0.0 dB
Transition Matrix (5x5 Heatmap)
Stationary Distribution (π)
Monte Carlo Convergence (Law of Large Numbers)
Uncertainty Decay (1/√N)
Samples: 0
CI Width: 0.0 dB
Law of Large Numbers
SE = σ / √N
Standard error decreases with sample size NMean SPL Heatmap (dB)
Uncertainty (95% CI Width)
Planner Insights
Optimization Results
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01
3D Monte Carlo Ray Tracing
How it works: The engine launches thousands of stochastic rays from a source. Each ray's trajectory is calculated using 3D mesh intersections (Trimesh/RTree). Unlike deterministic models, we sample the emission angle using a uniform distribution on a unit sphere.
Uniqueness: It captures "shadow zones" and complex scattering that 2D models miss. It creates a high-fidelity Acoustic Impulse Response by accumulating energy arrivals over time.
Application: Concert Hall Design, Indoor Localization
02
Markovian Street Canyons
PROBABILISTIC COUPLINGHow it works: We model an urban street as a system of coupled states (Left Facade, Right Facade, Ground, Sky). The 5x5 transition matrix P defines the probability of sound moving between these surfaces based on geometric aspect ratios (H/W) and absorption coefficients.
Uniqueness: Instead of tracing individual rays, we solve the Stationary Distribution (πP = π) to find long-term noise equilibrium. This is mathematically equivalent to infinite ray bounces.
Application: Urban Noise Mapping, Traffic Planning
03
Stochastic Convergence (LLN)
How it works: This tab demonstrates the Law of Large Numbers. As we increase the number of Monte Carlo samples (N), the running mean of the Sound Pressure Level (SPL) converges to the true expected value. We calculate the 95% Confidence Interval using the Standard Error (σ/√N).
Uniqueness: It provides a "Reliability Metric" for the simulation. It tells the researcher exactly how many rays are needed to achieve a desired precision (e.g., ±0.5 dB).
Application: Simulation Optimization, Error Budgeting
04
Macro-Scale Urban Grids
How it works: We scale the micro-level Markov models to a 4x4 city grid. Each cell represents a different urban profile (Dense, Suburban, Mixed) with randomized building heights and surface materials. We run parallel simulations to generate a city-wide noise heatmap.
Uniqueness: It bridges the gap between single-building acoustics and city-wide environmental impact. It visualizes not just the mean noise, but the uncertainty (Variance) of that noise.
Application: Smart City Sensors, Zoning Laws
Scientific Application & Future Scope
This research framework is unique because it combines Geometric Ray Tracing (short-term/direct sound) with Markovian Probability (long-term/reverberant field). It replaces heavy computational fluid dynamics with efficient stochastic approximations, making real-time urban acoustic planning possible on web-based platforms.