Dear David,

Rainfall spatial variability is a topic of research that several people have worked (and still work) on. Thiessen is a very basic method that may or may not be adequate depending on the gauge density, rainfall regime and type of applications (as already mentioned by others). You can certainly do better (how much better again depends) by looking at geostatistical approaches like ordinary Kriging or Kriging with external drift. You can find many publications devoted on the comparison of different methods (in terms of rainfall estimation) but also on their effect on runoff simulations.

Allow me to suggest some references that may provide some info on the topic:

- Work by Krajewski et al. 2003 (an analysis of small scale rainfall variability...) and similar work by the same authors offer info on the quantification of small scale rainfall variability and the issues associated with raingauge networks and their capacity to capture this variability.

- For an assesment of interpolation methods or methods that can account for the elevation gradients you can have a look (among others) at:

a) Bardossy and Pegram (2013): Interpolation of precipitation under topographic influence at different time scales (and references therein)

b)Ly et al (2011): Geostatistical interpolation of daily rainfall at catchment scale... (and references therein)

-Finally some more papers on the effect of rainfall spatial variability on runoff response you can find in the paper (and references therein) of

Zoccatelli et al (2010): Which rainfall spatial information ....

My apologies for the long response. Hopefully this may be of help.

The Thiessen methods are a kind of special case of IDW interpolation. Perhaps your area is having elevation difference, therefore, someone asked you about using Thiessen method.

Neither the IDW or Thiessen polygon approaches do not address the influence of elevation themselves (as with any interpolation method that does not explicitly include the influence of elevation - and other factors that influence the spatial distribution). They require that the rain gauges are suitably located to capture the influence of elevation on rainfall depth. Such methods work best if the rainfall data is normalised using a rainfall surface. You can then interpolate the normalised data, and combine with the rainfall surface. For example see

https://www.researchgate.net/publication/256464473_Evaluation_of_approaches_for_estimation_of_rainfall_and_the_unit_hydrograph?ev=prf_pub

The main problem with estimating the areal rainfall is the gauge density - particularly at finer time scales. At longer timescales (e.g. seasonal, annual), you are averaging across a number of events, and interpolating the rainfall data is more reliable. At a finer time scale (e.g. daily or finer), interpolation will not give reliable estimates as the spatial distribution of rainfall is very badly undersampled. This means that any method will have large uncertainty, and this needs to be acknowledged and taken into consideration when using the data to model streamflow.

Article Evaluation of approaches for estimation of rainfall and the ...

As Barry said, over longer time scales (seasonally, annual) either method will give reasonable answers, but I am not aware of any good method for interpolating a low density of rain gauges across a landscape on an event basis.

I'm wondering if it would be possible to used satellite derived precipitation products (after first ground truthing their model output with point field data) and see how that would compare with IDW and Thiessan. Just a thought...

I'm running 20+ year simulations at an hourly time step. From what I can gather, common practice is to use a local gauge with the most complete period of record. I've been working to incorporate all the data, given that there is so much spatial variability in precipitation and there is relatively little information for a given watershed. So I find the single gauge approach wholly unsatisfying.

I fully appreciate the influence of elevation on precipitation, but at least in my case this isn't an issue (Midwestern United States). I'm looking at ~10 hourly precipitation gauges in areas on the order of 2000 km2. It would be interesting to explore other data sets but I would look to retain the hourly time step in hopes of capturing higher-flow events. AFAIK there are not really any options other than the rain gauges.

Dear David, I am not sure, I understand you well. It seems to me that the problem you raise - your observation of radically changing precipitation rates over extremely short distances, and especially in short time steps (!) - is equal for both, Thiessen AND IWD. (Both interpolate between two stations, assuming them as "true" as you write.) I deal with 10 stations in a 100km2 catchment over ten years and in event mode (I calculate 5-min steps). In my particular area I note a very steady rainfall gradient with elevation (200-900m asl) and distance to the Sea. I can only incompletely account for luv and lee effects of hills. In my case Th iessen polygons don't account for the gradient. In addition, one must take into account the time it takes for a cloud to pass from the first to the last station when comparing values. However, I also find huge discrepancies on one and the same roof for short intervals (equilibrates over time).

I guess, will have to come to terms with the fact that reality is always infinite and therefore focus on the parameters and factors that are important for our specific purpose of study...

This is a very interesting problem that I am working on the edge of. It is easiest to say that spatial variability in measured precipitation is extremely spatially variable. For example in upstate NY there is only small correlation (R^2 of .3) between daily gage measured precip stations 30km apart which is extremely good compared to NM or S. CA where there is almost none. For larger basins in the NE (1000km2+) various interpolation methods all work pretty well at representing the ppt in the basin, but this relationship diminishes fast as you get to 40km2 and below with gages outside the basin (10km away). In fact, it appears that 10km is away, for my studies, an interesting distance the NE in which using a short term weather forecast can better represent the precipitation over a basin of 40km^2. If anyone is interested in playing with these ideas more, I would love to participate in any related study. Here is my latest article on the topic: http://onlinelibrary.wiley.com/doi/10.1002/hyp.10073/abstract

Dear David,

Rainfall spatial variability is a topic of research that several people have worked (and still work) on. Thiessen is a very basic method that may or may not be adequate depending on the gauge density, rainfall regime and type of applications (as already mentioned by others). You can certainly do better (how much better again depends) by looking at geostatistical approaches like ordinary Kriging or Kriging with external drift. You can find many publications devoted on the comparison of different methods (in terms of rainfall estimation) but also on their effect on runoff simulations.

Allow me to suggest some references that may provide some info on the topic:

- Work by Krajewski et al. 2003 (an analysis of small scale rainfall variability...) and similar work by the same authors offer info on the quantification of small scale rainfall variability and the issues associated with raingauge networks and their capacity to capture this variability.

- For an assesment of interpolation methods or methods that can account for the elevation gradients you can have a look (among others) at:

a) Bardossy and Pegram (2013): Interpolation of precipitation under topographic influence at different time scales (and references therein)

b)Ly et al (2011): Geostatistical interpolation of daily rainfall at catchment scale... (and references therein)

-Finally some more papers on the effect of rainfall spatial variability on runoff response you can find in the paper (and references therein) of

Zoccatelli et al (2010): Which rainfall spatial information ....

My apologies for the long response. Hopefully this may be of help.

Efthymios thanks much for your thoughtful answer. I think you are getting at exactly my question. I would expect Kriging is the "best" answer--but would need to spend some more time looking at building it into the larger context of my models.

A caution concerning kriging - it may be potentially the best method, but how well it performs depends on the quality of the variogram produced. Generally, kriging goes through all data points exactly - not because the interpolation is working well, but because that is how kriging is designed. Unfortunately, many applications of kriging use all the data to produce the surface, which means you have no way of evaluating the performance, nor the accuracy of the variogram.

The main concern is the undersampling of the rainfall. Any interpolation method will have a high degree of uncertainty due to the undersampling, and as a result I would recommend applying several different methods to get an idea of the contribution of the selection of the method to the overall uncertainty. This includes the determination of the variogram.

Hi David

I made some interpolation of rain data while working on pluvial flood forecasting in Rome (Italy). I used both deterministic and probabilistic methods. I found that using a probabilistic method (like kriging) is adeguate in case of high density of rain gauge stations, which I considered to be around 1 station for 50 squared km. In the case of lower station density, I retained that probabilistic methods were not suitable and I used Radial Basis Function or Local Polynomial. Moreover, I know that someone uses cokriging to relate rain to other variables, as the ground elevation, to fill areas with no rain information; unfortunately I'm not practical about this technique

David,

First, thanks for getting a conversation started discussing this topic. I'm inherently interested in this topic and appreciate all the replies.

I've been doing some digging into NEXRAD II data and it seems like a great way of adding some form of measured spatial distribution.

The steps to overcome in using the data include:

1) A NEXRAD station needs to be located in close proximity.

2) The need for grounded gauges

3) Mathematically adjusting reflective data (as provided by the Radar) to produce rainfall at the surface.

4) Potential affects of wind and other problems associated with translating from aerial to ground precipitation.

If anyone has comments to speak to NEXRAD (or some good literature sources) I think it would greatly help the discussion.

Hi everybody,

I understand that you are assuming that rainfall is in liquid state. Have anybody of you working with interpolation of hail data?

Thank you in advance

Hi David

The answer to your question very much depends on the region and scale your interested in. Here an interesting link to a work, which demonstrates that precipitation even on small scale can be highly variable due to topographic effects:

http://e-collection.library.ethz.ch/view/eth:7165?q=%28author:Sch%C3%A4ppi%29

Hi David,

Take a look in this publication:

"SPATIAL INTERPOLATION OF CLIMATOLOGICAL INFORMATION: Comparison of methods for the development of precipitation distribution in Distrito Federal, Brazil."

It is also available in my contributions.

There are good references to follow. Moreover, there are some research groups engaged in this topic. For instance, there is a Conference in May.

http://www.met.hu/en/omsz/rendezvenyek/homogenization_and_interpolation/introduction/

Dear David,

just some additional remarks to all the informative answers you already got.

- how many gauges needed for a "true" spatial representation of rainfall is depending on spatial and temporal scale. While there are a lot of studies dealing with spatial variability on the meso-scale, relatively little is know regarding the spatial variability of rainfall on the micro-scale (

Dear colleagues

We have analyzed your problema using the Corelation Distance Decay function in spanish conterminous land; there are many papers about it I enclose the last we have produced and another that has recently published by other colleagues.

Sincerely

Carlos

Sorry, the papers

And the second one

We may also consider the possible influence of spatial variablity of near-surface wind speed on capturing rate of the rain gauge. It has been found that the significant temporal decline of the near-surface wind speed has increased the capturing rate of the winter-time snowfall in NE China over the past 5 decades.

Hello,

Interpolation is weather and hydrological data is a major concern for many hydrologists and models.

please read the paper below.

http://popups.ulg.ac.be/1780-4507/index.php?id=10003#tocto1n2

Hello David,

Here are my few additional remarks: The variability of the rain fall is a function of the spatial domain (scale) we consider. Also, the characteristics of the rain fall depends on the type of precipitation (convective/stratiform). Given the domain size, type of precipitation, and number of point observaitons, its variability can be estimated from the rain fall characteristics. You may refer to the basic work on rain fall characteristics of Prof. Isztar zawadzki (http://www.mcgill.ca/meteo/facultystaff/zawadzki/) in his earlier publications. Hope it helps...

Good luck !

Arunchandra

I have found that Precipitation data with altitude or mean weighted altitude (mean catchment altitude) of the station gives a fair correlation, integrating it in to GIS software (Raster calculator) resolves your question. Of course it is very important to keep in mind that the study areas size and the number of observation points will significantly influence your results!

spatial variability in measured precipitation is firstly depends on rainfall climatology of the area, and this variability is also quite dependent on time duration of the measured precip. the shorter the time then the higher the variability, vice versa. if precip is accumulated over time, than spatial var will reduce.

as of the climatology, convective rains will have higher spatial variability compared to frontal type rainfalls. so, spatial var would very much depends on locations.

and for estimation of areal rainfall, simple objective methods are arithmetic mean or Thiessen polygon. more advanced methods includes interpolation techniques such as inverse distance method or kriging with measures of co-variances.

The question:

do I assume the distribution of observed precipitation values represent spatial variability, or independent estimates of the average value within the box?

answer:

here is mixing two areas, the math and physics, in math is used the interpolation to link the precipitation gauged data from local stations (1D) to precipitation field (2D). Depending several environment conditions is selected the interpolation method (i.e. not all the interpolation methods can be useful in a same study area) and the amount of station data.

In a physical matter, the 1D to 2D link is a stochastic matter, then is possible to use math tools for represent the precipitation field distribution from local stations data, but only the distributional reproduction (statistical and probabilistic distribution) can be gotten. If you want reply the real past event (or past sample space) you need a little information over the field. An example is the Quiroz et al (2011), where is used the NDVI field (at ground level measure) to reconstruct precipitation data, the same can be used for a spatial case. The spatial case will be submitted the next month. If you want a future scenario, then a stocastic precipitation field can be gotten from the past field reconstructed.

You can take a look to the recently published paper by Omranian and Sharif:

Evaluation of the Global Precipitation Measurement (GPM) Satellite Rainfall Products over the Lower Colorado River Basin, Texas

Areal Average Rainfall methods such as Thiessen, Arithmetic Mean, IDW, etc. are all geometrical methods that have some assumptions about the observed rainfall (Point Gauge) and it's spread around the rain gauge whereas none of these methods take in consideration, the mobile nature of rainfall within the catchment hence unknowingly introducing an error. For Example: The distribution area of rainfall would remain same for areal averaging method such as thiessen in a catchment under study (until and unless the number of rain gauges are changed) even though spatial variability of rainfall might differ with mobility of rainfall within the catchment.

In other words the answer to your question would be both yes and no. My point of concern would be "How much influence will that assumption have on your results?".

A better way to incorporate the spatial variability of rainfall is by calibrating rainfall station weights using observed rainfall and streamflow data. In this way no areal averaging method is used but a weight is assigned to each rainfall station, the product with its observed value would represent the spread of each station and sum of the spread from all stations together would represent the areal rainfall of the catchment.

Here are a few papers that I would recommend you to read:

1. Berndtsson, R., & Niemczynowicz, J. (1988). Spatial and Temporal Scales in Rainfall Analysis - Some Aspects and Future Perspectives. https://s3.amazonaws.com/academia.edu.documents/42070786/Spatial_and_temporal_scales_in_rainfall_20160204-16542-k27i9i.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1540545854&Signature=HKtSBgVyGZJuVV7qwo5nwUSekOc%3D&response-content-disposition=inline%3B%20filename%3DSpatial_and_temporal_scales_in_rainfall.pdf

2. Patrick, N., & Stephenson, D. (1990). Spatial variation of rainfall intensities for short duration storms.

https://www.tandfonline.com/doi/abs/10.1080/02626669009492471

3. Wijesekera, N.T.S, & Musiake, K. (1990). Streamflow Modelling of a Sri Lankan Catchment Considering Spatial Variation of Rainfall.

http://geoinfo.mrt.ac.lk/waterresources/publications/C002.pdf

http://geoinfo.mrt.ac.lk/waterresources/publications/C001.pdf

4. Sugawara, M. (1992). Chapter 5 – On the weights of precipitation stations. In J. O'Kane (Ed.), Advances in Theoritical Hydrology.

5. Chen, T., Ren, L., Yuan, F., Yang, X., Jiang, S., Tang, T., . . . Zhang, L. (2017). Comparison of Spatial Interpolation Schemes for Rainfall Data and Application in Hydrological Modeling.

https://www.mdpi.com/2073-4441/9/5/342/htm

6. Cristiano, E., Veldhuis, M.-c., & Giesen, N. (2017). Spatial and temporal variability of rainfall and their effects on hydrological response in urban areas – a review.

https://www.hydrol-earth-syst-sci.net/21/3859/2017/