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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