Question

The ratio of the largest to the smallest amplitude of waveforms that can be accurately recorded by a digital seismometer is reported as \(10^7\). Then, the dynamic range of the seismometer in dB is _______ (in integer).

Hint:

For amplitude-based measurements (like seismic waveforms or sound levels), the dynamic range is always calculated using \(20 \log_{10}\) of the ratio. For power-based measurements, \(10 \log_{10}\) is used.

Answer 1

Step 1: Recall formula for dynamic range in dB
The dynamic range \(DR\) of a system is given by: \[ DR = 20 \log_{10}\left(\frac{A_{\text{max}}}{A_{\text{min}}}\right) \] where \(\frac{A_{\text{max}}}{A_{\text{min}}}\) is the amplitude ratio. Step 2: Substitute given values
Given amplitude ratio = \(10^7\).
\[ DR = 20 \log_{10}(10^7) \] Step 3: Simplify logarithm
\[ \log_{10}(10^7) = 7 \] \[ DR = 20 \times 7 \] \[ DR = 140 \, \text{dB} \] Step 4: Final Answer
Thus, the dynamic range of the seismometer is: \[ \boxed{140 \, \text{dB}} \]