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Linear Approximations

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

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Related Experiment Video

Updated: Jul 16, 2026

Composition and Distribution Analysis of Bioaerosols Under Different Environmental Conditions

Composition and Distribution Analysis of Bioaerosols Under Different Environmental Conditions

Published on: January 7, 2019

A spatiotemporal machine learning framework for high-resolution PM_2.5 estimation using reconstructed satellite

Tinka Singh¹, Prabirkumar Saha², Ashok Singh Sairam³

¹Mehta Family School of Data Science and Artificial Intelligence, Indian Institute of Technology Guwahati, Guwahati, 781039, India.

Environmental Science and Pollution Research International

|July 15, 2026

Summary

Accurate monitoring of fine particulate matter (PM2.5) in North-East India is now possible with a new satellite and machine learning framework. This integrated approach overcomes cloud cover and data scarcity, providing 1km resolution PM2.5 estimates.

Keywords:

AOD reconstruction PM2.5 modelling Remote sensing Spatiotemporal learning Urban pollution

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Last Updated: Jul 16, 2026

Composition and Distribution Analysis of Bioaerosols Under Different Environmental Conditions

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Area of Science:

Environmental Science
Atmospheric Science
Data Science

Background:

Accurate monitoring of fine particulate matter (PM2.5) in North-East India is challenging due to sparse ground data, complex terrain, high humidity, and persistent cloud cover.
Satellite aerosol retrievals are often unreliable during the monsoon season, further limiting air quality assessment.

Purpose of the Study:

To develop an integrated multi-sensor satellite and machine learning (ML) framework for estimating daily surface PM2.5 concentrations at 1 km resolution.
To address the limitations of sparse observations and cloud cover in North-East India for PM2.5 monitoring.

Main Methods:

Integrated reconstructed multi-sensor aerosol optical depth with meteorological variables, land-surface characteristics, and limited ground measurements.
Employed a unified modeling architecture and evaluated multiple ML models, identifying a multi-layer perceptron as the best estimator.
Achieved 1 km spatial resolution for daily PM2.5 estimation.

Main Results:

The multi-layer perceptron model achieved a coefficient of determination (R2) of 0.82, significantly reducing prediction errors compared to models using raw satellite data.
The resulting PM2.5 fields showed distinct urban-rural gradients, transport-aligned hotspots, and seasonal variations.
Highest pollution accumulation occurred during winter and post-monsoon periods, while monsoon conditions suppressed concentrations.

Conclusions:

The developed framework provides a reliable method for high-resolution PM2.5 estimation in cloud-dominated, data-scarce regions.
Long-term analysis suggests a potential intensification of urban PM2.5 exposure based on current emission trends.
The study highlights the importance of integrated approaches for effective air quality management in challenging environments.