 50 m (left), compared with our modified formulation that includes the hillslope erosion (Section 6.1, right of Figure 9). Without hillslope, we observe the emergence of unrealistically sharp ridges and peaks. 9. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 4 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #UE2TM4 4. Analytical solutions of the stream power law #QR788H 4.1. The method of characteristics for the stream power law Score: 0.024 Related excerpt #P8246L: where a(s) = kA(s)^m . Note that we assume that A does not depend on time – we observed that, after some time, the drainage A stabilizes in the main river channels. A solution of Eqn. 3 was proposed in Earth sciences [Ste21] with the assumption that u is constant in space and varies in time, which is important for geomorphologists who study the erosional response to tectonic perturbations. We prefer an orthogonal approach where the uplift varies in space but not in time, as we expect the uplift to be easier to control for the user as a function of space alone [SPF + 23]. 10. Source: Physically-based analytical erosion for fast terrain generation (#DWXKYQ), Boris Gailleton, Guillaume Cordonnier, Petros Tzathas, Philippe Steer, p. 8 Context: #JDHNVB Physically-based analytical erosion for fast terrain generation #97YQQD 6. Other erosion factors #C4WS38 6.1. Hillslope erosion Score: 0.023 Related excerpt #SH59MV: An analytical solution including the hillslope erosion would require joining the stream power law (Eqn. 1) with Eqn. 23 which results in an advection-diffusion equation. Solving this equation needs a global 2D treatment, preventing our decomposition to a set of 1D solutions on the stream tree, and the complexity of the derivations and implementation of the solutions even in the 1D case challenges their usability in a terrain modeling framework. ### 9. Assistant tool calls #### Tool call: `read` ```json { "blocks": [ "JT9864" ], "expand": true } ``` #### Tool call: `read` ```json { "blocks": [ "MYA2EJ" ], "expand": true } ``` #### Tool call: `read` ```json { "blocks": [ "AWRWEN" ], "expand": true } ``` ### 10. Tool result: read DOCUMENT #96ZMGK Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion SECTION #JT9864 3.1. Geological Background EXCERPT #UZPNPP p. 2 Uplift and faults. Terrains result from the combined action of tectonic uplift of the Earth’s surface and erosion. Collisions between continental plates, as well as subduction of ocean plates under continental ones, cause the continental crust to shorten and thicken. This results in the growth of mountains along the main boundaries between plates. Faults and folds appear in regions where the crust undergoes the highest stress [WBF93]. EXCERPT #N99WSW p. 2 In geology, the term uplift is used to denote the local speed at which a mountain grows. Because the growth occurs between the series of parallel folds (and faults), considering the uplift as locally uniform between these folds is a valid approximation. The complex landform features found in nature are mainly the result of the interaction between the uplift factor and fluvial erosion, i.e., the action of water forming streams that carve the terrain while it grows. EXCERPT #7GS7F8 p. 2 Fluvial erosion is the erosion of the bedrock material and its transportation downhill by streams. It is caused by the shear stress exerted by running water and the sediment it contains onto the bed of a stream. The interaction between the fluvial erosion and the tectonic uplift has been studied for many years in geology and is usually modeled by the stream power equation [WT99]: EXCERPT #QTW2XX p. 2 \frac{dh(\mathbf{p})}{dt} = u(\mathbf{p}) - kA(\mathbf{p})^m s(\mathbf{p})^n \quad (1) EXCERPT #PGNGMP p. 2 The stream power equation states that the rate of change of surface topography h(\mathbf{p}) at a position \mathbf{p} is controlled by the balance between the surface uplift u(\mathbf{p}) and the fluvial erosion, which is a function of the local slope s(\mathbf{p}) and the drainage area A(\mathbf{p}) . The local slope s(\mathbf{p}) is defined as the surface topographic gradient: EXCERPT #6VBLVP p. 2 s(\mathbf{p}) = \nabla h(\mathbf{p}). EXCERPT #EKXATA p. 2 The constants m and n depend on rock strength, climate, and the topology of river networks. While the values of those parameters are poorly understood, the ratio m/n is constrained by the shape of the stream profiles and is thought of being m/n \approx 0.5 [WT99]. As in most geomorphological studies, we use n = 1 and m = 0.5 . Moreover, some geological studies attempt to tune these parameters by example [CB14] and a recent survey [Lag14] studies the limit of geological knowledge regarding the parameters of the stream power equation. EXCERPT #9XKDQM p. 2 Figure 3: A topographic map showing a drainage area A(p) in blue. A point p is marked on the map, and a stream network is visible flowing towards it. EXCERPT #HMTH3S p. 2 Figure 3: The drainage area A(\mathbf{p}) is defined as the planar region where streams flow to point \mathbf{p} . EXCERPT #GPZB7Q p. 2 The drainage area A(\mathbf{p}) is the upstream area draining through point \mathbf{p} , assuming that water flows along the topographic gradient (Figure 3). In our implementation, the terrain is represented by a geometric graph \mathcal{G} connecting points sampled over the terrain domain. The drainage area A(\mathbf{p}) is the area associated to the set of points \{\mathbf{q} \in \mathcal{G}\} strictly above \mathbf{p} such that there exists one path of strictly increasing height starting from \mathbf{p} and ending at \mathbf{q} . The factor k is an erosion constant that depends on many factors, such as lithology (the composition of the soil/bedrock), vegetation, climate, and climate variability. EXCERPT #MSXUGH p. 2 Note that, when applied at the right temporal and spatial scales (typically between 10^5 and 10^7 years and a few tens to hundreds of kilometers), the stream power equation does not only model erosion, but also captures the way a complex relief emerges from a supposedly flat part of the continental crust [How94]. ### 11. Tool result: read DOCUMENT #DWXKYQ Physically-based analytical erosion for fast terrain generation SECTION #MYA2EJ 3.1. Landscape dynamics in geology EXCERPT #5M4LES p. 2 Mountain formation is generally associated with tectonic plate convergence, mantle dynamics, and/or volcanic activity. In the context of tectonic convergence, the eventual collision of the plates leads to a thickening of the crust accommodated by visco-elasto-plastic deformation, brittle rupture (faults), and folding of the rock layers [Avo03, GCG + 09]. The thickening leads to a vertical upward motion of rocks towards the surface, referred to as rock uplift , EXCERPT #NYZKTL p. 2 which competes with erosion and can lead to a progressive increase in the surface altitude, called surface uplift [EM90]. Rates of rock uplift and erosion vary in space and time and achieve values up to a few millimeters per year in some mountain ranges. EXCERPT #74AXB7 p. 2 The uplift is counteracted by erosion, which impacts the slopes of the mountain and therefore its maximal elevation. Erosion comes from many factors: water, glaciers, landslides, wind, and even anthropic or biological impact. Many models in geomorphology consider only erosion by water, also called fluvial erosion . Indeed, the fluvial network is considered the backbone of landscapes, and fluvial incision dictates the rate of landscape erosion [Whi04]. While simple to model, fluvial erosion explains the main topographical characteristics of most mountain ranges and has been the dominant erosion factor over many geological periods - with the notable exception of the last million years, where the Quaternary saw an important increase in glacial erosion that leaves specific marks in high altitude [PMD01, ENPL09, SHV + 12]. EXCERPT #ESC38J p. 2 A first common modeling approximation is to consider surface evolution as a detachment-limited process (as opposed to transport-limited) where the evolution of surface elevation z is directly related to the competition between rock uplift and erosion rate, and not to the capacity of the rivers to transport or deposit sediments [How94]. In this setting, erosion is generally described using the stream power incision model [HK83, How94, WT99, Lag14], where erosion rate is a power law of the surface slope \|\nabla z\| , and drainage area A that acts as a proxy for the river discharge. The drainage area A(\mathbf{x}) is defined at a position \mathbf{x} as the area of the drainage basin - or catchment - upstream of \mathbf{x} . Coupled with the uplift u , the Stream Power Law expresses the rate of change of surface elevation: EXCERPT #ZGDN2Y p. 2 \frac{\partial z}{\partial t} = u - kA^m \|\nabla z\|^n, \quad (1) EXCERPT #2DEC4W p. 2 where k , m and n are erosion coefficients. Throughout the paper, we will use some of the common values: m = 0.4 and n = 1 . The choice of n = 1 , also commonly used in geomorphology, makes the equation linear and therefore simplifies the derivation of the analytical solutions. While this choice barely impacts the result as the valley profiles are mostly directed by the ration m/n , we acknowledge that the actual values of m and n remain an open question in geomorphology [Lag14]. EXCERPT #VZD2MD p. 2 Eqn. 1 is a linear hyperbolic Partial Differential Equation (PDE), which can be interpreted as an advection of the initial terrain z_0 = z(t = 0) along the rivers, at a velocity kA^m . This equation admits analytical solutions in 1D that can be computed thanks to the method of characteristics [Luk74, RTP13], and used to model 2D heightmaps [Ste21]. ### 12. Tool result: read DOCUMENT #96ZMGK Large Scale Terrain Generation from Tectonic Uplift and Fluvial Erosion SECTION #AWRWEN 6.4. Stream power erosion EXCERPT #5DA9P3 p. 6 Figure 9 shows the result of the stream power equation erosion, combined with small scale details. Because our method generates the coverage of the terrain by the river and lake network, it also provides hydrology network to any possible uplift input. EXCERPT #MNZNZU p. 6 The erosion parameters in the stream power erosion are not intuitive to set. Even in geology, the impact of these coefficients is not well-understood. The erosion coefficient k and the uplift u are both subject to a multiplication by dt , so only their ratio is relevant. However, its value has a strong influence on the mountain height. We made a series of experiments in order to find a relationship between this ratio and the maximum mountain height. It turned out this relation is linear and the height in kilometers follows the rule h_{max} = 2.244 u/k . EXCERPT #C77H2U p. 7 EXCERPT #6VYHRR p. 7 Figure 11: Comparison of two rendering methods. The top row shows a far view (left) and a close-up (right) of real-time tessellation. The bottom row shows the same views using procedural primitives. The far view shows a large mountain range with a blue body of water on the left. The close-up shows the texture of the mountain slopes, with the procedural primitives method appearing more detailed and less blocky than the tessellation method. EXCERPT #SCKX4X p. 7 Figure 11: Comparison of two of the rendering methods used in our approach. The images show a far view (left) and a close-up of real time tessellation (top) and procedural primitives (bottom). EXCERPT #9P2KJV p. 7 Figure 12: Different steps of the fluvial erosion algorithm. Four panels show the progression from 1 iteration to 300 iterations. At 1 iteration, a single blue river line is visible on a flat, light-colored terrain. As iterations increase to 10, 50, and 300, the terrain becomes increasingly rugged and mountainous, with the river network becoming more complex and branching. The terrain color transitions from light to darker shades as elevation increases. EXCERPT #F3BJFW p. 7 Figure 12: Different steps of our fluvial erosion algorithm. The mountains are formed at \sim 50 iterations, and the algorithm converges after 100 to 300 steps. EXCERPT #L66ZXE p. 7 In our approach, we allow only the changes to the drainage area exponent m . Indeed, only the ratio between m and n has a meaning that we can deduce from the equations. Let us suppose we have reached an equilibrium state in a region where u and k are constant over the space. The stream power Equation (1) becomes: EXCERPT #8RRPZD p. 7 \frac{dh(\mathbf{p})}{dt} = 0 = u(\mathbf{p}) - kA(\mathbf{p})^m s(\mathbf{p})^n EXCERPT #6ECN47 p. 7 From that, we have s and A proportional: EXCERPT #5E4PYR p. 7 s \sim A^{-m/n}. EXCERPT #Z2BRQU p. 7 We note x the distance between \mathbf{p} and the corresponding outflow, and d the distance between ep and a ridge. Then we assume that A is proportional to (d-x)^2 . Recall that s = dh/dx , so we obtain: EXCERPT #WW46WD p. 7 \frac{dh}{dx} \sim (d-x)^{-2m/n}. EXCERPT #AFMNNY p. 7 After integration, and imposing h(0) = 0 , we have: EXCERPT #33A4VS p. 7 h \sim \begin{cases} d^{1-2m/n} - (d-x)^{1-2m/n} & \text{if } m/n < 1/2 \\ \log(d) - \log(d-x) & \text{if } m/n = 1/2 \\ \frac{1}{(d-x)^{2m/n-1}} - \frac{1}{(d)^{2m/n-1}} & \text{otherwise} \end{cases} EXCERPT #P52B3D p. 7 We use the result of these equations to choose the proper ratio m/n to shape the desired river profile. EXCERPT #DZ5LAY p. 7 Figure 13: The uplift gradient has a strong influence on the resulting terrain. The central image shows a 3D terrain with a river network. To the left, two vertical color bars represent different uplift gradients: 'Constant' (uniform light gray) and 'Smooth' (gradient from light to dark gray). To the right, two circular icons represent different river profiles: 'Disc-shaped' (a solid dark gray circle) and 'Piecewise' (a circle with horizontal bands of different shades of gray). EXCERPT #JKB9A8 p. 7 Figure 13: The uplift gradient has a strong influence on the resulting terrain. EXCERPT #4MKD3J p. 7 The uplift has a strong influence on the resulting terrain and it is the main way the user affects the final shape. As shown in Figure 13, the main impact is in the gradient of the uplift values. The actual shape of the uplift does not have a strong influence, except on the boundaries. Having the same uplift shape, a slowly decreasing gradient leads to a very straight erosion in the gradient direction, whereas a set of steps of constant values gives more random valleys with sudden jumps on the gradient in the mountain heights. We can also observe that lower values of uplift lead to smaller valleys, and that the thermal erosion is important with regular slopes for high uplift values. EXCERPT #L93W85 p. 8 EXCERPT #U2YKYY p. 8 This figure consists of four panels arranged in a 2x2 grid. The top-left panel shows a grayscale fractal terrain map. The top-right panel shows the same terrain after stream power erosion, with blue lines indicating newly formed rivers and blue areas indicating lakes. The bottom-left panel is a 3D perspective view of the fractal terrain. The bottom-right panel is a 3D perspective view of the terrain after erosion, showing the same features as the top-right panel but in a three-dimensional format. Figure 14: A fractal terrain (left) after the application of the stream power erosion (right). Note the creation of new rivers and lakes. EXCERPT #T4N6WU p. 8 Figure 14: A fractal terrain (left) after the application of the stream power erosion (right). Note the creation of new rivers and lakes. EXCERPT #76252N p. 8 Stream power erosion without the uplift. The stream power erosion can be used without the uplift and it adds a global hydrological realism to an existing scene as shown on an example of a fractal terrain enhanced with erosion in Figure 14. As the erosion converges toward a flat terrain, it is necessary to use small time steps and to stop the simulation after only a few iterations. EXCERPT #JCR3MQ p. 8 Figure 15 shows that this method can model a large variety of landscape: starting from a simple height map formed with two strokes and a gradient, we obtain a plausible canyon. We chose a height dependent maximal slope for thermal erosion to obtain the succession of cliffs and slopes in the result. EXCERPT #TB8KR6 p. 8 The algorithm for lakes flow connection decreases greatly the number of iterations needed to obtain a plausible terrain in terms of hydrology. Without the lake connections, the water eventually finds a flow out of the main local minima, but after 4-5 more iterations than the total convergence of the erosion with lake connection (Figure 16). Even after many iterations, a large number of local minima are still adding some discontinuities in the hydrology network. EXCERPT #9VNYCT p. 8 The thermal erosion has an important influence on the shape of the valleys. It affects their regularity as shown by Figure 17. If the maximum slope given for the mountain is 30^\circ , which is the usual talus angle for thermal erosion [MKM89], our erosion model results in a layout of very regular geometric valleys. EXCERPT #EUK7YL p. 8 We experimented by forcing the maximum slope to follow a 3D Perlin noise with a high persistence, to account for local different EXCERPT #3EWW8D p. 8 This figure is divided into two main horizontal sections. The top section shows a wide, flat landscape with a single, prominent blue river carving its way through the terrain. A small inset in the bottom-left of this section shows a cross-section of the river valley. The bottom section is a close-up view of a canyon, showing steep, eroded slopes and a river flowing through the center. The terrain is rendered in shades of brown and tan, with the river in blue. Figure 15: A flat terrain initialized with a carving river (top left) has been eroded with height dependent slopes (close-up at bottom). EXCERPT #NPXCJ2 p. 8 Figure 15: A flat terrain initialized with a carving river (top left) has been eroded with height dependent slopes (close-up at bottom). EXCERPT #N8SX9N p. 8 This figure consists of two side-by-side grayscale terrain maps. The left map shows a terrain with several local minima (valleys) but without the complex network of rivers and lakes seen in the right map. The right map shows the same terrain after 175 iterations of erosion with lake flow computation, resulting in a much more complex and realistic hydrological network with many more rivers and lakes. Figure 16: Comparison of without (left) and with (right) lakes flows computation, after 175 iterations. EXCERPT #GRVPCK p. 8 Figure 16: Comparison of without (left) and with (right) lakes flows computation, after 175 iterations. EXCERPT #6N825C p. 8 rocks strength. We choose the minimal and maximal slope angles to be 6^\circ and 54^\circ respectively. This results in a more random distribution of the erosion patterns in valleys as shown in Figure 17. EXCERPT #R2E2MJ p. 8 Furthermore, we can obtain interesting features by procedurally adjusting the thermal slope. Figure 18 shows a landscape with small thermal slopes below a given height, but higher slopes above it. This adds cliffs to the crests, which is typical for many mountains. ### 13. Assistant ## The basic idea The **$\chi$ transform** does not change elevation. It replaces ordinary upstream distance $x$ with a **drainage-area-weighted distance** $\chi$. Why? In the stream-power incision model, $$ \frac{\partial z}{\partial t} = U-KA^mS^n, $$ where $U$ is uplift, $K$ is erodibility, $A$ is upstream drainage area, and $S$ is channel slope (#QTW2XX; #ZGDN2Y). Drainage area acts as a proxy for water discharge (#ESC38J), so large downstream rivers can produce the same incision with gentler slopes than small headwater streams. At steady state, uplift balances incision: $$ U=KA^mS^n. $$ Rearranging gives $$ S=\left(\frac{U}{K}\right)^{1/n}A^{-m/n}. $$ Thus channel slope varies systematically with area: $$ S\propto A^{-\theta}, \qquad \theta=\frac{m}{n}, $$ which is the slope–area relationship also derived in #L66ZXE–#5E4PYR. $\theta$ is called the **channel concavity**. ## Defining $\chi$ Starting from an outlet or base-level point $x_b$, define $$ \chi(x) = \int_{x_b}^{x} \left(\frac{A_0}{A(x')}\right)^{\theta_{\mathrm{ref}}} \,dx'. $$ Here: - $A(x')$ is upstream drainage area along the channel; - $A_0$ is an arbitrary reference area used to keep the units convenient; - $\theta_{\mathrm{ref}}$ is usually an assumed value for $m/n$, often around $0.4$–$0.5$ in the models in the corpus (#EKXATA; #2DEC4W). Each channel segment therefore contributes $$ d\chi= \left(\frac{A_0}{A}\right)^{\theta_{\mathrm{ref}}}dx. $$ Small headwater channels have small $A$, so ordinary distance is **stretched**. Large downstream channels have large $A$, so it is **compressed**. ## Why this linearizes river profiles Because $$ \frac{d\chi}{dx} = \left(\frac{A_0}{A}\right)^\theta, $$ the slope in $\chi$ space is $$ \frac{dz}{d\chi} = S\left(\frac{A}{A_0}\right)^\theta. $$ Substituting the steady-state stream-power relationship gives $$ \frac{dz}{d\chi} = \left(\frac{U}{KA_0^m}\right)^{1/n}. $$ If $U$, $K$, and $m/n$ are spatially uniform, this is constant. Therefore, $$ z(\chi)=z_b+M_\chi\chi, $$ with $$ M_\chi= \left(\frac{U}{KA_0^m}\right)^{1/n}. $$ So a concave longitudinal river profile in ordinary elevation–distance space becomes approximately a **straight line in elevation–$\chi$ space**. ## How it is used 1. **Comparing tributaries** Tributaries draining the same steady-state landscape should collapse onto roughly the same $z$–$\chi$ line. This removes much of the profile variation caused simply by different drainage areas. 2. **Estimating concavity** Try different values of $\theta_{\mathrm{ref}}$. The value that best aligns or linearizes tributary profiles estimates the effective ratio $m/n$. 3. **Measuring channel steepness** The slope $M_\chi$ of the $z$–$\chi$ plot is closely related to normalized channel steepness: $$ k_{sn}=SA^{\theta_{\mathrm{ref}}}. $$ High $M_\chi$ or $k_{sn}$ can indicate greater uplift, lower erodibility, or both—it cannot distinguish them without independent information. 4. **Finding transient features** Knickpoints and changes in uplift, lithology, or base level appear as breaks or curvature in an otherwise linear $z$–$\chi$ profile. 5. **Simplifying stream-power calculations** For the commonly used $n=1$ case, the stream-power equation becomes $$ \frac{\partial z}{\partial t} = U-KA_0^m\frac{\partial z}{\partial\chi}. $$ The area dependence has disappeared from the erosion term. In $\chi$ coordinates, profile evolution resembles advection at a constant rate. This connects closely to the corpus’s interpretation of the $n=1$ stream-power equation as elevation being advected upstream along river paths (#VZD2MD). ## Computationally For a discrete river graph, $\chi$ is accumulated upstream from each node’s receiver: $$ \chi_i = \chi_r+ \frac{\Delta x_{ir}}{2} \left[ \left(\frac{A_0}{A_i}\right)^\theta+ \left(\frac{A_0}{A_r}\right)^\theta \right], $$ where $r$ is the downstream receiver of node $i$. This fits naturally with the same drainage-tree ordering used to accumulate drainage area and evaluate stream-power erosion upstream from the boundaries (#S3XW5N). The shortest intuition is: **$\chi$ measures distance in units of erosional difficulty**. Headwater distance counts more because small drainage area provides less erosive power; downstream distance counts less because large rivers have more erosive power.