GATE Geomatics Engineering Common Syllabus

What does the regression coefficient $b$ of $y$ on $x$ physically represent?

It is the slope — the average change in the dependent variable $y$ per unit change in the independent variable $x$: $$b_{yx} = \frac{\text{Cov}(x,y)}{\sigma_x^{2}} = r\frac{\sigma_y}{\sigma_x}$$

Define the correlation coefficient $r$ and give its formula.

The (Pearson) correlation coefficient measures the strength and direction of the linear relationship between two variables: $$r = \frac{\sum (x-\bar{x})(y-\bar{y})}{\sqrt{\sum (x-\bar{x})^{2}\,\sum (y-\bar{y})^{2}}} = \frac{\text{Cov}(x,y)}{\sigma_x \sigma_y}$$

What is the range of the correlation coefficient and how is each extreme interpreted?

$-1 \leq r \leq +1$. $r = +1$ perfect positive linear correlation; $r = -1$ perfect negative linear correlation; $r = 0$ no linear correlation. Values near $\pm 1$ indicate strong linear association.

What is the relationship between the correlation coefficient and the two regression coefficients?

The correlation coefficient is the geometric mean of the two regression coefficients: $$r = \pm\sqrt{b_{yx}\cdot b_{xy}}$$ The sign of $r$ matches the (common) sign of the regression coefficients.

What does the coefficient of determination $r^{2}$ represent?

$r^{2}$ is the proportion of the total variance in the dependent variable that is explained by the regression (the independent variable). E.g., $r^{2}=0.81$ means 81% of the variation in $y$ is accounted for by $x$.

Distinguish correlation from regression.

Correlation measures the degree/strength of mutual linear association between two variables (symmetric, a single coefficient $r$). Regression establishes a cause-effect predictive equation of one variable on another (asymmetric; $b_{yx} \neq b_{xy}$ in general) and is used for estimation/prediction.

What is a 'statistically significant value' (statistical significance)?

It is a result judged unlikely to have occurred by random chance alone, evaluated against a chosen significance level $\alpha$ (e.g., 0.05 or 5%). If the test statistic's probability ($p$-value) is less than $\alpha$, the result is statistically significant and the null hypothesis is rejected.

Define the significance level $\alpha$ and the confidence level in hypothesis testing.

$\alpha$ is the probability of rejecting a true null hypothesis (Type I error), commonly 5% or 1%. The confidence level is $1-\alpha$ (e.g., 95% or 99%), the probability that the conclusion is correct.

State the purpose of the Chi-square ($\chi^{2}$) test.

The Chi-square test is a non-parametric test used to (i) test goodness-of-fit between observed and expected (theoretical) frequencies, and (ii) test independence/association between categorical variables. In surveying it tests whether observed errors follow the assumed (e.g., normal) distribution and checks variance.

Write the Chi-square goodness-of-fit statistic and define its terms.

$$\chi^{2} = \sum_{i=1}^{k} \frac{(O_i - E_i)^{2}}{E_i}$$ where $O_i$ = observed frequency, $E_i$ = expected frequency, and $k$ = number of categories. A small $\chi^{2}$ indicates good agreement with the expected distribution.

How are degrees of freedom found for a Chi-square goodness-of-fit test with $k$ classes?

Degrees of freedom $= k - 1 - m$, where $k$ is the number of categories and $m$ is the number of parameters estimated from the data. For a simple goodness-of-fit with no estimated parameters, $\text{df} = k-1$.

How is the Chi-square test decision made using critical values?

Compare the computed $\chi^{2}$ with the tabulated critical value $\chi^{2}_{\alpha,\,df}$. If $\chi^{2}_{\text{calc}} > \chi^{2}_{\text{critical}}$, reject the null hypothesis (significant difference); otherwise accept it (observed agrees with expected).

What is the basic concept of remote sensing?

Remote sensing is the science of acquiring information about an object, area, or phenomenon through the analysis of data obtained by a sensor that is not in physical contact with the target — typically by measuring reflected or emitted electromagnetic radiation.

List the basic components/stages of a remote sensing system.

(1) Energy source/illumination, (2) radiation and the atmosphere, (3) interaction with the target, (4) recording of energy by the sensor, (5) transmission, reception and processing, (6) interpretation and analysis, and (7) application of the information.

Differentiate active and passive remote sensing.

Passive sensors record natural energy (e.g., reflected sunlight or emitted thermal radiation) and depend on an external source. Active sensors supply their own energy, emit a pulse, and record the reflected/backscattered return (e.g., RADAR, LiDAR), and can operate day or night.

What is the electromagnetic spectrum?

The electromagnetic spectrum is the continuous, ordered range of all electromagnetic radiation arranged by wavelength (or frequency), from short-wavelength gamma rays and X-rays through ultraviolet, visible, infrared, microwave, to long-wavelength radio waves.

Give the relationship between the speed of light, wavelength, and frequency of EM radiation.

$$c = \lambda \nu$$ where $c \approx 3\times 10^{8}\ \text{m/s}$ is the speed of light in vacuum, $\lambda$ is wavelength, and $\nu$ is frequency. Wavelength and frequency are inversely related.

State the approximate wavelength range of the visible portion of the spectrum and its colour limits.

The visible region spans approximately $0.4$ to $0.7\ \mu\text{m}$ (400–700 nm), from violet/blue (~0.4 μm) through green and to red (~0.7 μm).

List the principal spectral regions used in optical remote sensing with approximate ranges.

Visible: $0.4$–$0.7\ \mu\text{m}$; Near-infrared (NIR): $0.7$–$1.3\ \mu\text{m}$; Shortwave/Mid-IR (SWIR): $1.3$–$3\ \mu\text{m}$; Thermal IR: $3$–$14\ \mu\text{m}$; Microwave: $1\ \text{mm}$–$1\ \text{m}$.

What are 'atmospheric windows' in remote sensing?

Atmospheric windows are wavelength ranges where the atmosphere is relatively transparent and transmits electromagnetic radiation with little absorption (e.g., visible and certain IR/microwave bands). Sensors are designed to operate within these windows.

Define 'spectral signature'.

A spectral signature is the characteristic pattern of reflectance (and/or emittance) of a feature as a function of wavelength. Each material — vegetation, water, soil — reflects, absorbs, and emits EM energy uniquely, allowing its identification from remote sensing data.

Describe the typical spectral reflectance behaviour of healthy green vegetation.

Low reflectance in blue and red (chlorophyll absorption), a small peak in green (hence green appearance), a sharp 'red edge' rise, and high reflectance in the near-infrared (due to leaf internal/mesophyll structure). This high NIR–low red contrast underlies vegetation indices like NDVI.

How does the spectral reflectance of clear water typically behave?

Water reflects in the blue-green region but absorbs strongly in the near-infrared and beyond, so it appears very dark (near-zero reflectance) in NIR bands. This makes NIR effective for delineating water bodies.

What are the four types of resolution in remote sensing?

Spatial resolution, spectral resolution, radiometric resolution, and temporal resolution.

Define 'spatial resolution'.

Spatial resolution is the smallest ground area (size of one pixel / ground sample distance) that a sensor can distinguish — the level of spatial detail. A smaller pixel (e.g., 1 m) means higher spatial resolution; a larger pixel (e.g., 1 km) means coarser detail.

Define 'spectral resolution'.

Spectral resolution is the ability of a sensor to distinguish fine wavelength intervals — defined by the number and narrowness (bandwidth) of the spectral bands. More, narrower bands (e.g., hyperspectral) give higher spectral resolution and better material discrimination.

Define 'radiometric resolution'.

Radiometric resolution is the sensitivity of a sensor to differences in signal strength — the number of brightness/grey levels it can record, expressed in bits. An $n$-bit sensor records $2^{n}$ levels (e.g., 8-bit = 256 levels); higher bit depth means finer radiometric resolution.

Define 'temporal resolution'.

Temporal resolution is the revisit time — the frequency with which a sensor images the same area (e.g., every 16 days). High (fine) temporal resolution means more frequent coverage, important for monitoring dynamic phenomena and change detection.

Explain the typical trade-off between spatial and spectral/temporal resolution.

Sensor design involves trade-offs: increasing spatial resolution (smaller pixels) reduces the energy per pixel, often forcing wider bands (lower spectral resolution) or a narrower swath that lengthens revisit time (lower temporal resolution). High spatial detail and high temporal frequency usually cannot be maximized simultaneously.

How does radiometric resolution affect an image's appearance and information content?

Higher radiometric resolution (more bits/grey levels) lets the sensor record finer differences in reflected/emitted energy, producing images that show subtle tonal variations and contrast. Low radiometric resolution gives few grey levels and a coarse, contrast-limited image.