Past exam of the year 2 of the earth sciences (geology) course of the University of Oxford / 2022 / Paper 1 / 10 / c

Sampling at $0.2$ Hz means $\Delta t=5$ s. The target times are therefore $5$, $10$ and $15$ s, all within the supplied observation range. For bracketing observations, linear interpolation gives $d(t)=d_i+(d_{i+1}-d_i)(t-t_i)/(t_{i+1}-t_i)$. ThusThe estimated displacements are $7.00$, $12.99$ and $8.98\mu \mathrm{m}$. Interpolation assumes approximately linear behavior within each interval; it does not recover any unobserved high-frequency motion.

Regular timing supplies a common interval and a well-defined frequency grid for a standard discrete Fourier transform and Fast Fourier transform. It also makes sample-index lags correspond to fixed physical delays in cross-correlation, autocorrelation and digital filtering. These methods estimate frequency content and characteristic periods, separate instrument or environmental noise from signals, identify arrivals and time delays, and compare ground-motion spectra.

A uniform interval establishes a Nyquist frequency; frequencies above it alias unless filtered before acquisition. Interpolating irregular data onto a regular grid does not undo prior aliasing or fill large gaps with new measurements, and it can alter spectra and create correlated errors. Methods that directly use irregular times are possible, but the ordinary FFT and fixed-lag filter formulas cannot simply treat the original irregular observations as evenly spaced.

Successive jumps occur at times separated by $T=6$ s. Between $-3$ and $3$ s the plotted function is $f(t)=2-t$, periodically extended, with maximum $5$ and minimum $-1$. Its mean is therefore $2$ m, and $f(t)-2=-t$ is an odd function over that centered period.

For the Fourier series convention$a_0=4$ m and $a_r=0$ for every $r\ge 1$; the nonconstant content consists of sine terms. If $a_0$ denotes the mean directly in the alternative convention $f=a_0+\sum \cdots$, its value is $2$ m. Stating the convention resolves the otherwise ambiguous notation.

For completeness, integrating $-t$ against the sine basis gives $b_r=6(-1{)}^r/(\pi r)$ m. At a jump the Fourier series converges to the midpoint $2$ m. The nonzero mean does not destroy the oddness argument for the mean-subtracted function; declaring the original function itself odd would incorrectly force its mean to vanish.