# Path Average Rain Rate Estimation ``` python cml, baseline = open_cml_sample(), open_cml_sample("baseline") ``` ## k-R law for Network Management System (NMS) min/max sampling The rain rate (*R*) can be estimated from the specific attenuation (*k*) expressed in *d**B* · *k**m*−1 using an exponential law known as the k-R relationship: *k* = *a**R**b* where *a* and *b* are parameters that depend on the frequency, polarisation and drop size distribution (DSD). However, the available attenuation is the Path Integrated Attenuation: *P**I**A* = ∫0*L**k*(*l*) *d**l* where *L* is the length of the link. We are interested in the path-averaged rain rate *R̄*, and the above relationship can be approximated as: $$ PIA = \int_0^L k(l) \\ dl = \int_0^L a R(l)^b \\ dl \stackrel{b \approx 1}{=} a \bar{R}^b L $$ The further *b* deviates from 1, the more imprecise this relationship becomes. If attenuation sampling is not instantaneous (e.g., 15 minutes), PIA is also time-averaged; thus *R̄* becomes the time-mean path-averaged precipitation rate. Considering that the sample data is located in Douala, and that the *a* and *b* coefficients are dependent on the DSD, which varies by region, we will first develop the coefficients estimated by Alcoba (2019). ------------------------------------------------------------------------ source ### alcoba_2019_africa_coefs ``` python def alcoba_2019_africa_coefs( )->Dataset: ``` *k-R relationship coefficients for Africa from [Alcoba (2019)](https://theses.hal.science/tel-02955598/).* ``` python alcoba_2019_africa_coefs().to_dataframe() ```
a b
frequency
7000 0.000197 1.8540
8500 0.006600 1.2897
11000 0.019500 1.1951
11500 0.023000 1.1775
13000 0.034700 1.1333
14500 0.047300 1.1022
15000 0.051700 1.0943
18000 0.078100 1.0654
19000 0.087300 1.0600
22000 0.117100 1.0471
23000 0.128100 1.0428
> **Warning** > > While different coefficient assignment algorithms are available (see > below), they all based on some form of nearest value. If your > frequencies differ significantly from those listed, you may need to > use a downscaling algorithm (such as interpolation) to generate > coefficients for your specific frequencies before estimating rainfall.
In many scenarios, NMS only save the min and max values for the sampling interval. Thus rain rate has to be estimated from those values. Overeem et al. (2013), developped an algorithm that computes time mean path averaged rain rate for 15 minute interval sampling based only on minimum and máximum values. The algorith asumes that the TSL is constant for each time. Thus máximum and mínimum attenuations are computed as follows: *A**m**i**n* = (*T**S**L* − *R**S**L**m**i**n*) − *b**a**s**e**l**i**n**e* *A**m**a**x* = (*T**S**L* − *R**S**L**m**a**x*) − *b**a**s**e**l**i**n**e* Then rain rate is estimated as follows. $$ \bar{R}\_{min | max} = a \left(\frac{A\_{min|max} - WAA}{L}\right)^b H(A\_{min|max} - WAA) \quad $$ *H*(*x*) is the Heaviside step function, defined as: $$ H(x) = \begin{cases} 0 & \text{if } x \leq 0 \\ 1 & \text{if } x \> 0 \end{cases} $$ The baseline and WAA have already been developed in other submodules, meaning that PIA can easily be obtained. Therefore, to maintain the modular approach, the focus will be on obtaining the *R̄* from the PIA. $$ \bar{R}\_{min | max} = a \left(\frac{PIA\_{min|max}}{L}\right)^b H(PIA\_{min|max}) \quad $$ Where PIA is defined as: *P**I**A**m**i**n*|*m**a**x* = *A**m**i**n*|*m**a**x* − *W**A**A* Finally we can complute *R̄* as a weighted average of *R̄**m**a**x* and *R̄**m**i**n*: *R̄* = *α**R̄**m**a**x* + (1 − *α*)*R̄**m**i**n*   Overeem et al. (2013) estimated and optimal value of *α* = 0.33 by calibrating the model using 12 days of data from the Netherlands. We use this value as a reasonable default; however, it should be calibrated for each specific region and deployment. ------------------------------------------------------------------------ source ### get_overeem_et_al_2013_min_max_nms_tprate ``` python def get_overeem_et_al_2013_min_max_nms_tprate( pia:Dataset, # ds containing the minimum and maximum Path Integrated Attenuations (PIA) cfs:Dataset, # ds containing `a` and `b`, k-R law coefficients as variable and `frequency` as dim cfs_assign_method:str='nearest', # xr.sel method used to map the pia frequencies to the cfs freqs alpha:float=0.3, # weighting factor between max and min sampling rain rates name_min:str='pia_min', # Name of the variable containing minimum PIA name_max:str='pia_max', # Name of the variable containing maximum PIA )->Dataset: ``` *Compute the rain rate using the Overeem et al. (2013) min/max sampling approach.* ``` python tprate = get_overeem_et_al_2013_min_max_nms_tprate(pia, alcoba_2019_africa_coefs()) tprate ``` 
<xarray.Dataset> Size: 18MB
Dimensions:      (cml_id: 126, sublink_id: 6, time: 2964)
Coordinates: (10)
Data variables:
    tprate       (cml_id, sublink_id, time) float64 18MB 0.0 0.0 0.0 ... 0.0 0.0
![](13_rain_rate_files/figure-commonmark/cell-7-output-1.png) #### References - Alcoba Kait, M. 2019. A contribution to rainfall observation in Africa from polarimetric weather radar and commercial microwave links. Ph.D. dissertation, Université Paul Sabatier – Toulouse III, 202 pages. https://tel-02955598 (accessed 2025‑11‑21). - Overeem, A., H. Leijnse, and R. Uijlenhoet, 2013: Country-wide rainfall maps from cellular communication networks. Proceedings of the National Academy of Sciences, 110, 2741–2745, doi:10.1073/pnas.1217961110. - Chwala, C., and H. Kunstmann, 2019: Commercial Microwave Link Networks for rainfall observation: Assessment of the current status and future challenges. WIREs Water, 6, doi:10.1002/wat2.1337.