Scale-recursive estimation for merging precipitation data from radar

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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, D08104, doi:10.1029/2008JD010709, 2009

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Scale-recursive estimation for merging precipitation data from radar and microwave cross-track scanners H. Van de Vyver1 and E. Roulin1 Received 3 July 2008; revised 7 November 2008; accepted 10 February 2009; published 17 April 2009.

[1] This paper presents an application of scale-recursive estimation (SRE) used to

assimilate rainfall rates within a storm, estimated from the data of two remote sensing devices. These are a ground-based weather radar and a spaceborne microwave cross-track scanner. The rain rate products corresponding to the latter were provided by the EUMETSAT Satellite Application Facility on Support to Operational Hydrology and Water Management. In our approach, we operate directly on the data so that it is not necessary to consider a predefined multiscale model structure. We introduce a simple and computationally efficient procedure to model the variability of the rain rate process in scales. The measurement noise of the radar is estimated by comparing a large number of data sets with rain gauge data. The noise in the microwave measurements is roughly estimated by using upscaled radar data as a reference. Special emphasis is placed on the specification of the multiscale structure of precipitation under sparse or noisy data. The new methodology is compared with the latest SRE method for data fusion of multisensor precipitation estimates. Applications to the Belgian region show the relevance of the new methodology. Citation: Van de Vyver, H., and E. Roulin (2009), Scale-recursive estimation for merging precipitation data from radar and microwave cross-track scanners, J. Geophys. Res., 114, D08104, doi:10.1029/2008JD010709.

1. Introduction [2] Rainfall is a highly variable process, and a difficult one to measure. A variety of devices exist for measuring rainfall which cover a range of scales and they tend to trade off accuracy and spatial coverage. Rain gauges are the only source of direct rainfall measurements but unfortunately, rain gauge networks are typically sparse. The primary source of land-based measurement is the radar because it produces reliable measurements over extended areas. Infrared measurements, provided by geostationary satellites, deliver a broad and continuous coverage but they are considerably less accurate. Microwave sensors on low-Earth orbit fall somewhere between these extremes. [3] Among the methods currently used to measure rainfall, no single technique can provide an accurate measurement of global precipitation. Therefore rainfall estimates should be improved by synthesizing data from different sources. This paper presents a new methodology which is based on scale-recursive estimation (SRE) for the stochastic assimilation of rainfall rates at different scales. The SRE technique, whose essence is derived from Kalman filtering, is able to optimally merge observations of a process at different scales while explicitly accounting for their uncertainties. SRE was first introduced by Chou et al. [1994] in signal processing, which can produce the best estimate of the field at any desired scale. Since then, many researchers 1

Royal Meteorological Institute of Belgium, Brussels, Belgium.

Copyright 2009 by the American Geophysical Union. 0148-0227/09/2008JD010709$09.00

have explored the applicability of SRE to other subjects such as soil moisture [Kumar, 1999], solute travel time distributions [Daniel et al., 2000], imaging and remote sensing problems [Fieguth, 1995], assimilation of remote sensing data [Daniel and Willsky, 1997], and estimation of satellite altimetry [Fieguth et al., 1995]. The application of SRE to precipitation data was first performed by Primus et al. [2001], and further investigated by Tustison et al. [2002], Gupta et al. [2006] and Bocchiola [2007]. Besides the ability to compute optimal rain rate estimates, SRE is of particular interest in quantitative precipitation forecasts (QPF) [see, e.g., Bocchiola and Rosso, 2006; F. Kong et al., Application of scale-recursive estimation to ensemble forecasts: A comparison of coarse and fine resolution simulations of a deep convective storm, paper presented at 20th Conference on Weather Analysis and Forecasting/16th Conference on Numerical Weather Prediction, American Meteorological Society, Seattle, Washington, 2004]. [4] In most of the papers on rainfall one uses a multiplicative cascade model because it naturally fits into the SRE methodology. To transform the multiplicative cascade model into an additive form (i.e., the space-state equation) one needs to consider the logarithm of the rain rates. Consequently, this introduces the difficulty in handling zero rain rates, a problem which has recently been treated by Gupta et al. [2006] and Bocchiola [2007]. In short, Gupta et al. [2006] propose to operate directly on the data, so that it is not necessary to consider a predefined multiscale model. Bocchiola [2007] introduced a SRE methodology which excludes the zero rainfall. In this paper we combine the great advantages of both approaches and we add a new

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[7] The representation of a multiscale process on the inverted tree is achieved via a governing state-space equation that specifies how the multiscale stochastic process evolves from coarse scales (sg) to fine (s) scales. The scalar version is of the form xðsÞ ¼ AðsÞxðsgÞ þ BðsÞwðsÞ;

Figure 1. The node s is shown in relationship to its parent, sg, and its children, sa1, sa2, sa3, and sa4. methodology to model the variability of the rain rate process in scales. We apply our suggestion to real multisensor data. Data from two measurement sources, microwave satellite measurements and ground-based radar measurements, were assimilated to produce a multiscale estimated field. The microwave measurements are based on observations made by the advanced microwave sounding unit (AMSU) instrument on board NOAA satellites. [5] This paper is organized as follows. In section 2, a brief overview of SRE is presented while the technical details are included in the Appendix. In section 3 we locate the problem of modeling the zero intermittency of rainfall and summarize some existing approaches to treat this difficulty. Next, we describe our contribution which consists of a combination of both approaches together with a new parameter estimation procedure. Section 4 provides a detailed description of the data used and how they are transformed (e.g., convolution, normalization, . . .) in such a way that they fit into our SRE methodology. In section 5 some applications to intense observations in the Belgian region show the relevance of the new methodology. Special emphasis is placed on the specification of the multiscale structure of precipitation under sparse or noisy data. Finally, in section 6, some conclusions are drawn.

2. Scale-Recursive Estimation [6] The algorithm described in this section has already been applied in the above cited papers but we repeat the main points to provide insight into the problem. A multiscale process can be represented on a tree as shown in Figure 1. All the grids cover the same area, but each one corresponds to a different resolution. Every node can be related to nodes at finer and coarser scales. In particular, a node s has a ‘‘parent,’’ sg, and ‘‘children,’’ sai, i = 1,. . ., n. The nodes at the finest scale are often called ‘‘leaf nodes’’ and the scale which contains only one pixel is referred to as the ‘‘root node.’’

ð1Þ

where x(s) is the zero state mean of the system, A(s) and B(s) are parameters that control the scale-to-scale variability of the process, and w(s)  N(0, 1) is a noise component which is independent of the state. The term B(s) w(s) is often called ‘‘process noise.’’ In general, the state is a multidimensional vector composed of several variables. However, in this work we are concerned with estimating rainfall, which is a scalar quantity so that theory will be treated in the scalar case. [8] In order to determine the uncertainty of the estimates, we consider the variance P(s) = E[x2(s)]. One easily shows that P(s) evolves according to a Lyapunov equation on the tree PðsÞ ¼ A2 ðsÞPðsgÞ þ B2 ðsÞ:

ð2Þ

Likewise, the reverse process, i.e., from fine scale (s) to coarse scale (sg), is covered by xðsgÞ ¼ FðsÞxðsÞ þ w*ðsÞ;

ð3Þ

where w*(s)  N(0, Q(s)) and F(s) takes the form FðsÞ ¼

PðsgÞ AðsÞ: PðsÞ

ð4Þ

The upward process variance Q(s) is given by QðsÞ ¼ PðsgÞ  F 2 ðsÞPðsÞ:

ð5Þ

We wish to incorporate the measurements z(s) at a certain scale into the estimation algorithm. The measurement model equation is of the form zðsÞ ¼ CðsÞxðsÞ þ vðsÞ;

ð6Þ

where the state x(s) is observed via a noisy measurement z(s) and where the measurement noise v(s) has distribution N(0, R(s)). [9] Note that including a lognormal model in SRE automatically implies the scale- and node-invariant parameter A(s) = 1. In the context of nonparametric models, Gupta et al. [2006] also assumed that A(s) = 1. However, we prefer to use the approach from Frakt and Willsky [2001], in which it was shown that A(s) depends only on the state covariances and parent-child cross covariances:

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AðsÞ ¼

Pðs; sgÞ : PðsgÞ

ð7Þ

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Analogously to the above cited papers, C(s) is assumed to be equal to 1 since the measurement and the state represent the same quantity, i.e., precipitation. This may not always be the case and, in general, C(s) can be a complex, often nonlinear, relationship between the measured quantity and the state of the system. [10] The state-space equation (1), variance propagation equation (2), and measurement model equation (6) form the basic framework of scale-recursive estimation (SRE). The multiscale estimates are computed from an upward sweep in which information is passed from one scale to the next coarsest scale, and a downward sweep which proceeds from coarse to fine scales. The idea of combining the upward and downward sweep finds its origins in the Rauch-TungStriebel (RTS) algorithm and was extended to tree-like data structures by Chou et al. [1994]. Technical details of the SRE algorithm are given in Appendix A.

3. SRE for Precipitation Data 3.1. SRE and Intermittence [11] The process and measurement noise parameters can be determined when the rainfall process is modeled by a multiplicative cascade model. In short, one obtains a process value at a certain scale by multiplying the process value at the parent node with a random cascade weight. The lognormal distribution is well suited to incorporate a multiplicative cascade model for spatial rainfall into the SRE methodology. However, there are some shortcomings when using this approach. Because of intermittency, the spatial rainfall fields contain zero values so that it is not possible to work in the lognormal space. Another problem is that, by definition, multiplicative cascade models produce fields which are nonzero everywhere within the modeling domain. Motivated by these restrictions, Gupta et al. [2006] proposed a methodology that performs simultaneously system identification and SRE. To achieve normality, which is required for optimal Kalman filtering, they have suggested working in a power space (13). After the application of the SRE algorithm, the precipitation field is obtained with the inverse transformation. Furthermore, they have pointed out that satisfactory results can be obtained when the amount of zeros at the highest resolution is not higher than the order of 30%. It is well known that rainfall is generally modeled by a mixed distribution that consists of a continuous part and a discrete part at zero rain rate. Any attempt to model mixed distributions by a continuous distribution must fail since the two object are not compatible. When applying Gupta’s approach to rainfall fields with a higher percentage of zeros, one should identify the zeros that come from inside the storm versus the ones that are outside the storm. In practical application, it is a very cumbersome task to define such an area over which merging can be done. Another approach that deals with zero intermittency was suggested by Bocchiola [2007]. This is called the ‘‘tree pruning technique,’’ which means that the multiscale trees are restricted only to nonzero rainfall. In the case of zero rainfall, one guesses that a zero value is already a correct estimation. This is in agreement with the intuitive idea that it is easier to detect rainfall in a cell, rather than to quantify it. As such, in the upward sweep, the nonzero children are merged to produce an estimation for the parent node. In the downward sweep,

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when a parent node is zero, the children will be set zero. Otherwise, the wet children will be estimated, while the dry children are not taken into account. After having applied SRE and the inverse transformation, the estimated rain rates have to be weighted according to the fraction of dry descendants. [12] If a conflict occurs between images at different resolution (i.e., a dry node has at least one wet child), a twofold approach is available. If the coarse sensor is more reliable, the children nodes are ‘‘dried.’’ If the fine sensor can be trusted, the parent node is assumed to be unreliably measured and its value is set to unknown. Then the Kalman filtering step will not be considered on that node. [13] Tree pruning does not spoil the SRE procedure. In fact, SRE can be carried out on asymmetric and unusually shaped trees, with no limitations to its reliability [see Basseville et al., 1992; Fieguth et al., 1995; C. B. Atkins et al., Tree-based resolution synthesis, paper presented at PICS 1999: Image Processing, Image Quality, Image Capture, Systems Conference, Society for Imaging Science and Technology, Savannah, Georgia, 25– 28 April 1999]. Following Appendix A, it is evident that for regular trees (i.e., at a certain scale every node has the same branching number), all the nodes of the same scale have the same uncertainty. For a pruned tree, however, the estimate variances vary from node to node. [14] In our work we combine the tree pruning technique with the nonparametric model of Gupta et al. [2006]. In contrast to the work of Bocchiola [2007] where a lognormal distribution (thus a multiplicative cascade model) is used, we will work in the power space. Numerical experiments in section 4.3 suggest that with a power transformation it is possible to approximate the Gaussian distribution more closely compared to the log transformation. 3.2. Parameter Estimation [15] Before running the SRE algorithm, one must specify the process and measurement noise parameters. One should take into account that the quality of SRE is strongly dependent on the estimated parameters. It is straightforward to obtain the measurement noise parameters. However, for process noise parameters (i.e., B(s) in (1)) this is not a trivial task and they are almost never known a priori. Until now, the research on parameter estimation of multiscale processes is very limited, the few existing algorithms assume a restricted model structure [Digalakis and Chou, 1993; Fieguth and Willsky, 1996; Tustison et al., 2002] or use ad hoc fitting methods [Fieguth et al., 1995]. An expectation-maximization (EM) algorithm has been used in maximum-likelihood parameter estimation of nonhomogeneous trees by Kannan [1997]. An application of such an EM-SRE algorithm to rainfall was recently given by Gupta et al. [2006]. [16] In our work the variances of the average normalized radar measurements are used to model the variability of the rain rate process in scales. Numerical tests suggest that this methodology is competitive with the procedures of the above cited works. Radar measurements alone were used to estimate process noise parameters because radar estimates are considered to be more reliable than microwave measurements. We denote the radar-scale descendants of a node s as sdi, i = 1,. . ., nd. Here, nd indicates the number of

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Table 1. Estimated Rain Rate Cascade and Measurement Error Variances (Event 2) Variance Process noise

Measurement noise

Scale 0 1 2 3 4 5 6 3 6

(microwave) (radar) (microwave) (radar)

Grid Size (km km) 240 48 24 12 6 3 0.6 12 0.6



240 48 24 12 6 3 0.6 12 0.6

Estimated Variance P*1 (s) P*2 (s) P*3 (s) P4*(s) P*5 (s) P*6 (s)

= 0.1298 = 0.1633 = 0.1936 = 0.2116 = 0.2132 = 0.2015 0.3 0.07

7 scales (including the root node); the tree geometry is given in Table 1. In addition, the estimates P*l(s) (see section 3.2) at each scale of the model are also included. The present tree geometry was chosen in order to have as many scales as possible. Of course, several other geometries are possible. In section 5.3 we will investigate the dependency of SRE on the tree geometry. Figure 2. The quadrangle denotes the estimation region. The circle denotes the radar coverage. descendants which have measurements. We denote the normalized radar measurements by z(s). Then the average radar data xl(s), located at node s of scale l, are obtained as xl ðsÞ ¼

nd X

zðsdi Þ:

ð8Þ

i¼1

The prior (unconditioned) variance P*l (s) is then estimated as Pl*ðsÞ ¼ E½xl ðsÞxl ðsÞ:

Remark that at the leaf nodes, Pl*(s) is the unconditioned variance of the measurements. In previous papers, the process noise variances, i.e., B(s), were determined for the computation of the propagation factor F(s); see (4). However, in our approach we directly compute F(s) via FðsÞ ¼

Plþ1 * ðsgÞ AðsÞ; Pl*ðsÞ

so that there is no need to obtain the propagation factor via (2). In section 5.1 several numerical tests were performed with excellent results.

4. Data Preparation [17] Two intense rainfall events in the Belgian region were selected: (1) event 1 on 9 July 2007 at 1544 UTC and (2) event 2 on 29 October 2007 at 1326 UTC. Both events coincide with an overpass of a National Atmospheric and Oceanic Administration (NOAA) satellite. For the estimation region we selected a quadrangular area within (3.8 – 7.2 E, 48.8– 51.0 N) because there is a good radar coverage available; see Figure 2. In our SRE algorithm we have used

4.1. Description of the Measurement Devices [18] Radar observations were provided by the radar of Wideumont which is operated by the Royal Meteorological Institute of Belgium (RMI). The radar is a single-polarization C band weather radar from Gematronik. It performs a 5-elevation scan every 5 min with reflectivity measurements up to 240 km. Rain rates were reported in 1 mm/h at a resolution of 0.6 0.6 km2. In our case study we selected a quadrangle with size 240 km. The center coincides with the radar station. [19] Satellite microwave observations were available in the framework of EUMETSAT Satellite Application Facility on Support to Operational Hydrology and Water Management (H-SAF); see http://www.meteoam.it/modules.php? name=hsaf. These data are based on microwave cross-track scanners which are, in these particular test cases, AMSU-A and AMSU-B instruments. The processing chain of this product has been developed by the Institute of Atmospheric Sciences and Climate of the National Research Council of Italy (ISAC-CNR) [see Mugnai et al., 2006]. It comprises AMSU-A spatial resolution enhancement to the resolution of AMSU-B, then a geometric correction for the variable viewing angle, and finally the precipitation rate retrieval

Table 2. Effective Field of View of AMSU-B as a Function of the Scan Position n n

Fxn (km)

Fyn (km)

1 5 10 15 20 25 30 35 40 45

51.6 39.8 30.7 25.2 21.7 19.3 17.6 16.6 16.0 16.0

25.3 23.3 21.3 19.7 18.5 17.6 16.9 16.4 16.0 16.0

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half maximum (FWHM). The FOVs are denoted by Fxn and Fyn, with x the direction of flight, y the scan direction and n the scan position (n = 1,. . ., 90). Some values of the FOVs are listed in Table 2. The microwave measurements have to be averaged following the AMSU-B antenna pattern. This is described by a Gaussian surface ! ðx  xi Þ2 ðy  yi Þ2 ; gi ðx; yÞ ¼ exp   a b

ð9Þ

where (xi, yi) are the Belgian Lambert coordinates of the center of the FOV, and the parameters a and b can be determined from the FWHM: Fxn a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 lnð2Þ

Figure 3. Gaussian filter for AMSU-B scan position 1 (scan edge). using an approach based on an Artificial Neural Network (ANN). The ANN is trained using the results of a mesoscale numerical weather prediction model. At the present, this product is undergoing an extensive validation program in the H-SAF project by comparing the output with weather radar and rain gauge data from all over Europe. Preliminary validation results are provided by La´bo´ et al. [2008] and Van de Vyver and Roulin [2008]. 4.2. Convolution of the Microwave Data [20] To make the radar and microwave measurements fit into the tree structure we propose to reconfigure the microwave data with constant resolution 12 12 km2. As in the adopted validation methodology, we take the footprints of AMSU-B into account. The sizes of the AMSU footprints vary since they depend on the scan angle. The elliptic fields of view (FOV) correspond to the full width at

Fyn b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 2 lnð2Þ

ð10Þ

Figure 3 shows the Gaussian pattern for scan position n = 1 (scan edge). [21] According to the CNR guidelines, the convolution of the ground data with the Gaussian filter should be performed on an area where the response is not less than 1% of the maximum. Consequently, the relevant region of a footprint is an extension of a FOV to a larger ellipse with sizes d and e: a d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 lnð100Þ

b e ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 2 lnð100Þ

ð11Þ

Finally, for a given set of microwave rain rates ri at (xi, yi), the average microwave measurement r at a node with center (xc, yc), consists of the following weighted mean P

gi ðxc ; yc Þri i r¼ P ; gi ðxc ; yc Þ

for all i for which

i

ðxi  xc Þ2 ðyi  yc Þ2 þ

1: d2 e2

Figure 4. (left) Microwave measurements of event 2. (right) Tilted to a grid 12 12 km2 via (12). 5 of 14

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Table 3. Statistics of the Original (Within the Estimation Box) and Convoluted Microwave Observations (Event 2)a Original Convoluted

Max.

Mean

Variance

2.730 2.716

0.894 0.886

0.763 0.680

a

Statistics are in mm/h.

[22] An example of such a convolution (for event 2) is displayed in Figure 4. Figure 4 (left) shows a satellite image of the precipitation field where each rectangle presents an AMSU footprint. The convoluted data in a 12 12 km2 grid via (12) are shown in Figure 4 (right). In Table 3 we present some relevant statistical quantities such as the maximum precipitation, mean and variance. Because of the construction procedure, the statistics of both fields are very similar. 4.3. Normalization of the Data [23] To ensure that the process noise and measurement noise from the basic equations (1) and (6) are normally distributed, it is sufficient to require that the multiscale process is normally distributed. In practice, approximate Gaussianity is achieved by applying a transformation to the original data and then working in the transformed space. After the application of the SRE algorithm, the precipitation field is simply obtained with the inverse transformation. Henceforth, the result of the inverse transformation will be called the ‘‘real field.’’ For example, Primus et al. [2001] and Tustison et al. [2002] have considered the log space. Alternatively, Gupta et al. [2006] proposed to apply a power transformation to the data. It is convenient to rescale the rain rate r(s) at node s by X(s) = r(s)/r, where r is the average value of r(s) at a certain scale. We assume that the expected value r over the modeled region is the same at every scale. For every data set we have to search for a power p such that the result of the power transformation xðsÞ ¼ X p ðsÞ  E½X p ðsÞ;

ð13Þ

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approximates a normal distribution as closely as possible. The cumulative distributions of the normalized data via (13) of the rain rate measurements (radar and microwave) are shown in Figures 5 and 6. In addition, the cumulative Gaussian distribution is also plotted which indicates that the normalized data closely follow the body of the Gaussian distribution. Numerical experiments reveal that, in general, Gaussianity is better approximated with a power transformation compared to the case when the usual log transformation is used. This is clearly illustrated in Figure 7 in which we have displayed the lognormalized data of event 2. It can be seen that the normal distribution is poorly approximated. [24] Remark that the range of the space (13) is dependent on p. It may happen that p is different for radar and microwave measurements. In order to reduce this discrepancy we multiply the normalized microwave data such that the variance equals the variance of the upscaled radar measurements at microwave scale. Similar approaches to reduce discrepancies between both measurements are commented on by Negri et al. [1994] and Primus et al. [2001]. 4.4. Measurement Noise Parameters [25] A hypothesis is required for the measuring devices. A study of the variation of multiplicative lognormally distributed noise is discussed by Smith and Krajewski [1993]. They have considered a range of radar measured storms at different locations, and compared these with rain gauge data. Similar results are found by Kitchen and Blackall [1992]. More recently, Bocchiola [2007] has estimated the lognormal observational noise in the framework of the SRE methodology. [26] The radar used in our case study underestimates the 24 h accumulated rainfall with 15% [see Delobbe, 2007]. To estimate the measurement noise variances at finer time resolution, radar data are compared to gauge data from the Hydrological Service of the Ministry of Equipment and Transport of the Walloon Region (SETHY). Most of them are tipping bucket systems providing hourly rainfall accumulations. For the validation of the radar observations, the 5-min precipitation are summed to produce 1-h precipitation

Figure 5. Microwave and radar cumulative probability distributions for normalized rain rate measurements of event 1. 6 of 14

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Figure 6. Microwave and radar cumulative probability distributions for normalized rain rate measurements of event 2. accumulation products. For each gauge location, we make the average of these 1-h precipitation from the nine closest radar pixels. We have analyzed 244 images from January 2007 to May 2008, containing intense hourly precipitation measurements. As said, the range of the normalized space via (13) is strongly dependent on p, so we perform the validation for a variety of values of p = 0.1, 0.2, . . ., 1. [27] Microwave measurements result in a higher level of observational noise, with an uncertainty in the order of 80%. Since the time span between two images on the same location is irregular and often larger than one hour, we cannot use the SETHY rain gauges as a reference. Alternatively, we propose the following validation strategy: we use the convoluted microwave data (section 4.2) on a 12 12 km2 grid. Reference data are obtained by averaging radar measurements via (8) to the same 12 12 km2 grid. However, when considering radar measurements as refer-

ence data, we have to take in mind that, besides measurement errors, some additional errors from the upscaling procedure are present. Anyway, microwave observations carry a higher order of measurement errors and an alternative for their validation is hardly available. [28] The error variances of the radar and microwave measurements as a function of p are plotted in Figure 8.

5. Estimation Results 5.1. Numerical Performance of SRE [29] We start our numerical tests by analyzing the accuracy of the new algorithm. In Table 4 the mean estimation  variance, P(s), at each scale is reported, as no clue is available for the true precipitation field. The key result of SRE is the increase in accuracy of the final rain rate estimates. The average benefit (i.e., on the whole grid)

Figure 7. Microwave and radar cumulative probability distributions for log normalized rain rate measurements of event 2. 7 of 14

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Table 5. Final Benefit (14) Scale Event 1 Event 2

Figure 8. Estimated measurement error variances as a function of p (see (13)). in terms of accuracy of rainfall estimates can be quantified  i.e., the difference between the initial as P*l(s)  P(s), measurement variance (listed in Table 1) and the final average estimation variance. The final benefit at scale l is then given by GAl ¼

 Pl*ðsÞ  PðsÞ : Pl*ðsÞ

ð14Þ

The final benefit of SRE, listed in Table 5, indicates very reliable estimates. Such a high benefit is also reported in the analysis of Bocchiola [2007]. In addition, the standard measurement residuals at radar resolution ðsÞ ¼

zðsÞ  ^xðsÞ ; PðsÞ

ð15Þ

are tested for consistency with the proposed model. The condition to have an unbiased estimator and the hypothesis of least square variance are respectively m(e(s)) = 0 and s2(e(s)) = 1. Table 6 indicates that both conditions are reasonably well respected. For event 2, the estimated field at several resolutions is shown in Figure 9. [30] We want to compare our methodology with the recently presented SRE-EM approach of Gupta et al. [2006]. Recall that Gupta’s approach produces satisfying results when the percentage of zeros is not higher than the order of 30%. Event 2 is well suited for the comparison since it contains 22.8% zeros. Unfortunately, event 1 contains 68.7% zeros so that in this case Gupta’s SRE-EM  (s) at Each Scale When Table 4. Mean Estimate Variances P Applying SRE

1

2

3

4

5

6

0.96 0.96

0.96 0.97

0.97 0.98

0.84 0.98

0.83 0.98

0.88 0.88

method must fail. For the new method, the average estimate  variances P(s) at each scale are listed in Table 7. The estimate variance P(s) provided by the SRE-EM method, which is also included in Table 7, is node-invariant at each scale because of the regular tree geometry. It can be seen that the estimate variances P(s) provided by the SRE-EM method are slightly different or smaller than our values. On the other hand, the RMSE of both real fields does not reveal a significant difference. An inspection of the field learns that this difference is located only at the second decimal. For rainfall, such a gain is not of any practical relevance. However, from a computational point of view, the new methodology is much more efficient than the SRE-EM method since we need to run the SRE algorithm only once. For comparison, for small observation errors, the number of runs in the SRE-EM method is typically around 10 to 20. Furthermore, this number grows rapidly for larger observation errors. 5.2. Influence of Observational Noise to Process Noise Estimation [31] We investigate the accuracy of the process parameters from noisy observations. We closely follow the Monte Carlo analysis of Tustison et al. [2002] and Bocchiola [2007]. They generated several synthetic cascades with known variances and they apply their SRE methodology to back estimate the process and measurement statistics. As suggested by Bocchiola [2007], the outcome of such a simulation depends on the particular cascade geometry. We consider the same cascade geometry as in our test case (see Table 8). A total number of 1000 synthetic fields are generated with variance P*6 (s) = 0.2015. Then we add to the leaf nodes some observational noise with variances R(s) = 0.07, 0.2 and 0.3. At each scale, we compare the variance of the ‘‘true’’ field (i.e., P*l(s)) with the estimated variance of the field when observational noise has been included. The ~ l(s). The quantities latter variance is further denoted as P ~ l ðsÞÞ mðPl*ðsÞ  P

and

~ l ðsÞÞ; s2 ðPl*ðsÞ  P

ð16Þ

define the accuracy of the SRE algorithm in system identification. The results are shown in Table 8. The main conclusion is that the influence of observational noise added at the finest scale is very small on coarser scales. A similar test was performed for A(s) (7) because it also plays an important role in the coarse- to fine-scale model. The outcome of our simulations is presented in Table 9. Clearly, observational noise does not have a noticeable influence on A(s).

Scale 1 Event 1 Event 2

a

8.16E-4 5.66E-3

3

4

5

6

1.37E-3 4.68E-3

1.63E-3 4.49E-3

9.85E-3 5.29E-3

1.31E-2 5.13E-3

1.15E-2 2.40E-2

Read as 8.16E-4 as 8.16 104.

a

Table 6. Consistency Tests for the Residual (15)

2

Event 1 Event 2

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m(e(s))

s2(e(s))

4.78E-2 3.19E-4

0.74 0.89

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Figure 9. Estimated precipitation field of event 2: (top left) 3 3 km2, (top right) 6 6 km2, (bottom left) 12 12 km2, and (bottom right) 24 24 km2.

Table 8. Statistics of the Difference Between the Estimated Variances Without and With Measurement Noisea Scale

Table 7. Estimate Variances of the Estimates (A6) When Applying SRE (Event 2) Scale 1

2

3

4

5

6

 via section 3.2 5.66E-3 4.68E-3 4.49E-3 5.29E-3 5.13E-3 2.40E-2 P(s) P(s) via EM 1.55E-2 1.04E-3 7.11E-3 3.72E-3 2.44E-3 2.75E-3 [Gupta et al., 2006] RMSE 1.41E-2 7.36E-3 9.22E-3 1.18E-2 1.13E-2 9.26E-2

1 Rrad(s) m s2 Rrad(s) m s2 Rrad(s) m s2 a

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= 0.07 6.46E-6 3.79E-11 = 0.20 2.75E-5 1.60E-10 = 0.30 4.87E-5 2.00E-10

See (16).

2

3

4

5

6

3.89E-5 2.79E-10

1.74E-4 5.40E-10

6.89E-4 5.72E-9

2.81E-3 1.47E-8

7.0E-2 2.02E-7

8.91E-5 4.25E-10

3.79E-4 1.74E-9

1.52E-3 3.91E-9

6.13E-3 1.33E-8

1.5E-1 1.24E-6

2.13E-4 1.76E-9

7.80E-4 2.55E-9

3.02E-3 8.55E-8

1.20E-2 5.74E-8

3.00E-1 2.15E-6

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Table 9. Same as Table 8, but for A(s)

Table 11. Estimate Variances of the Estimates (A6) When Applying SRE (Event 2) With Different Geometriesa

Scale 1

2

3

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4

5

6

Scale

Rrad(s) = 0.07 m 4.62E-3 7.78E-4 1.31E-3 4.46E-4 1.46E-5 2.62E-5 s2 2.31E-3 2.21E-3 3.94E-4 1.13E-4 2.76E-5 2.59E-7 Rrad(s) = 0.20 m 3.50E-3 1.74E-3 1.77E-4 2.92E-4 3.26E-4 1.85E-5 s2 4.71E-3 3.13E-3 8.05E-4 2.04E-4 5.21E-5 3.90E-7 Rrad(s) = 0.30 m 1.92E-3 3.26E-3 6.87E-4 6.29E-4 3.37E-4 9.30E-6 s2 4.70E-3 4.64E-3 1.17E-3 3.18E-4 6.08E-5 5.24E-7

[32] Next, we examine the influence of observational noise on SRE for a real world application. According to the Monte Carlo analysis, adding some observational noise to the radar measurements should increase the estimated variance P6*(s). We put P6*(s) = 0.27 and we compare the real field with the original field (i.e., in which we used P6*(s) = 0.2015; see Table 2). For every scale, the RMSE of both real precipitation fields is recorded in Table 10. Obviously, the influence of observational noise on process noise estimation does not seriously affect the final result. Similar conclusions hold for lognormal multiplicative models [see Tustison et al., 2002]. 5.3. Dependency of SRE on the Tree Geometry [33] It is likely that the estimation variances depend on the particular tree geometry, i.e., the number of nodes, the branching numbers and the number of layers. To our knowledge, this issue was not investigated in the past. Here we examine the performance of SRE when we remove some layers in the present tree geometry. In Table 11 we present the estimation variances in the case of (1) the present geometry, (2) without scale 5, (3) without scale 4 and (4) without scale 4 and 5. In addition, we have performed the analysis on 12 extra geometries (not shown here for sake of shortness). We observe that when removing a certain scale, the average estimation variances of the ascendant nodes decrease, while they increase for the descendant nodes. A theoretical explanation has still to be found. On the basis of the above analysis, it is recommended that the nodes of the scale of interest have as many children as possible, while the corresponding parent node should have a low number of children. When developing SRE for operational use, the designer must have a clear idea of the scale(s) of interest when choosing an optimal tree geometry.

 P(s)

a

1

2

3

4

5

6

5.66E-3 5.61E-3 5.73E-2 5.38E-2

4.68E-3 4.52E-3 4.65E-3 3.94E-3

4.49E-3 3.82E-3 4.00E-3 1.40E-3

5.29E-3 2.61E-3 -

5.13E-3 6.05E-3 -

2.40E-2 3.03E-2 2.41E-2 3.64E-3

A dash (‘‘-’’) means that this scale is removed in the SRE algorithm.

[2001], two patterns of measurement withholdings were investigated: a banded withholding pattern and a scattered withholding pattern. The banded withholding pattern mimics the actual situation when the field of view is blocked over entire sectors. We have omitted a rectangular data set with sizes 240 45 km2 (see Figure 11) and we try to reconstruct the lost data with the SRE algorithm. To demonstrate the influence of the microwave data we compare our SRE methodology with the case in which only radar observations are included. We also consider the case in which only microwave data are used. Obviously, the estimation variance increases rapidly when we move away from the divide. A useful method to illustrate the relevance of the SRE algorithm is to plot the mean estimation variance as a function of the distance from the divide. This is done in Figure 10. An optimal benefit of data fusing is observed within the region located at 5 – 10 km from the divide. The good accuracy obtained in this region is due, in part, to information obtained from microwave measurements available at a coarse scale. The final estimate features a noticeably greater accuracy than those of the devices alone. The uncertainty gradually increases because the small-scale variability of the field is lost. Consequently, a greater amount of smoothing is present. The reconstructed field against the merged field (based on complete data) at 3 3 km2 resolution is shown in Figure 11. Although estimates at the withheld nodes do not capture the small-scale variability revealed in the measurements they follow the

5.4. Effect of Missing Data on SRE [34] We address the question of how well the SRE algorithm reproduces the precipitation field if some radar measurements are withheld. As done by Primus et al.

Table 10. RMSE of the Real Field for Event 2 When Using P6*(s) = 0.2015 and P*6 (s) = 0.27a Scale RMSE

1

2

5.38E-3

5.39E-3

3

4

5

6

5.94E-3 1.74E-2 4.32E-2 8.07E-2 a RMSE of the real field is in mm/h. P6*(s) = 0.2015 is the original choice, and P6*(s) = 0.27 is perturbed by measurement noise.

Figure 10. Banded withholding pattern. The estimate variance P(s) at radar scale is shown as a function of the distance from the divide.

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Figure 11. Banded withholding pattern (shown at 3 3 km2 resolution). (left) Original precipitation field. (right) Estimated precipitation field when the data within the rectangle are removed. general trend of the data. The correlation coefficient of the reconstructed field and the radar measurements within the rectangle (on which the data were withheld) is 0.76. For comparison, within the same rectangle the correlation coefficient of the merged field (based on complete data) and the radar measurements is 0.98. [35] We continue the experiments by checking the algorithm’s performance on a scattered withholding pattern (randomly sampled). In our tests the availability of the radar data ranges from 10% to 90%. In all cases we have found the same estimated variances P*l (s) as for the original data set. This is because in randomly withholding, it is unlikely that the field will exhibit drastically different features than the original field. In contrast to the case with a banded withholding pattern, the estimation variances are varying from node to node. In Table 12 we show the average  estimation variance P(s) at radar scale both in the nonwithheld nodes and withheld nodes. We observe that pixels that have missing values have higher uncertainty. The cumulative distribution function of the estimate variances, shown in Figure 12, provides a better overview of the uncertainty. The overall uncertainty is smaller compared to the banded withholding pattern. This is because more data pixels are available in the immediate surrounding of withheld pixels. Even if 90% of the radar pixels are randomly withheld, the SRE algorithm is able to reproduce the original data set in a very satisfactory way.

until now, Gupta et al. [2006] and Bocchiola [2007] have offered an acceptable solution for this issue. To be more precise, the tree pruning technique of Bocchiola [2007] consists in restricting the tree geometry only to nonzero rainfall. Gupta et al. [2006] have explored nonparametric models in order to avoid the usual lognormal multiplicative cascade models. Normality of the SRE processes, which is necessary for optimal Kalman filtering, is then obtained by applying a power transformation to the data. Our numerical tests reveal that a power transformation is to be preferred over the log transformation. [37] With respect to the present literature on multiscale data merging our methodology features some new developments. In this paper we have combined the great advantages of both approaches: on one hand we have adopted the tree pruning technique, and on the other hand we have considered a power transformation. To model the variability in scales we have introduced a simple procedure which forms, from a computational point of view, an attractive alternative to Gupta’s EM-SRE. A real world application shows the relevance of the new methodology. Data from two measurement sources, microwave satellite measurements and ground-based radar measurements, were assimilated to produce a multiscale estimated field. Before running the SRE algorithm it is required to specify the observational error of the measurement devices and the variability of Table 12. Estimate Variances P(s) at Radar Resolution (Event 2)a

6. Conclusions [36] The power of SRE resides in its ability to compactly model a rich class of phenomena and to efficiently address complications that arise in many one-dimensional and multidimensional problems (e.g., spatially irregular data, nonstationarities, and others). SRE of precipitation data is difficult because of the zero intermittency of rainfall and

Missing Observations Available Withheld

10%

30%

50%

70%

90%

2.41E-2 3.67E-2

2.44E-2 3.76E-2

2.50E-2 3.89E-2

2.60E-2 4.14E-2

2.85E-2 4.81E-2

a The first (second) row of a cell represents the SRE algorithm in the region where some radar measurements are available (withheld).

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Figure 12. Cumulative distribution function for the estimate variance P(s) at radar scale. the rain rate process in scales. The measurement noise of the radar is estimated by comparing a large amount of data sets with rain gauge data. The noise in the microwave measurements is estimated by using upscaled radar data as reference. [38] For the new SRE methodology, the following topics were investigated. Monte Carlo simulations are performed to assess the sensitivity of the variability of the rain rate to observational noise. As a result, observational noise increases the uncertainty of the variability of the coarsest scale but it does not have a significant effect on the final estimation in the real space. We conclude that SRE can be considered as a robust methodology. The performance of SRE is clearly dependent of the tree geometry. An optimal estimate for the nodes of any scale can be obtained when these nodes have as many children as possible, while the corresponding parent node should have a low number of children. We have demonstrated our approach on large scale problems of current interest in quantitative precipitation estimation (QPE). The SRE algorithm can quickly compute estimates and corresponding variances on a 400 400 grid. Because the complexity of the SRE problem scales only linearly with the problem size, this approach is practical for climate and global scale estimation. Numerical experiments indicate that the SRE algorithm is capable of reconstructing missing data in a very satisfactory way. This is confirmed by statistical tests on the estimate variance. [39] Further research is needed prior to an operational use of SRE and the production of improved QPE for hydrological applications. The methodology should be tested on series of satellite and radar images covering a larger variety of meteorological situations. Efforts should also deal with the extension of the SRE framework to space-time multiscale estimation and fusing. Other applications include, for instance, the repair of damaged images.

of measurements, define ^x(sjs) = ^x(sjZ(s)) and the associated error variance P(sjs) = E[(x(s)  ^x(sjs))2jZ(s)]. Likewise, the definition of ^x(sjs+) and P(sjs+) is evident. Smoothed estimates that are conditioned on all the measurements, i.e., ^x(sjZ0), are for notational convenience abbreviated as ^x(s), and similarly for P(s). The unconditioned variance at scale s, i.e., the variance of the states before applying SRE, is presented by P*(s). [41] Smoothing is performed in two steps: an upward sweep starting at the leaves to compute filtered estimates ^x(sjs) and P(sjs), followed by a downward sweep starting at the root node to compute smoothed estimates ^x(s) and P(s).

Appendix A:

where K(s) is an estimator gain given by

Details of the SRE Framework

A1. Initialization [42] Starting the upward sweep requires an initialization step. For each node s at the finest scale, the following prior values are assigned: xðsjsþÞ ¼ 0; ðA1Þ PðsjsþÞ ¼ P*ðsÞ:

This choice is justified by the fact that the global mean of the process is zero by definition. About the variances, since no measurements have yet been used, the error variance is set to the variance of the process.

A2. Upward Sweep [43] The first step in the upward sweep is to incorporate the measurements z(s). This is accomplished with the scalar version of the classical Kalman filter ^xðsjsÞ ¼ ^xðsjsþÞ þ KðsÞð zðsÞ  ^xðsjsþÞÞ; ðA2Þ PðsjsÞ ¼ PðsjsþÞð1  KðsÞÞ;

[40] Before describing SRE, we need to introduce some definitions and notations. Define Zs = {z(s)js = s or s is a descendant of s} and Z+s = {z(s)js is a descendant of s}. Z0 is the root node and comprises all the observations. For a set 12 of 14

KðsÞ ¼

PðsjsþÞ : PðsjsþÞ þ RðsÞ

ðA3Þ

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The case K(s) = 0 refers to a measuring device which is completely unreliable, since the error variance, R(s), must be infinite. From (A2) it is clear that the measurements offer no additional information about the state. The other limiting case, K(s) = 1, represents a perfect measurement since R(s) = 0. In that case, it can be seen from (A2) that the measurements are considered as the estimated state. If no measurements are available, the values of the state and its variance are not updated. [44] Secondly, we perform the scale propagation step with (3). Let the children of a node s be denoted by sai, i = 1,. . ., n. By taking the conditional mean of (3) we obtain ^x(saijsai) to predict x(s) from each child ^xðsjs ai Þ ¼ Fðs ai Þ ^xðs ai js ai Þ; ðA4Þ Pðsjs ai Þ ¼ F 2 ðs ai Þ Pðs ai js ai Þ þ Qðs ai Þ:

[45] Thirdly, every child sai provides an estimate to the state of its parent node s. A merged estimate ^x(sjs+) can be obtained by combining the upward propagated estimates ^x(sjsai), i = 1,. . ., n ^xðsjsþÞ ¼ PðsjsþÞ

n X

P1 ðsjs ai Þ ^xðsjs ai Þ;

i¼1

PðsjsþÞ ¼

1

ð1  nÞ P* ðsÞ þ

n X

!1 1

P ðsjs ai Þ

ðA5Þ ;

i¼1

where it is recalled that P*(s) is the unconditional variance at node s. The merged estimate ^x(sjs+) from (A5) consists of a weighted sum of the estimates of its children. The weights are P1(sjsai) which indicates that the larger the uncertainty of an estimate the smaller the influence of that estimate in the merging step. [46] Starting at the finest scale, the update, upward propagation, and merge steps are processed recursively until we arrive at the root node. At the end of the upward sweep, the estimates are conditioned on all the measurements at this node and its descendants.

A3. Downward Sweep [47] The downward sweep is a coarse to fine scale evolution, starting at the next finer scale from the root scale, and continuing to the finest scale. Both the downward sweep estimates of the state an its variance are produced as the sum of the upward estimate and a weighted difference between the upward and downward estimates of its parent. The operations for this propagation have the same form as the classical temporal smoothing solution ^xðsÞ ¼ ^xðsjsÞ þ J ðsÞ ð^xðs gÞ  ^xðs gjsÞÞ; ðA6Þ PðsÞ ¼ PðsjsÞ þ J 2 ðsÞ ð Pðs gÞ  Pðs gjsÞÞ;

where J(s) is a weighting coefficient, which is given by J ðsÞ ¼ FðsÞ

PðsjsÞ : Pðs gjsÞ

ðA7Þ

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At the end of the downward sweep the estimate at each node in the tree has been conditioned on all the measurements. [48] Acknowledgments. The authors wish to thank the anonymous referees for carefully reading the manuscript. We are particularly indebted to a referee whose comments and suggestions greatly improved this paper. The authors thank Joris Van den Bergh, Laurent Delobbe, and Nicolas Clerbaux. This research was performed in the framework of the Satellite Application Facility on Support to Operational Hydrology and Water Management and was supported by EUMETSAT, the Belgian Federal Scientific Policy Office, and ESA PRODEX Program.

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Smith, J. A., and W. F. Krajewski (1993), A modeling study of rainfall ratereflectivity relationships, Water Resour. Res., 29, 2505 – 2514. Tustison, B., E. Foufoula-Georgiou, and D. Harris (2002), Scale-recursive estimation for multisensor Quantitative Precipitation Forecast verification: A preliminary assessment, J. Geophys. Res., 107, 8377, doi:10.1029/ 2001JD001073, [printed 108(D8), 2003]. Van de Vyver, H., and E. Roulin (2008), Belgian contribution to the validation of the precipitation products of the Hydrology-SAF: Methodology developed and preliminary results, in 2008 EUMETSAT Meteorological

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Satellite Conference [CD-ROM], 8 pp., Eur. Organ. for the Exploit. of Meteorol. Satell., Darmstadt, Germany. 

E. Roulin and H. Van de Vyver, Royal Meteorological Institute of Belgium, Avenue Circulaire 3, B-1180 Brussels, Belgium. (hvijver@ oma.be)

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