# Explanation of OASPL and SEL

I have data of different SPL for different 1/3 center frequencies. Is it correct to sum the pressures at a mean square basis of those SPL values (and then convert them back to SPL) to obtain the OASPL (Overall Sound Pressure Level)? Furthermore, how to compute a SEL (Sound Exposure Level) from different OASPL over a time period? Just acoustically sum those OASPL values and divide the sum by the number of time steps? Many thanks in advance!

Yes, you can calculate it by summing all of the 3rd-octave band pressures. Keep in mind, that if the units are pressure then simple sum and conversion to the decibel scale is enough. However, if you have 3rd-octave band levels, then you have to calculate the anti-log of those values, sum them and then convert back to SPL.

Regarding the Sound Exposure Level, you can calculate it from the equivalent SPL $L_{eq}$:

$$SEL = L_{eq} + 10\log_{10}T$$

Where $T$ is the duration of the measurement in seconds. There is a simple intuition behind it. The $L_{eq}$ can be understood as the level of a constant signal, which has the same energy as the varying signal you’ve measured, with the same length. The $SEL$, on the other hand, is the same value, but for a signal that lasts 1 second.

I assume that you will calculate the sum of all pressures (in linear scale) for each of the time instants - you’ve called them OASPL (I don't like this name). Then all it takes to calculate the $SEL$ is to average them, convert to decibel scale and add the duration logarithm of the measurements duration.

Here is a complete example, so you can grasp what I am talking about. To keep it simple, I will use octave bands instead (8 Hz - 16 kHz), which doesn't actually change anything.

Here are some unweighted levels ($L_{Zeq}$), logged every 0.1 second with total duration being 4.8 seconds.

67.2, 61.3, 46.9, 57.1, 57.7, 51.2, 44.4, 37.7, 35.5, 31.4, 23.2, 19.7
66.7, 60.0, 52.5, 58.0, 56.6, 50.5, 44.8, 38.6, 35.3, 32.7, 24.6, 20.0
64.7, 60.6, 65.8, 56.6, 57.0, 53.3, 45.4, 38.8, 35.0, 34.5, 24.6, 19.4
60.8, 60.5, 66.4, 55.9, 55.7, 52.7, 45.6, 38.7, 33.9, 29.8, 21.7, 19.6
57.3, 63.3, 54.8, 57.4, 58.1, 52.1, 43.8, 39.2, 34.5, 33.5, 26.5, 20.2
53.8, 67.2, 50.7, 56.0, 54.7, 50.4, 43.4, 37.7, 35.5, 35.4, 31.5, 23.8
53.3, 65.4, 51.0, 56.2, 55.0, 51.6, 45.4, 38.6, 35.0, 32.3, 28.5, 21.7
53.6, 61.7, 48.2, 56.1, 56.8, 52.5, 45.0, 39.7, 39.6, 35.5, 27.2, 21.1
52.4, 61.9, 49.7, 56.7, 56.1, 51.5, 45.2, 39.0, 35.5, 31.5, 24.8, 20.5
56.6, 62.2, 44.2, 57.8, 56.7, 50.8, 46.5, 58.0, 51.3, 47.9, 43.4, 40.9
57.6, 63.0, 44.7, 58.1, 55.1, 51.0, 58.3, 48.4, 33.6, 27.8, 22.0, 20.5
60.7, 61.6, 52.7, 56.5, 57.9, 56.7, 46.3, 36.9, 33.3, 29.2, 23.7, 19.9
61.2, 62.3, 51.5, 57.1, 57.2, 51.4, 57.9, 56.7, 49.4, 48.8, 51.0, 53.1
60.4, 59.7, 50.6, 56.7, 55.1, 55.9, 51.7, 38.5, 33.5, 27.5, 21.6, 19.4
60.6, 60.4, 52.8, 55.3, 55.9, 53.2, 43.7, 53.8, 52.0, 50.9, 47.6, 49.8
59.7, 67.1, 48.7, 56.0, 54.9, 51.1, 58.6, 55.0, 41.9, 36.7, 39.2, 28.4
60.5, 68.0, 46.3, 56.6, 55.9, 54.3, 55.9, 55.0, 61.3, 44.5, 43.2, 29.2
61.0, 66.4, 43.3, 57.6, 54.4, 56.9, 52.8, 50.3, 51.1, 36.5, 32.8, 20.5
62.3, 61.2, 44.5, 57.9, 52.3, 57.3, 45.3, 40.0, 43.8, 39.9, 34.5, 28.1
64.7, 59.2, 49.2, 59.0, 58.8, 51.0, 47.5, 52.3, 47.5, 44.9, 38.8, 28.6
66.0, 59.6, 53.2, 62.6, 57.2, 50.6, 46.5, 45.2, 40.7, 35.6, 27.2, 20.0
68.8, 60.3, 51.3, 63.7, 51.7, 48.2, 46.5, 38.6, 34.6, 28.7, 25.3, 21.9
70.2, 60.2, 49.5, 60.2, 54.6, 52.2, 47.8, 38.8, 33.0, 29.3, 29.0, 26.1
69.6, 58.6, 48.9, 60.9, 54.1, 51.0, 45.4, 38.1, 34.3, 31.1, 29.3, 24.1
68.8, 58.7, 47.3, 61.5, 53.2, 48.4, 44.8, 37.6, 35.9, 32.3, 29.2, 25.6
67.4, 61.4, 49.4, 61.3, 55.3, 49.8, 47.2, 40.1, 36.7, 32.8, 26.8, 20.9
68.4, 61.8, 61.3, 60.9, 52.6, 47.5, 45.0, 38.5, 36.3, 32.3, 25.7, 19.8
68.2, 61.9, 64.3, 61.5, 51.5, 49.3, 45.7, 37.8, 33.9, 28.3, 22.6, 19.2
68.9, 59.6, 61.1, 60.4, 54.2, 50.9, 44.2, 39.0, 34.1, 27.9, 22.5, 19.7
69.4, 59.5, 57.3, 59.8, 55.0, 51.4, 45.7, 38.8, 33.6, 29.3, 28.1, 20.7
69.3, 60.5, 54.1, 60.3, 51.7, 50.4, 45.2, 37.9, 34.3, 29.1, 25.4, 20.6
68.8, 59.4, 50.0, 60.3, 50.5, 51.7, 44.7, 36.5, 34.0, 29.3, 25.5, 20.0
67.5, 56.2, 49.4, 60.4, 53.2, 51.0, 47.0, 37.4, 33.5, 28.2, 23.0, 19.5
66.3, 53.8, 51.4, 61.4, 52.6, 50.2, 46.1, 37.0, 33.9, 29.8, 23.5, 20.2
65.1, 58.7, 52.3, 61.1, 51.2, 50.6, 45.6, 38.6, 34.5, 28.8, 21.4, 19.6
67.2, 59.6, 50.0, 60.9, 52.5, 48.4, 45.9, 38.9, 34.3, 31.0, 23.0, 19.9
67.1, 60.3, 50.9, 60.2, 53.4, 52.0, 45.7, 39.2, 35.3, 35.2, 27.2, 22.3
66.9, 61.2, 51.2, 60.4, 52.7, 52.1, 46.7, 38.4, 34.7, 31.1, 24.6, 20.8
65.3, 63.6, 52.1, 59.2, 55.6, 50.0, 46.9, 38.5, 34.4, 35.2, 28.3, 21.2
61.9, 65.7, 52.7, 59.4, 56.6, 50.3, 46.8, 39.7, 53.0, 46.8, 32.8, 26.1
59.7, 65.8, 52.6, 61.1, 54.0, 50.1, 44.1, 39.3, 45.0, 41.1, 28.8, 21.7
57.9, 63.7, 48.4, 60.6, 52.9, 51.5, 46.4, 41.3, 37.6, 35.3, 27.2, 20.9
57.4, 62.6, 48.4, 60.3, 55.2, 52.2, 45.7, 37.3, 34.6, 30.4, 25.4, 20.5
58.3, 61.1, 51.0, 59.9, 53.3, 48.7, 45.7, 37.3, 35.9, 31.5, 29.2, 21.6
60.9, 59.3, 51.4, 59.5, 53.3, 50.5, 46.7, 38.7, 36.3, 31.7, 28.6, 21.5
63.1, 59.7, 48.5, 60.1, 54.9, 51.4, 45.5, 37.8, 37.2, 34.4, 30.3, 22.8
62.5, 57.9, 48.4, 59.9, 54.9, 49.4, 45.1, 39.4, 40.3, 35.4, 30.5, 22.4
60.9, 59.6, 48.8, 59.9, 56.4, 50.7, 43.9, 46.2, 41.0, 39.5, 32.6, 25.0


Since this measurement had been performed using an SLM, I know the exact value of $SEL$, which is $75.3 \mathrm{dB}$.

Let's read it and calculate the instantaneous $L_{Zeq}$ for each frame (the thing that you call OASPL).

#!/usr/bin/env python

import numpy as np

if __name__ == "__main__":
bands = [8.0 , 16.0, 31.5, 63.0, 125.0, 250.0, 500.0, 1000.0, 2000.0, 4000.0, 8000.0, 16000.0]
oct_L = np.genfromtxt('octave.csv', delimiter=',')

oct_P = np.power(10, oct_L/10)   # Anti-log to go back to linear domain
frame_P = np.sum(oct_P, axis=1)  # Instantaneous pressure in each octave band

Leq_dt = 10*np.log10(frame_P)    # Instantaneous SPL - OASPL

Leq = 10*np.log10(np.mean(frame_P))  # Equivalent SPL

T = oct_L.shape[0] * 0.1
SEL = Leq + 10*np.log10(T)

print(Leq_dt)
print(SEL)


That returns the $SEL$ equal to 75.3 dB and per frame equivalent levels:

[ 69.   68.5  69.6  68.9  66.4  68.1  66.8  64.6  64.6  66.3  66.5  66.4
67.7  65.4  66.1  69.1  70.3  68.7  66.5  67.7  68.8  70.6  71.2  70.6
70.   69.5  70.5  71.   70.6  70.6  70.5  70.   68.8  68.   67.6  68.9
68.9  68.9  68.6  68.6  68.2  66.6  66.1  65.3  65.4  66.6  65.9  65.8]