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Influence of Humidity on the Accuracy of
Low-Cost Particulate Matter Sensors
Norbert Streibl
October 2017
Contents
1 summary 2
2 correlation of mass of particulate matter and humidity 4
3 possible choices for empirical growth functions 5
4 empirical parametrization of growth functions 6
5 growth function according to Hänel 7
6 growth function according to Soneja et. al. 9
7 growth function according to Skupin and comparison 11
8 results of humidity corrections 12
9 possible directions for future research 15
1
1 summary
Air borne particulate matter P M absorbs humidity from the air. Therefore the measured values of PM-
sensors systematically increase, when the relative humidity rh is high1. By using a growth function
gf (rh)this condensation effect may be estimated by using the formula PMwet =gf (rh)·PMdry . Low-cost
sensors typically measure wet particulate matter PMwet , whereas expensive sensors for professional
purposes often measure dried particles PMdr y. For comparison of these values the growth function
gf (rh)should be known.
The relative humidity is mainly determined by the large-scale weather conditions; on top of this there
exists a daily temperature effect: Usually it is colder in the night than in the day and on sunny days the
PM -sensor may even be exposed to the sun. Therefore during the night the ambient air is nearer to
the dew point than during the day. The weather dependent relative humidity is overlaid by a 24h-cycle
induced by temperature. This suggests a “big data approach” to identify the humidity dependent growth
function: The growth function can be identified either by minimization of the overlaid 24h-cycle of the
PM -signal or by minimization of the correlation between the PM-signal and the rh-signal.
In this study growth functions were determined for PM-sensors from the OK-Lab sensor network2which
consists of inexpensive NOVA SDS011 sensors3. According to specification these may be used for
rh < 70%. For rh ≈80%the empirical growth factors scatter between 1.5up to 3. For rh ≈90%growth
factors are in the range 2to 5. Low-cost sensors for wet particulate matter overestimate the amount of
particulate matter significantly compared to sensors for dry particulate matter. Therefore a correction of
the condensation effect is necessary and it will be large and quite uncertain. Besides the measurement
uncertainty of the PM-sensor and the measurement uncertainty of the rh-sensor a significant “correction
uncertainty” must be considered. Nevertheless the correction brings the best low-cost sensors into the
same range of magnitude as professional measurement stations4.
The empirical growth functions seem to display significant seasonal variations: during spring time in
April and May an increased growth was found. It may be speculated that this is caused by plant pollen.
It has not yet been investigated, whether there is also a variation between city, suburbs and countryside.
We considered OK-Lab sensors firstly in the suburban countryside in and around the small city Leonberg
15km from Stuttgart and secondly at the periphery of Stuttgart located at Pragsattel near the edge of
the Stuttgart basin. Thirdly the inner-city sensor operated by the government agency LUBW for dry
particulate matter at Stuttgart Neckartor in the middle of the Stuttgart basin is used as a reference. The
humidity corrected PM -values of OK-Lab sensors increase - as it may be expected - from the countryside
to the suburbs and are lower than the inner city reference sensor. Without humidity correction the raw
PMw et -signals are on humid days higher than the reference sensor for PMdry . At the beginning of
2017 strong episodes of pollution have been observed at the reference sensor near Stuttgart Neckartor.
These are well visible with the low-cost sensors in the surrounding area, even on the countryside, due
to large-scale meteorological conditions such as inversions. During late summer and in the fall the
nightly humidity is near dew point and the air becomes dry on sunny days. Therefore a 24h-cycle in
humidity as well as in wet particulate matter PMwet is distinctly observable and can be compensated
by applying an empirical growth function. Adjacent sensors show comparable results after humidity
compensation. However, the empirical growth function must be determined individually for each sensor,
as there is no universally valid compensation function - the local environmental conditions as well as
systematic deviations of the humidity sensors are individual. Further improvements may be achievable,
if a time-variant (i.e. sliding) parametrization of the empirical growth function would be employed.
In summary: The raw measured values PMwet from low-cost sensors require correction for humidity rh
in order to become comparable to reference sensors for PMdry . Due to the uncertainties of the relevant
sensor signals and the strong dependency on humidity, however, the corrected results remain rather an
indication than a highly accurate measurement. On the other hand overall trends are well reproduced.
1Bernd Laquai pointed this out to the author in the context of low-cost PM sensors
2https://www.madavi.de/ok-lab-stuttgart/
3hyperlink: datasheet NOVA SDS011
4https:(https://www.stadtklima-stuttgart.de/index.php?luft_messdaten_feinstaubwerte)
2
Figure 1: foggy and dusty sunrise on February 11th, 2017 near the sensor Leonberg Silberberg #2
Figure 2: signals of sensor Leonberg Silberberg #2 on February 11th, 2017. Air temperatures (green)
rise from nightly 0◦Cup to 35◦Caround 14:00 hours while exposed to the sun. Humidity (blue) falls
from 82%to 14%while PM10wet (gray) also decreases from 135µg
m3to less than 40µg
m3. After correction
with an empirical growth function gf (rh)=(1−rh)−0.49 a slow continuous decrease of the estimated
PMdry is observed over the day.
Location Operator PM 10wet P M10dr y
Leonberg Silberberg #1 OK-Lab 101µ g
m351µ g
m3
Leonberg Silberberg #2 OK-Lab 106µ g
m353µ g
m3
Leonberg Gartenstadt #2 OK-Lab 69µ g
m334µ g
m3
Stuttgart Pragsattel OK-Lab 159µ g
m367µ g
m3
reference: Stuttgart Neckartor LUBW - 89µ g
m3
Table 1: comparison of the median PM-values for several sensors on February 11th, 2017
3
2 correlation of mass of particulate matter and humidity
During fall season, when not much particulate matter pollutes the countryside, and when the nightly tem-
perature is near dew point, and when many days are warm and sunny in Germany, then the temperature-
and humidity-signals have a visible 24hperiodical component. The humidity signal oscillates between
90%in the night and less than 30%when exposed to the sun.
Figure 3: temperature and humidity between 24.09. and 30.09.2017 at sensor Leonberg Silberberg #2
The corresponding PM -signals also oscillate, although they look significantly more noisy. The values for
dry air are in the single digit range PMdry <10µg
m3.
Figure 4: particulate matter between 24.09. and 30.09.2017 at sensor Leonberg Silberberg #2
The reference sensor downtown Stuttgart operated by the LUBW and located near the Neckartor 5is
exposed to a lot of street traffic and measures dry particulate matter of the class PM10.During the same
week its daily average values range between 29and 40µg
m3and were mostly near 34µg
m3. Therefore single
digit values out at the countryside seem quite plausible.
5see: https://www.stadtklima-stuttgart.de/index.php?luft_messdaten_feinstaubwerte
4
Figure 5: P M2.5wet versus rh between 24.09. and 30.09.2017 at sensor Leonberg Silberberg #2
The scatter plot of particulate matter against humidity shows a certain correlation, because the data
points are not scattered chaotically in a cloud. They seem to accumulate along a curve, which is rather
flat for humidities rh < 50%.For higher humidity the curve increases strongly. It looks as if for rh > 90%
fivefold higher PM-values may occur, than under dry conditions. This indicates a growth function with a
steep slope towards higher humidity. In the lower right corner lies a cluster of measured values, where
the PM -value is very low, while humidity is very high: these measurements were obtained after a strong
rain had cleaned the air from particulate matter.
3 possible choices for empirical growth functions
It is assumed that a growth function exists, which relates the measurements of wet and dry particulate
matter:
PMw et =gf (rh)·PMdr y (1)
The relative humidity is in the calculation formulas normalized 0≤rh ≤1, i.e. a value rh =1 corre-
sponds to 100%humidity. If wet particulate matter was measured and the growth function was somehow
estimated, then a correction for humidity becomes possible:
PMdry =PMwet
gf (rh)(2)
From the physics of condensation it is not obvious, that a growth function actually exists: the increase
of the measured values depends on the size distribution of the particulate matter and how much of the
finer dust becomes visible to the sensor after swelling due to the condensation of humidity. If the size
distribution is irregular and there are peaks at certain sizes, then the growth will also become irregular.
Nevertheless literature gives several parametrized formulas for empirical growth functions:
gfH¨
anel =1
(1−rh)β(3)
gfSonej a =1+α·rh2
1−rh (4)
5
gfcombo =1+α·rh2
(1−rh)β(5)
gfSk upin =
α
(1−rh)βrh ≥0.7
1
(1−rh)β−log(α)
log(0.3)
rh < 0.7(6)
By properly choosing the parameters αand βthese functions can be fitted to measured data. Common
to all growth functions is thatgf (0)=1 and gf (rh →1)1. The growth functions according to Hänel
and to Skupin were found in the thesis of A. Skupin6, the function according Soneja from an scientific
article7and the combination function is an own attempt to combine Soneja with Hänel.
4 empirical parametrization of growth functions
The measured values of the OK-Lab sensors are stored on a public server8. These signals were pro-
cessed by taking the following steps:
download time series from server, which consists of time stamp, PM 10,PM2.5, temperature,
humidity and some additional data
eliminate non-plausible data, where values are outside a reasonable range and/or items are miss-
ing
resample the measured data onto a regular time raster with 1sample per minute (repeat last valid
data set until a new and valid data set becomes available)
process signals, e.g. apply humidity corrections or perform Fourier transformations to analyze
spectral behavior or calculate correlations
smooth data by median filtering and, if desired, resample to a more coarse raster with 1sample
per hour or per day
The identification of an empirical growth function is based on one of the two following hypotheses:
temperature, humidity and the growth of particulate matter contain a periodic component with
a24hperiod. The absolute value of the corresponding normalized Fourier-coefficient becomes
minimal, as soon as the influence of humidity is compensated in the best possible way. Therefore
the parameters of the empirical growth functions are identified by minimization of the function
fom(α,β)=10·log10
´∞
−∞
PM (t)
gf(α ,β)·e2π i t
24h·dt
´∞
−∞
PM (t)
gf(α ,β)·dt
(7)
the signals for dry particulate matter and humidity are not expected to be correlated. The nor-
malized correlation factor between PMdry and rh becomes minimal, as soon as the influence
of humidity is compensated in the best possible way. Therefore the parameters of the empirical
growth functions are identified by minimization of the function
fom(α,β)= PN
i=1(rhi− hrhi)·P Mi
gf(α ,β)−DP Mi
gf(α ,β)E
rPN
i=1(rhi− hrhi)2·PN
i=1P Mi
gf(α ,β)−DP Mi
gf(α ,β)E2(8)
whereby the brackets denote an arithmetic mean value hxi=1
NPN
i=1xi
6Anne Skupin: “Optische und mikrophysikalische Charakterisierung von urbanem Aerosol bei (hoher) Umgebungsfeuchte”,
University Leipzig (2013)
7Sutyajeet Soneja et al.: Humidity and Gravimetric Equivalency Adjustments for Nephelometer-Based Particulate Matter Mea-
surements of Emissions from Solid Biomass Fuel Use in Cookstoves; Int. J. Environ. Res. Public Health 2014, 11, 6400-6416
8https://www.madavi.de/ok-lab-stuttgart/
6
The growth functions according to Hänel and Soneja depend on one parameter each. Therefore the
parameter can easily identified by plotting the figure of merit fom as a function of the parameter. More
complicated functions are minimized numerically by using the simplex method, which is reasonably fast
to compute.
5 growth function according to Hänel
Hänel gave the growth function gf =(1−rh)−βwherein the “Hänel exponent” βserves as fitting parame-
ter. For the sensor Leonberg Silberberg #2 different results are obtained depending on the time range of
the measurements and depending on the figure of merit, i.e. whether the normalized Fourier-coefficient
for the 24hperiod or the normalized correlation coefficient between particulate matter and humidity are
minimized. In the following analysis the value for PM2.5was used, because this signal looks less noisy
than PM 10.
Figure 6: Fourier coefficient for the 24hperiod as a function of the Hänel exponent β
The normalized Fourier-coefficient typically changes by 10dB, that is by an order of magnitude. Only in
January the variation was less decisive, when the weather was humid and not very sunny and the PM-
signals where dominated by strong pollution events. During every month, however, a clearly defined
minimum occurs albeit at different Hänel exponents: they vary between 0.38 and 0.69. For the time
range between 01.01.2017 - 30.09.2017 the optimum Hänel exponent takes the value β=0.49.
7
Figure 7: correlation coefficient as a function of the Hänel exponent β
The normalized correlation coefficient typically changes by more than an order of magnitude. However,
in January no minimum occurs at all, probably because the correlation coefficient 0.06between PMwet
and rh is already too small for a minimization. January was dominated by strong pollution events, such
as the pollution by fireworks at new years day and a huge pollution episode caused by a meteorological
inversion, that lead to an environmental alarm in Stuttgart (“Feinstaubalarm”). During the other months
a clearly defined minimum occurs albeit at different Hänel exponents: these vary between 0.27 and
0.70. For the time range between 01.01.2017 - 30.09.2017 the optimal Hänel exponent takes the value
β=0.43.
Figure 8: monthly variation of the Hänel exponent at sensor Leonberg Silberberg #2
The Fourier and the correlation method yield Hänel exponents in a similar range. Also the seasonal
variation, which is clearly observable, looks similar. One might speculate, whether the spring maximum
of the Hänel exponent around April and May is caused by plant pollen with a hygroscopic property
different from ordinary dust.
8
Figure 9: empirical growth functions according to Hänel for the different months
The empirical growth functions vary from month to month. Below a humidity rh = 70% the growth is
below a factor of 2. Incidentally the sensor NOVA SDS011 is specified only for humidities rh < 70%.
For humidities rh > 90%growth factors of 5may occur. Under humid conditions the sensor has a huge
systematic deviation.
6 growth function according to Soneja et. al.
A similar analysis was conducted for Soneja’s growth function gfSonej a =1+α·rh2
1−rh with the “Soneja
weight” αas fitting parameter:
Figure 10: Fourier coefficient for 24hperiod as a function of the Soneja weight α
The normalized Fourier-coefficient typically changes again by 10dB, that is by an order of magnitude.
Only in January the variation was less, when the weather was humid and not very sunny and the PM-
signals where dominated by strong pollution events. During every month, however, a clearly defined
9
minimum occurs albeit at different Soneja weights: they vary between 0.15and 0.45. For the time range
between 01.01.2017 - 30.09.2017 the optimal Soneja weight takes the value α=0.256.
Figure 11: correlation coefficient as a function of the Soneja weight α
The normalized correlation coefficient typically changes by more than an order of magnitude. However,
in January no minimum occurs just as in the case of the Hänel growth function. During the other months a
clearly defined minimum occurs albeit at different Soneja weights: these vary between 0.1and 0.45. For
the time range between 01.01.2017 - 30.09.2017 the optimum Soneja weight takes the value β=0.218.
Figure 12: monthly variation of the Hänel exponent at sensor Leonberg Silberberg #2
Comparison of the monthly values for the Soneja weights shows a behavior similar to the Hänel exponent
with a spring time peak around April and May.
10
Figure 13: empirical growth functions according to Soneja et.al. for the different months
The empirical growth functions according to Soneja are quite similar to those after Hänel indicating a
systematic deviation of a factor 5 under very humid conditions.
7 growth function according to Skupin and comparison
The two fitting parameters of the growth function according to Skupin were numerically identified by
using the simplex method. However, the related growth curves look slightly less plausible than those
according to Hänel or Soneja. A probable reason might be that the two fitting parameters are derived
from noisy data with not too many data at low humidities rh < 70%. A similar effect occurred, when it
was tried to identify the two fitting parameters of the combo growth function. It seems that parameter
identification for growth functions with more than one parameters is less robust.
Skupin investigated in her thesis large data sets derived from state of the art sensors in Leipzig and
gives mean values and limits for the fitting parameters of her growth function. It is instructive to plot the
resulting growth functions and compare to our empirical growth function according to Hänel.
Figure 14: range of growth functions according to Skupin and Hänel
11
If the empirical growth functions according to Hänel and Soneja for the long time range from 01.01.2017
until 30.09.2017 and for the Fourier- and the correlation method are plotted, again a significant scattering
of results becomes visible. Again a large deviation occurs for high humidities in all considered cases.
If we use any empirical growth function for the correction of wet P M-signals, the correction uncertainty
will be rather high due to this scatter.
Figure 15: comparison of empirical growth functions for sensor Leonberg Silberberg #2
Around 80%humidity the growth factor may range between 1.5and 3. Around 90%humidity the growth
factor may range between 2and 5.
8 results of humidity corrections
In the following study the growth function according to Hänel was chosen and the fitting parameter βwas
determined by minimization of the normalized Fourier-coefficient using the numerical simplex optimiza-
tion method over the long time range from 01.01.2017 until 30.09.2017. The optimal Hänel exponents
for different sensors were determined individually for each sensor. In this way humidity corrected time
series for dry particulate matter P M10dry were estimated for exemplary sensors.
The raw data PM10wet (gray) and the humidity corrected data PM10dr y (blue) show at the sensor
Leonberg Silberberg #2 in January a comparatively high pollution and in August a comparatively low
pollution in the hourly median value.
12
Figure 16: raw and humidity corrected PM10-signals at sensor Leonberg Silberberg #2 for January and
August 2017
In January we see the aftermath of the new years day fireworks and a strong meteorological inversion
beginning around 20.01.2017, both inducing heavy air pollution. The flat signal around 10.01.2017
was caused by a temporary sensor blackout. In August the pollution is generally low and overlaid
by the above mentioned 24h-cyclic growth of the particulate matter that leads to an oscillating signal.
This periodic component is significantly reduced by the humidity correction. Actually a somewhat better
compensation would be possible if the higher August value of the Hänel exponent would be used instead
of the long term value.
Next the median of the PM-signal over one day was calculated for different sensors.
Figure 17: long time run (01.01.2017-30.09.2017) of the raw and humidity corrected PM-signals at the
sensors Leonberg Silberberg #2 and Stuttgart Pragsattel and the reference sensor Stuttgart Neckartor
operated by the LUBW
The blue bars show the reference sensor for PMdry at Stuttgart Neckartor. It is expected to yield the
highest signal, because of its location down in the Stuttgart basin and because of a high local traffic.
The gray curves show the raw data PMwet from the low-cost sensor Leonberg Silberberg #2, which is
13
situated in the countryside outside the Stuttgart basin, and Stuttgart Pragsattel, which is situated at the
edge of the Stuttgart basin. Both gray curves are often higher than the blue bars, although the pollution is
expected to be lower at these locations, because of the growth due to humidity. The green curves show
the humidity corrected values, which are always lower than the reference sensor at its highly polluted
location. All curves reflect the strong pollution episodes at the beginning of the year 2017.
Figure 18: dry particulate matter in the countryside, in the periphery and downtown
Overexposing the curves for dry particulate matter PMdry for the countryside sensor (Leonberg Silber-
berg #2), the sensor at the edge of the Stuttgart basin (Stuttgart Pragsattel) and the reference sensor
down in the Stuttgart basin (Stuttgart Neckartor, operated by LUBW) shows several facts: In the bot-
tom of the basin the pollution is highest. Outside the city the air becomes cleaner, because the air
is exchanged more effectively and because there is less traffic. Nevertheless under specific meteo-
rologic conditions, namely under inversion, the air pollution by particulate matter extends over a huge
geographic area far out of the city.
Figure 19: raw and humidity corrected PM-signals at adjacent sensors in Leonberg
Three sensors in Leonberg are quite near to each other: the distance between Leonberg Silberberg #1
and #2 is less than 100mand the sensor Gartenstadt #2 is about 3k m away. After humidity correction the
14
sensor signals are reasonably similar and significantly below the reference sensor Stuttgart Neckartor.
However, not all OK-Lab sensors fit this picture. Therefore another calibration or a quality selection of
the sensors might be considered. The sensor specific Hänel exponents are quite different and therefore
cannot be transfered from one sensor to another:
location Hänel exponent PM10med ian
wet P M10median
dry PM10sigma
wet P M10sigma
dry
Leonberg Silberberg #1 0.30 11.9 7.3 49.6 28.7
Leonberg Silberberg #2 0.49 11.9 6.4 43.9 23.9
Leonberg Gartenstadt #2 0.37 11.8 6.5 46.6 22.4
Stuttgart Pragsattel 0.28 19 10.8 58.1 22.9
Stuttgart Neckartor (LUBW) 1n/a 28 n/a 28.2
Table 2: individual Hänel exponents and statistics (PM-values in µg
m3) for some OK-Lab sensors from
01.01.2017 until 30.09.2017
The overall statistics of the humidity correction looks quite plausible: the long time median for PM10
decreases significantly after correction from wet to dry values. The medians increase from countryside
over city periphery to downtown. Even more significantly, the standard deviations are about cut in half by
the humidity correction and become comparable between the OK-Lab sensors and the reference sensor.
Another improvement in the correction of the growth caused by humidity may be possible, if not a long
time value of the Hänel exponent is employed, but instead a sliding short time value.
9 possible directions for future research
Are the seasonal variations in fig. 8 and 12 real? they seem to indicate that in spring the hy-
groscopy of particulate matter differs from other seasons
Is there a dependence on location or composition of particulate matter or weather conditions?
Probably the Hänel exponent varies also between countryside, inner city, locations nearer to sea
(salt) etc. as well as on weather conditions
Why do sensors differ so much in their Hänel exponents? One might suspect at first glance,
that the Hänel exponents should not be so much a function of the individual sensor, but rather of
the physics of condensation and the hygroscopy of the particulate matter. Or do we see here the
normal scatter and all we can say about low-cost sensors is, that we must expect a Hänel exponent
somewhere between 0.2and 0.6?
Is there a time lag between the time series of humidity and the increase of measurement values of
the particulate matter due to swelling of the particles? One might expect some time constants due
to the physics of condensation.
Some sensors seem not to allow for plausible humidity corrections. Why? Are they just bad?
How do inner city sensors, e.g. those near Stuttgart Neckartor, actually compare to the reference
sensor?
Would a sliding (time-variant) parametrization of the growth function yield better results?
It is quite simple to come up with many more interesting research questions ...
15
