# Adjustment of state space models in view of area rainfall estimation

Abstract: This paper uses state space models and the Kalman filter to merge weather radar and rain gauge measurements in order to improve area rainfall estimates. Particular attention is given to the estimation of state space model parameters because precipitation data clearly deviates from the normal distribution, and the commonly used maximum likelihood method is difficult to apply and does not perform well. This work is based on 17 storms occurring between September 1998 and November 2000 in an area including part of the Alenquer river hydrographical basin. Based on these data, the work aims to investigate the importance of the parameters estimation method to the accuracy of mean area precipitation estimates. It was possible to conclude that the distribution-free estimation methods produce, in general, better mean area rainfall estimates than the maximum likelihood. Copyright © 2010 John Wiley & Sons, Ltd.

## Summary (3 min read)

### 1. INTRODUCTION

- Many problems in meteorology and hydrology need an accurate measurement of the total rainfall amount in a certain area.
- Alternatively, weather radar may outline accurate rainfall isopleths , but their point estimates are not so good due to errors of either a meteorological or instrumental nature.
- Krajewski (1987) and Severino and Alpuim (2005) apply an optimal interpolation method based on Kriging and CoKriging.
- There are several ways of designing the state space model that relates the gauges observations, the radar observations and their bias.

### M. COSTA AND T. ALPUIM 2

- This is called the measurement equation whereas the equation governing the time variation of the calibration factor is called the transition equation.
- Equation (1) describes the relationship between a rain gauge and a radar cell in the same site, but it can be used to produce a mean-field bias, replacing Gt and Rt by the vectors comprising all the gauges in a certain area and the corresponding radar cells.
- The estimation problem has been addressed since the first works on radar calibration appeared.
- Ahnert et al. (1986) propose intuitively based estimation methods, but nothing is proved about their statistical properties.
- In many applications, the state space models’ parameters are estimated by maximum gaussian likelihood via the Newton-Raphson method (Harvey, 1996) or, more often, the EM algorithm (Shumway and Stoffer, 1982).

### M. COSTA AND T. ALPUIM 4

- There are five rain gauges located in the area being studied, namely, Abrigada (A), Olhalvo (O), Penedos (P), Meca (M) and Merceana (Mr), which correspond to a reasonably high density of gauge/20km2 .
- Brown et al. (2001) consider a circular region with an approximate density of one gauge per 500 km2.
- Furthermore, because the original radar data corresponded to 10-minute periods, the authors used the average of the six available measurements in each hour as the hourly data in each cell of 2kmx2km.
- The rain gauges in Abrigada and Olhalvo are used to calibrate the radar whereas the remaining three gauges are used to compute the mean area precipitation, usually known as the “ground truth”, in order to assess the quality of the adjusted models.
- The gauges used for calibration are not used for assessment, since the two processes should be independent.

### 3. THE KALMAN FILTER ALGORITHM

- The Kalman filter technique, Kalman (1960), has been used in many different scientific areas to describe the evolution of dynamic systems.
- The first one considers that in each site where a rain gauge is available the relationship between the gauge, the corresponding radar cell measurement and the bias is described by the set of two equations Later, a field of calibration factors is calculated for each radar cell, with the help of interpolation methods.
- It has fewer parameters to estimate and may also produce good results when applied to small areas or to storms where the precipitation pattern is spatially homogeneous.
- After estimation of the parameters, the Kalman filter may be applied to produce an estimator of the state variable bt, at each time t.

### 4. ESTIMATION OF THE PARAMETERS

- There are several methods that may be used to estimate the state space models parameters.
- Meteorological and hydrological data are, generally, not normally distributed.
- This fact, together with the complexity of the ML method for the models in use, shows the need to look for alternative estimation procedures.
- The authors present two different distribution-free methods which are based on the generalized method of moments.

### M. COSTA AND T. ALPUIM 6

- Thus, the log-likelihood is given by the sum of the loglikelihoods for each storm.
- The use of numerical methods to maximize this likelihood function may be a difficult and complex task, very often without satisfactory results.
- This problem may occur either because the numerical iterative techniques do not converge or because of the existence of multiple critical points.

### 4.2 Distribution-free estimation for the parameters – method M1

- Costa and Alpuim (2010) propose consistent distribution-free estimators for the parameters of the univariate model (2)-(3) based on the generalized method of moments.
- These estimators have an explicit analytical expression and do not assume any specific distribution for the errors.
- Considering the multi-factor model as described in equations (2)-(3), the method estimates first the mean of the calibration factor, ][ tbE=µ , and then the autoregressive coefficient φ.
- Finally, note that the estimators for the bias mean and autocovariance are averages calculated over all pairs of observations for which the radar is not null.
- The maximum likelihood method may not converge at all and produces estimates falling outside the parameters space more frequently than this method, especially for small values of the autoregressive coefficient.

### M. COSTA AND T. ALPUIM 8

- Alpuim (1999) suggested noise variance estimators for univariate and multivariate state space models, designated in the text by D(k).
- The authors need, however, to estimate the autoregressive parameter of the state equation.
- Thus, in order to apply this method, the authors combine the D(k) estimators of noise variances with the mean and autoregressive parameters estimators as in method M1.
- Usually the indices are taken as k =1 and 2=l , but if these values produce estimates outside the parameters space, other values should be used.
- They tend, however, to produce estimates outside the parameters space more often than method M1.

### 5. ASSESSMENT OF THE RADAR-RAIN GAUGE ADJUSTMENT

- This section describes how to assess the several models’ performance according to two criteria: point and mean area precipitation.
- First, radar and gauge measurements have different natures: while radar represents an area mean value (radar cell), gauges measure precipitation in a point (gauge location).
- This is considered to give the best quality pattern because, due to its nature, the true value of mean area precipitation cannot be directly measured.
- The “ground-truth” is evaluated using interpolation methods in a reasonable thin grid of rain gauges.
- In this work the authors used also the Thiessen polygons and the inverse square distance methods.

### 6. RESULTS AND CONCLUSIONS

- Rain gauges located at Abrigada and Olhalvo were used for the estimation and calibration processes while the remaining gauges were kept for the validation process.
- Also, from inspection of figure 3 the authors may conclude that the ratios G/R at Olhalvo have a large variability, larger than at Abrigada, and this characteristic is reflected in the distribution-free estimates: method M1 produces estimates, both for the mean and the error variance, larger at Olhalvo than at Abrigada.
- The analysis of the isopleths of accumulated adjusted radar estimates provides a good appreciation of the final effect of the calibration process.
- --- FIGURE 6 --- Figure 6: Accumulated precipitation, in mm, for the storm of November 2 of 2000 for the radar non calibrated estimates and for the calibrated estimates considering the multi-factor model and the three methods of estimation, M1, M2 and ML.
- For the multi-factor model, the two different interpolation methods were compared.

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### Additional excerpts

...따라서 관측 대상지역에 대한 제약이 적다(Michelson and Koistinen, 2000; Brown et al., 2001; Costa and Alpuim, 2010)....

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...to compare this conclusion with the results in Anagnostou et al. (1998), where adaptive recursive methods involving error correlations compare favorably with maximum likelihood, although applied to different measurement and transition equations....

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...For a more detailed description see Hamilton (1994) or Harvey (1996)....

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