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Lightning is one of the most spectacular phenomena in nature. It is produced when there is a breakdown in the resistance in the electric field between the ground and an electrically charged cloud. By simple observation, we observe that precipitation, especially the most intense, is often accompanied by lightning. Given this observation, lightning has been employed to estimate convective precipitation since 1969. In early studies, mathematical models were deduced to quantify this relationship and used to estimate precipitation. Currently, the use of several techniques to estimate precipitation is gaining momentum, and lightning is one of the novel techniques to complement the traditional techniques for Quantitative Precipitation Estimation. In this paper, the authors provide a survey of the mathematical methods employed to estimate precipitation through the use of cloud-to-ground lightning. We also offer a perspective on the future research to this end.

The estimation of precipitation is important because of its impact on several aspects of human life [

Rain gauges and radars have traditionally been used to monitor and estimate precipitation [

Satellites data also has been proposed as an alternative to complement ground-based MPE. It has almost global coverage [

Considering that the problem of sensor coverage is recurrent, even with multisensor QPE products (either ground or space-based), other alternative techniques to address this problem are required.

Figure

Convective storm in Midland, Texas, lightning-QPE correlation example, from [

When studying precipitation, it is appropriate to focus on the basic types, convective and stratiform. Convective precipitation occurs predominantly in the form of localized rain showers and thunderstorms and may have greater intensity. Stratiform precipitation tends to be of larger-scale and long-lasting, with lesser intensity. Disaggregated precipitation data can be used for parametrizing climate models which simulate convective and stratiform (large-scale) precipitation separately [

There is scientific evidence that the occurrence of lightning is linked with convective precipitation. This can be physically explained by the fact that cloud electrification processes need the presence of supercooled water, ice particles, and larger heavier graupel to coexist in a region experiencing moderately high updraft velocities [

On the other hand, lightning data have also been used in models as a proxy for deep convection to improve parameters related to cold clouds and precipitation [

In view of the scientific evidence, lightning based methods have been proposed as a complement or alternative for QPE [

In practice, many regions of the world have ground lightning detection networks [

This paper attempts to contribute to the discussion of the methods to estimate QPE by using lightning. We present a brief survey of the mathematical models used over the years and provide an overview of future research geared towards development of a system that may provide a basis to incorporate lightning-derived precipitation into the current multisensor precipitation products.

Let

In much of the previous work investigating the LPR, the mathematical model was a simple linear least squares regression [

The feature vector

In [

Based on the power law relationship that is employed for the relationship between liquid water reflectivity and precipitation rate used to estimate precipitation by radars [

Overall, previous results demonstrate that LPR is reliable when one compares relatively large regions and/or longer time periods. However, if this relationship is tested at higher resolutions (such as those of the new precipitation products), lightning events and convective precipitation may not be colocated. On the other hand, some results report a time lag between lightning and precipitation [

Let

Cumulative total domain Mean Squared Error: simple linear model, STI model, and Kalman filter, from [

The STI model is a set of linear parameters fixed in time. Physically LPR changes from one storm to another or even within the same convective event. Therefore, a fixed time model may work better when the thunderstorms are close to the average but fails for non “typical storms.” This suggests the need to develop a method to model the LPR changes in time. Let

The

Equations (

In order to apply the Kalman filter for estimation, some conditions must be assumed, such as a zero mean Gaussian distribution of the parameters variations. Also, fixed and known covariance matrices

Figure

A random field

The construction of optimal predictors on a single and partial realization of a random field is based on some form of stationarity. A random field is called second-order stationary if

For processes for which the above conditions do not hold (i.e., covariance function does not exist) another hypothesis is introduced. A random field is called

Let

The semivariogram is a measure of dissimilarity between a pair of observations. As a function, it provides information on spatial continuity and variability of a random field. The inference on the shape is based on empirical semivariogram

The spherical model with range

The cubic model with range

The dampened hole effect model:

The most common algorithm of geostatistical estimation is the so-called

Ordinary kriging is a minimal variance estimator, given by

The ordinary kriging approach is the simplest practical model for geostatistical estimation. There are several modifications of ordinary kriging; some deal with a nonconstant mean like

The ordinary kriging is a punctual estimator, while the QPE is a spatial estimation. Therefore, it is necessary to modify the method. Let

It can be shown that this is equivalent to using ordinary kriging in each grid point, thus averaging the estimated values. However, there are several important differences. First, since the distribution of the unknown values is itself unknown, it is not possible to predict the appropriate choice of the grid to obtain a given error tolerance when averaging the kriged estimates. On the contrary, it is possible to estimate an appropriate grid when numerically integrating a known function. Secondly, this approach results in many estimation errors (one for each kriged estimate) and many kriging standard deviations. In contrast, it blocks kriging results in a single kriging standard deviation for a given area and controllable numerical integration errors.

Kriging with external drift is a method to merge two sources of spatial information: a primary variable that is precise but only known at few locations and a secondary variable that is available in the spatial domain [

For QPE several authors have used block kriging with external drift with elevation as the secondary variable. In the reviewed literature, there is a dearth in the research on the use of lightning data to carry out kriging with external drift. Nevertheless, there is physical and modeling evidence that justifies the assumption that the expected value of the QPE is dependent on the lightning values.

Quantitative precipitation estimates often have significant uncertainty. Stochastic precipitation models provide an alternative framework for Quantitative Precipitation Estimation [

In [

Based on this idea, one approach to perform probabilistic QPE based on lightning data is to compute an empirical climatological cumulative distribution function of precipitation in grids with observed precipitation and using a lightning based kriging with external drift to estimate a locally weighted regression for each grid. This model will be used to estimate a conditional cumulative distribution function of precipitation at each grid.

Another approach to probabilistic QPE is to model the conditional probability distribution function of each grid

The benefit of using a dynamic model to track LPR is evident in Figure

Precipitation map for a Mesoscale Convective System in southern Arizona. Dots are lightning strikes; grids with crosses do not have precipitation sensor coverage, from [

In the clustering problem, we are given a training set (the lightning locations at one interval time in this case) and want to group the data into a few

Since storms change with time, it is important to capture the main transitions. In recent years, there has been an increasing interest in tracking scenarios in which a very large number of coordinated objects evolve and interact. It should be noted that clusters can be thought of as extended objects that produce a large number of observations. In recent work [

Assume that at time

In this paper, several approaches for estimating QPE from lightning measurements were reviewed. We also reviewed the existing techniques for storm tracking, since these methods can be used in conjunction with linear models, allowing a better parametrization of the models for convective events.

Linear models assume implicitly that the data used to parametrize the model is independent, normal, and homogeneous in variance. The most simple models are suitable for a first approach or when the data are expensive or limited. On the other hand, the STI models provide a powerful tool to easily express a heuristic knowledge about the space-time relation of lightning with QPE. Dynamical STI models allow adjustment of the STI model in response to changes in LPR from one storm to another (or even in the same convective event). However, for these models to be effective it is necessary to carry out an adjustment phase of the tracking parameters, which is critical for the quality of the estimation.

Geostatistical methods generate smooth interpolated surfaces, where the estimation errors depend strongly on the assumed probability distribution, derived from the variogram model. Kriging methods are suitable when there is sufficient data to establish (and statistically verify) a variogram function. On the other hand, discriminant models (such as conditional random fields) require less assumptions about the distribution of the data and the structure of the model, so it is possible to reduce estimation errors. However, discriminative models do not offer clear representations of relations between lightning and convective precipitation. These models are suitable in large regions, with a large amount of historical data.

As mentioned at the end of Section

This infrastructure represents a great opportunity to investigate the relationship between convective precipitation and the occurrence of lightning at a global level. It Investigates the differences that may exist in LPR depending on the geographic location as well as the nature of the different convective events. The development of new algorithms and mathematical models for the estimation of convective precipitation as well as those that emerge from other investigations will be of great importance to develop systems of prediction of severe storms, to study the physical relationship between LPR for convective events of different nature, or simply to complement existing methods and techniques for estimating precipitation.

The authors declare that there are no conflicts of interest regarding the publication of this paper.