• Ahijevych, D., E. Gilleland, B. G. Brown, and E. E. Ebert, 2009: Application of spatial verification methods to idealized and NWP-gridded precipitation forecasts. Wea. Forecasting, 24, 14851497, doi:10.1175/2009WAF2222298.1.

    • Search Google Scholar
    • Export Citation
  • Berg, R., 2013: Tropical Cyclone Report Hurricane Isaac (AL092012) 21 August–1 September 2012. Tech. Rep. AL092012, National Hurricane Center, 78 pp. [Available online at http://www.nhc.noaa.gov/data/tcr/AL092012_Isaac.pdf.]

  • Chen, Y., E. E. Ebert, K. J. E. Walsh, and N. E. Davidson, 2013: Evaluation of TRMM 3B42 precipitation estimates of tropical cyclone rainfall using PACRAIN data. J. Geophys. Res. Atmos., 118, 21842196, doi:10.1002/jgrd.50250.

    • Search Google Scholar
    • Export Citation
  • Davis, C., B. Brown, and R. Bullock, 2006: Object-based verification of precipitation forecasts. Part I: Methodology and application to mesoscale rain areas. Mon. Wea. Rev., 134, 17721784, doi:10.1175/MWR3145.1.

    • Search Google Scholar
    • Export Citation
  • Done, J., C. A. Davis, and M. Weisman, 2004: The next generation of NWP: Explicit forecasts of convection using the Weather Research and Forecasting (WRF) model. Atmos. Sci. Lett., 5, 110117, doi:10.1002/asl.72.

    • Search Google Scholar
    • Export Citation
  • Ebert, E. E., 2009: Neighborhood verification: A strategy for rewarding close forecasts. Wea. Forecasting, 24, 14981510, doi:10.1175/2009WAF2222251.1.

    • Search Google Scholar
    • Export Citation
  • Ebert, E. E., and J. L. McBride, 2000: Verification of precipitation in weather systems: Determination of systematic errors. J. Hydrol., 239, 179202, doi:10.1016/S0022-1694(00)00343-7.

    • Search Google Scholar
    • Export Citation
  • Gilleland, E., D. Ahijevych, B. G. Brown, B. Casati, and E. E. Ebert, 2009: Intercomparison of spatial forecast verification methods. Wea. Forecasting, 24, 14161430, doi:10.1175/2009WAF2222269.1.

    • Search Google Scholar
    • Export Citation
  • Huffman, G. J., and Coauthors, 2007: The TRMM Multisatellite Precipitation Analysis (TMPA): Quasi-global, multiyear, combined-sensor precipitation estimates at fine scales. J. Hydrometeor., 8, 3855, doi:10.1175/JHM560.1.

    • Search Google Scholar
    • Export Citation
  • Keil, C., and G. C. Craig, 2007: A displacement-based error measure applied in a regional ensemble forecasting system. Mon. Wea. Rev., 135, 32483259, doi:10.1175/MWR3457.1.

    • Search Google Scholar
    • Export Citation
  • Keil, C., and G. C. Craig, 2009: A displacement and amplitude score employing an optical flow technique. Wea. Forecasting, 24, 12971308, doi:10.1175/2009WAF2222247.1.

    • Search Google Scholar
    • Export Citation
  • Lin, Y., and K. E. Mitchell, 2005: The NCEP Stage II/IV hourly precipitation analyses: Development and applications. 19th Conf. on Hydrology, San Diego, CA, Amer. Meteor. Soc., 1.2. [Available online at https://ams.confex.com/ams/pdfpapers/83847.pdf.]

  • Mass, C. F., D. Ovens, K. Westrick, and B. A. Colle, 2002: Does increasing horizontal resolution produce more skillful forecasts? Bull. Amer. Meteor. Soc., 83, 407430, doi:10.1175/1520-0477(2002)083<0407:DIHRPM>2.3.CO;2.

    • Search Google Scholar
    • Export Citation
  • Patricola, C. M., M. Li, Z. Xu, P. Chang, R. Saravanan, and J.-S. Hsieh, 2012: An investigation of tropical Atlantic bias in a high-resolution coupled regional climate model. Climate Dyn., 39, 24432463, doi:10.1007/s00382-012-1320-5.

    • Search Google Scholar
    • Export Citation
  • Patricola, C. M., R. Saravanan, and P. Chang, 2014: The impact of the El Niño–Southern Oscillation and Atlantic meridional mode on seasonal Atlantic tropical cyclone activity. J. Climate, 27, 5311–5328, doi:10.1175/JCLI-D-13-00687.1.

    • Search Google Scholar
    • Export Citation
  • Romero, R., C. Doswell III, and R. Riosalido, 2001: Observations and fine-grid simulations of a convective outbreak in northeastern Spain: Importance of diurnal forcing and convective cold pools. Mon. Wea. Rev., 129, 21572182, doi:10.1175/1520-0493(2001)129<2157:OAFGSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Speer, M. S., and L. M. Leslie, 2002: The prediction of two cases of severe convection: Implications for forecast guidance. Meteor. Atmos. Phys., 80, 165175, doi:10.1007/s007030200023.

    • Search Google Scholar
    • Export Citation
  • Szunyogh, I., 2014: Applicable Atmospheric Dynamics: Techniques for the Exploration of Atmospheric Dynamics. World Scientific, 608 pp.

  • Weisman, M. L., W. C. Skamarock, and J. B. Klemp, 1997: The resolution dependence of explicitly modeled convective systems. Mon. Wea. Rev., 125, 527548, doi:10.1175/1520-0493(1997)125<0527:TRDOEM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Wernli, H., M. Paulat, M. Hagen, and C. Frei, 2008: SAL—A novel quality measure for the verification of quantitative precipitation forecasts. Mon. Wea. Rev., 136, 44704487, doi:10.1175/2008MWR2415.1.

    • Search Google Scholar
    • Export Citation
  • Zinner, T., H. Mannstein, and A. Tafferner, 2008: Cb-TRAM: Tracking and monitoring severe convection from onset over rapid development to mature phase using multi-channel Meteosat-8 SEVIRI data. Meteor. Atmos. Phys., 101, 191210, doi:10.1007/s00703-008-0290-y.

    • Search Google Scholar
    • Export Citation
  • View in gallery
    Fig. 1.

    Illustration of the effect of the proposed constraint on the pyramid matching algorithm: (a) the forecast field, (b) the verifying analysis, (c) the optical flow for the original algorithm, (d) the morphed field for the original algorithm, (e) the optical flow for the modified algorithm, and (f) the morphed field for the modified algorithm.

  • View in gallery
    Fig. 2.

    Illustration of the idealized example of section 3. The full domain consists of 32 × 32 elementary parcels. The “forecast” feature and the “observed” feature (gray square in the lower-left corner) consist of 8 × 8 elementary parcels. The forecast feature is gradually shifted along the main diagonal in the direction indicated by the arrow. The dots labeled by along the diagonal indicate the location of the center of the forecast feature for the different values of l.

  • View in gallery
    Fig. 3.

    The metrics (black) and (blue) as a function of l in the idealized example of section 3 for . (For , the black and green symbols overlap.) Also shown is (green) as function of l when the value of is computed by using the criterion of section 3b for the selection of F [the value of F selected by the algorithm is also shown as function of l (red)].

  • View in gallery
    Fig. 4.

    Illustration of the morphing process for the idealized example of section 3 for : (a) the original forecast feature and the optical flow, and (b) the morphed forecast feature. Each color identifies a distinct section of the forecast feature and the image of that section in the morphed forecast feature.

  • View in gallery
    Fig. 5.

    Illustration of the morphing process for the idealized example of section 3 for and a gradually increasing value of F (F increases from left to right). (from top to bottom) The forecast feature and the optical flow for 1, and 2; the resulting morphed features; the forecast feature and the optical flow for , 4, and 5; and the resulting morphed features.

  • View in gallery
    Fig. 6.

    As in Fig. 5, but for .

  • View in gallery
    Fig. 7.

    The metrics as a function of F for and .

  • View in gallery
    Fig. 8.

    Illustration of the result of morphing for geometric case 5. Gray shades indicate the silhouette of the verification feature.

  • View in gallery
    Fig. 9.

    As in Fig. 8, but without correction for the amplitude error before performing morphing.

  • View in gallery
    Fig. 10.

    The CRCM model domain, which is also used for the computation of the verifications scores. The subdomain where stage IV analysis data are available is mark by a gray shade.

  • View in gallery
    Fig. 11.

    Example 1: (a) the forecast, (b) the verifying data, (c) the morphed forecast, and (d) the forecast corrected for the location error (shifted by ). In (a),(c), and (d), gray shading indicates the silhouette of the precipitation feature in the verification data.

  • View in gallery
    Fig. 12.

    As in Fig. 11, but for example 2. The red arrow in (d) shows .

  • View in gallery
    Fig. 13.

    As in Fig. 11, but for example 3. The red arrow in (d) shows .

  • View in gallery
    Fig. 14.

    As in Fig. 11, but for example 4. The red arrow in (d) shows .

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A Morphing-Based Technique for the Verification of Precipitation Forecasts

Fan HanTexas A&M University, College Station, Texas

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Istvan SzunyoghTexas A&M University, College Station, Texas

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Abstract

This paper describes a morphing-based approach for the verification of precipitation forecasts. This approach employs a pyramid matching algorithm to morph the precipitation features in a forecast into features that match the related precipitation features in the verifying analysis (observations) as closely as possible. The algorithm computes an optical flow (vector field) that maps the original forecast features into the morphed forecast features. The optical flow also provides quantitative information about the error in the location of the forecast features. This information, combined with information about the error in the prediction of the total precipitation over the verification domain, is used to quantify the structure error in the precipitation forecast. The proposed approach has three novel aspects compared to the published morphing-based verification strategies. First, it imposes a constraint on the pyramid matching algorithm to prevent overconvergence toward strong precipitation features during morphing. Second, it introduces an objective criterion for the selection of the subsampling parameter to avoid splitting or distorting features due to an arbitrary maximum displacement limit. Third, the proposed definitions of the location and structure errors are new. The behavior of the proposed multivariate verification metrics is investigated by applications to both idealized and numerical forecast examples.

Corresponding author address: Fan Han, Department of Atmospheric Sciences, Texas A&M University, 3150 TAMU, College Station, TX 77843-3150. E-mail: fanhan172739775@tamu.edu

Abstract

This paper describes a morphing-based approach for the verification of precipitation forecasts. This approach employs a pyramid matching algorithm to morph the precipitation features in a forecast into features that match the related precipitation features in the verifying analysis (observations) as closely as possible. The algorithm computes an optical flow (vector field) that maps the original forecast features into the morphed forecast features. The optical flow also provides quantitative information about the error in the location of the forecast features. This information, combined with information about the error in the prediction of the total precipitation over the verification domain, is used to quantify the structure error in the precipitation forecast. The proposed approach has three novel aspects compared to the published morphing-based verification strategies. First, it imposes a constraint on the pyramid matching algorithm to prevent overconvergence toward strong precipitation features during morphing. Second, it introduces an objective criterion for the selection of the subsampling parameter to avoid splitting or distorting features due to an arbitrary maximum displacement limit. Third, the proposed definitions of the location and structure errors are new. The behavior of the proposed multivariate verification metrics is investigated by applications to both idealized and numerical forecast examples.

Corresponding author address: Fan Han, Department of Atmospheric Sciences, Texas A&M University, 3150 TAMU, College Station, TX 77843-3150. E-mail: fanhan172739775@tamu.edu

1. Introduction

Increasing model resolution has lead to qualitatively more realistic precipitation forecasts (e.g., Weisman et al. 1997; Romero et al. 2001; Speer and Leslie 2002; Done et al. 2004). The greater qualitative realism of the forecast precipitation fields, however, does not necessarily translate into quantitatively more accurate forecasts. In fact, studies using traditional precipitation verification metrics have failed to show consistent improvements with increasing model resolution (e.g., Mass et al. 2002; Done et al. 2004). This result may be due to limitations of the traditional, point-to-point verification techniques rather than to lack of forecast improvement. In particular, such techniques can indicate large errors in situations where generally well-predicted precipitation events of high spatial variability are slightly misplaced. This problem has motivated the search for verification techniques that can separate the location (displacement) error from errors in the structure and the total amount of the predicted precipitation (e.g., Ebert and McBride 2000; Ebert 2009; Davis et al. 2006; Wernli et al. 2008; Gilleland et al. 2009).

Some approaches for the separation of the three error components are based on identifying distinct spatial patterns of precipitation as objects, and comparing the properties of the matching objects in the forecasts and the analyses (observations). The difference between the spatial locations of the matched objects is the location error, the difference between the spatial structures of the matched objects is the structure error, and the difference between the total amounts of precipitation in the matched objects is the amplitude error. Finding algorithms for the identification and matching of the objects, and defining proper measures of the three error components are challenging tasks.

Our strategy for the identification and matching of the forecast and analyzed objects is based on a technique proposed by Keil and Craig (2007, 2009, hereafter KC07 and KC09, respectively). Their technique employs an optical flow method to morph the precipitation forecast field into a field that resembles the precipitation analysis (observation) field as closely as possible. It treats precipitation as a passive scalar and carries out morphing by first computing a vector field, called the optical flow, which is then used to rearrange the original forecast precipitation field. Because a successful morphing corrects the location error efficiently, a measure of the location error can be defined based on the optical flow.

There are three novel aspects of our study. First, we demonstrate that the morphing algorithm has a tendency to move precipitation into a highly localized region around the location of the largest forecast error; leading to the complete removal of precipitation from other regions, including those where precipitation is observed and reasonably well predicted. We impose a constraint on the morphing process that greatly reduces the potential severity of the resulting overconvergence of precipitation. Second, we introduce an objective criterion for the optimal choice of the subsampling parameter, the key free parameter of the morphing algorithm. Finally, we define a new measure of both the location error and the structure error, and a new integral measure of the location, structure, and amplitude errors.

The structure of the paper is as follows. Section 2 describes both the original and our modified version of the morphing algorithm; section 3 defines the components of the proposed verification metrics; section 4 validates the proposed approach by applications to cases from the Spatial Verification Methods Intercomparison Project (ICP); section 5 shows validation results for high-resolution tropical cyclone (TC) forecasts; and section 6 offers our conclusions.

2. The morphing algorithm

The optical flow method of KC07 and KC09 is a pyramid matching algorithm. This section provides a detailed description of the algorithm (sections 2a and 2b) and its modification that we propose to prevent overconvergence of the precipitation during morphing (section 2c).

a. Notation and synopsis

The pyramid matching algorithm morphs the forecast precipitation field into an approximate image of the analyzed (observed) precipitation field iteratively. The algorithm treats both the forecast and the analyzed precipitation fields as collections of elementary parcels (pixels). When the two fields are provided on a grid, the value of the precipitation for an elementary parcel can be obtained by averaging the gridpoint values that fall within the area of the parcel. To simplify notation, we assume that the elementary parcels are d × d squares. Before providing a step-by-step description of the algorithm, we introduce notation and provide a synopsis of the algorithm.

The symbol denotes the elementary parcel located at position before morphing, where are the zonal and the meridional indexes of location, respectively. We denote the forecast and analyzed (observed) total precipitation in by and , respectively.1 The algorithm calculates the morphing vector , which points from the location of each parcel of the original forecast image to the new location of the parcel in the morphed forecast image. The resulting vector field is the optical flow. The precipitation is moved from to with the elementary parcel.

There are two special situations in which the computation of the morphed field requires additional considerations. First, when a parcel is moved, there may not be another parcel to move into its original location. Such a hole can be filled with a parcel with no precipitation. Second, multiple elementary parcels may arrive at the same location. To cope with this special situation, KC07 and KC09 implemented a strategy that bounded the precipitation of the morphed field by the minimum and the maximum of the original precipitation field. We follow a different strategy by obtaining the value of precipitation in the morphed field by the summation of the precipitation of all parcels that arrive at the same location. Our strategy has the advantage that it preserves global precipitation in the verification region, but has the disadvantage that it makes the algorithm more sensitive to the overconvergence. The motivation for our choice of strategy is the experience that models tend to produce overly smooth precipitation fields due to their limited resolution and the effects of subgrid parameterizations. Thus, we want to allow morphing to introduce finer structures into the forecast precipitation field, possibly including a new maximum and/or a new minimum, without changing the global precipitation in the verification region. We believe that this can be done, because the proposed modification of the algorithm that will be described in section 2c greatly reduces the sensitivity to overconvergence.

b. Computation of the optical flow

The most important parameter of the algorithm is the subsampling parameter F. For a particular value of F, the optical flow and the related morphed image is computed in iterations. Each iteration computes a partial optical flow (k = F, …, 0) and moves the elementary parcels accordingly. The first iteration computes the partial optical flow for ; then, the following iterations compute the optical flow for decreasing values of k. The indexes i and j refer to the position of the elementary parcel in the original forecast field. In essence, the couple plays a role analogous to that of a particle label (the independent variable) of Lagrangian fluid dynamics [e.g., section 1.1 of Szunyogh (2014)]. Because
e1
the last iteration completes the morphing and provides the morphed image.

The partial optical flow is computed in the following steps:

  1. Both the image of the forecast and the image of the analyzed precipitation are coarse grained by averaging for elementary parcels. That is, the coarse-grained fields are covered by coarse-grained parcels, , of size × . The precipitation for the coarse-grained parcels is computed by averaging the values for the elementary parcels that form them. The precipitation for the coarse-grained parcels is denoted by for the forecast field and by for the analyzed (observed) field.

  2. The vector of the partial optical flow is computed for all coarse-grained parcels by

    1. shifting the position of each parcel of the coarse-grained forecast image by (by none or one position in the coarse-grained image) in both the zonal and the meridional directions;

    2. computing
      e2
      for each of the nine shifted positions and defining the components of by the values of and that minimize D; and
    3. making the vectors for all elementary parcels included in the coarse-grained parcel associated with label equal to .

  3. The intermediate optical flow produced by step 2 is used to obtain an intermediate (full resolution) morphed forecast image from the full resolution image produced by the preceding iteration. The next iteration operates on the morphed forecast image produced by this step.

Notice that F determines the longest distance over which an elementary parcel of the forecast field can be matched to an elementary parcel of the analyzed (observed) field by the pyramid matching algorithm. This distance is
e3
in both horizontal directions. For instance, for , , and for , . KC07 and KC09 suggested selecting F by considering the dynamical scale of the verified precipitation events.

Finally, it should be noted that in KC07’s and KC09’s implementation of the algorithm, a Gaussian filter was applied to the partial optical flow at each level of the algorithm. In our implementation of the algorithm, we do not apply such a filter. In addition, KC07 and KC09 morphed both the forecast feature into an image of the analyzed image and the analyzed image into an image of the forecast feature. This approach allowed them to define error measures that did not depend on the direction of morphing. Our error measures, which are based on morphing the forecast feature only, do not have this symmetry property. We investigated the effect of morphing the analyzed feature instead of the forecast feature on our verification scores and found that the effect was weak.

c. Modification to the algorithm to prevent overconvergence

The optical flow often, but not always, behaves like a fluid flow. In the exceptional cases, the optical flow is such that some elementary parcels leapfrog other elementary parcels. (An example for this behavior will be shown in section 3.) Even at locations where it behaves like a fluid flow, the optical flow is not necessarily a simple translational flow, because parcels can merge when they arrive at the same location. Thinking of the elementary parcels as infinitesimal parcels, this property can be stated by saying that the optical flow is not divergence free. (The merged parcels are squeezed into the volume of a single elementary parcel.) This behavior of the optical flow can be advantageous in situations where the model fails to represent small-scale divergent processes of the atmosphere correctly: the morphing can introduce the related observed structures into the forecasts, allowing for a more accurate comparison of the overall structure of the precipitation fields. In some situations, however, convergence can be an artifact of morphing. Such undesirable convergence can occur in situations where morphing attempts to correct for a large local error in the forecast by moving precipitation from those parts of the surrounding area where precipitation is both observed and, in general, well predicted.

An example for the aforementioned problem is shown in Fig. 1: Fig. 1a shows the original forecast field, Fig. 1b shows the verifying analysis, Fig. 1c shows the morphing vector, and Fig. 1d shows the morphed forecast field. A comparison of Figs. 1b and 1d shows that morphing shifts all precipitation from the Gulf of Mexico to the east over land, where the analyzed precipitation is the heaviest but no precipitation is forecast. Such a shift would only be acceptable if no precipitation was observed in the Gulf of Mexico, which is not the case. Next, we show that this undesirable behavior of the algorithm can be prevented by a modification of the algorithm.

Fig. 1.
Fig. 1.

Illustration of the effect of the proposed constraint on the pyramid matching algorithm: (a) the forecast field, (b) the verifying analysis, (c) the optical flow for the original algorithm, (d) the morphed field for the original algorithm, (e) the optical flow for the modified algorithm, and (f) the morphed field for the modified algorithm.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

We start the description of the modification of the algorithm by introducing the notion of a local hit rate. We call this hit rate local, because we define it for each coarse-grained parcel (). To define the local hit rate, we also introduce the notations for the number of elementary parcels in which the presence of precipitation is correctly predicted and for the number of elementary parcels in which the forecast fails to predict the presence of precipitation. Note that is the total number of elementary parcels in which precipitation is observed within the coarse-grained parcel . The local hit rate is defined by
e4
According to this definition, : the higher the local hit rate, the lower the probability that precipitation is present in both the forecast and the analysis (observations) within only by chance. Hence, the higher the local hit rate, the less justifiable it is to move to a new location. This argument motivates us to impose the condition that the parcel can be moved to a new location only if , where is a prescribed threshold value.

Once k becomes small, the number of elementary parcels in the coarse-grained parcels also becomes small. Hence, the local hit rate becomes a poor estimate of the probability of the forecast precipitation and the observed precipitation being related. Thus, the constraint imposed on the morphing process in the form of the threshold ε should be applied only in the top-most iterations (for the largest values of k) of the algorithm. For the cases we used for testing the proposed modification, we found that imposing the constraint only at the top level (for ) of the pyramid, combined with a threshold value of , always had the desired effect on the behavior of the algorithm. For the case shown in Fig. 1, this effect can be seen in Figs. 1e and 1f: comparing Figs. 1c and 1e and Figs. 1d and 1f, it can be seen that the constraint imposed on the morphing algorithm prevents the complete removal of the forecast precipitation from the Gulf of Mexico.

3. Definition of the components of the morphing-based error measure

First, we define the location error by the magnitude of the mean of the morphing vectors (section 3a). Then, we define our selection criteria for the subsampling parameter with the help of the location error (section 3b). Next, we adopt a definition of the amplitude error from Wernli et al. (2008), which is independent of the result of morphing (section 3c). Finally, we define the structure error by first correcting the forecast for both the amplitude and the location errors, then computing the root-mean square difference between the analyzed and the error-corrected forecast fields (section 3d).

a. Location error

Our measure of the location error is formally defined by
e5
where
e6
is the mean of the morphing vectors. We have filtered out the morphing vectors for which the precipitation in the full-resolution forecast field is zero. Here n is the number of elementary parcels for which the morphing vector is nonzero after filtering. The measure defined by Eq. (5) is not the same as the following measure:
e7
which was suggested by KC07 to quantify the displacement error based on the morphing vectors. The error measures and would be identical, only if the optical flow was always a translational vector field.

To illustrate the difference between the measures and , we consider an idealized test case: the verification domain consists of 32 × 32 elementary parcels for which d is equal to 1 dimensionless unit. In addition, both the “forecast” and the “analyzed” precipitation features are composed of 8 × 8 elementary parcels of uniform precipitation (Fig. 2). The “observed” precipitation feature is located in the lower-left corner of the verification domain, while the displacement of the forecast precipitation feature increases gradually: in step l, the location of the center of the forecast precipitation feature is shifted by both to the right and upward (diagonally) compared to the location of the center of the observed precipitation feature. We carry out morphing experiments and compute and for .

Fig. 2.
Fig. 2.

Illustration of the idealized example of section 3. The full domain consists of 32 × 32 elementary parcels. The “forecast” feature and the “observed” feature (gray square in the lower-left corner) consist of 8 × 8 elementary parcels. The forecast feature is gradually shifted along the main diagonal in the direction indicated by the arrow. The dots labeled by along the diagonal indicate the location of the center of the forecast feature for the different values of l.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

Figure 3 shows the value of and for the different values of l, when the fixed value of is used for the subsampling parameter. In the parameter range , morphing results in a perfect match of the morphed forecast feature and the verifying analysis feature. (Notice that for , .) For this range of l, , that is, the magnitude of the location error is equal to the distance between the center of mass of the observed feature and the center of mass of the forecast feature. For the same range of l, indicates somewhat larger errors than . Figure 4 illustrates the cause of this difference between and for . The figure shows that the optical flow is not a simple translational flow in this case, even though a perfect match could also be achieved by a translation in the diagonal direction. Formally, the vector of this translation would be equal to the vector that points from the center of mass of the forecast feature to the center of mass of the observed feature. While , .

Fig. 3.
Fig. 3.

The metrics (black) and (blue) as a function of l in the idealized example of section 3 for . (For , the black and green symbols overlap.) Also shown is (green) as function of l when the value of is computed by using the criterion of section 3b for the selection of F [the value of F selected by the algorithm is also shown as function of l (red)].

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

Fig. 4.
Fig. 4.

Illustration of the morphing process for the idealized example of section 3 for : (a) the original forecast feature and the optical flow, and (b) the morphed forecast feature. Each color identifies a distinct section of the forecast feature and the image of that section in the morphed forecast feature.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

b. Selecting the subsampling parameter

For , both and decrease as l increases. This is an undesirable property of the two measures, because they should always indicate a larger error when the forecast feature is farther away from the observed feature. The ultimate cause of this paradoxical result is that we use a fixed value () of the subsampling parameter, regardless of the distance between the forecast feature and the observed feature: because the pyramid matching algorithm cannot move elementary parcels to a distance larger than , for , an increasing number of elementary parcels stay at their original locations as l increases. Since the morphing vectors for the elementary parcels that stay at their original locations are zero vectors, and decrease as l increases.

Figures 5 and 6 are shown in support of our last argument. The former shows the optical flow and the morphed forecast feature for at different values of F: when , morphing results in a perfect match of the features; but, when , the morphing breaks down, because an increasing number of elementary parcels cannot be reached by morphing. Figure 6 shows the same behavior of the morphing algorithm, but for . The only difference between the two cases is that for , the morphing breaks down when (). The results shown in Figs. 5 and 6 suggest that if the value of F is sufficiently large, morphing can match the forecast feature and the observed features perfectly. In addition, for a sufficiently large the value of F, does not change when F is further increased (Fig. 7). This property of the morphing algorithm provides a testable condition for the optimal selection of F: in a practical computation, F should be increased until no longer increases. However, F should not be increased indefinitely because at one point, becomes larger than the distance between any two points in the verification region. The largest allowable value of F can be determined by prescribing the largest allowable value of .

Fig. 5.
Fig. 5.

Illustration of the morphing process for the idealized example of section 3 for and a gradually increasing value of F (F increases from left to right). (from top to bottom) The forecast feature and the optical flow for 1, and 2; the resulting morphed features; the forecast feature and the optical flow for , 4, and 5; and the resulting morphed features.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for .

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

Fig. 7.
Fig. 7.

The metrics as a function of F for and .

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

Intuition suggests that the objective criterion for the proper selection of F, which we found by the examination of an idealized example, can also be used in a real-life application; and indeed, the results of section 3f will show that our expectation is correct. The only added complication in a realistic situation is that morphing can never be perfect. This will require the introduction of additional criteria to test whether or not an acceptable match has been achieved.

c. Amplitude error

We measure the amplitude error by the following ratio:
e8
For a perfect forecast of the total precipitation in the verification domain, . If the forecast underpredicts the total precipitation, is negative and the smallest possible value of −1 occurs when precipitation is observed but not predicted in the verification domain. Likewise, a positive value of indicates an overprediction of the precipitation, and the largest possible value of 1 indicates that precipitation is predicted, but not observed in the verification domain.

d. Structure error

Our definition of the structure error requires first obtaining a forecast precipitation field (), which is calculated from the original precipitation forecast by correcting it for both the location error and the amplitude error. The location error is corrected by shifting all elementary parcels of the original forecast field () by . The amplitude error is corrected by making the assumption that it is dominantly due to a multiplicative forecast bias. That is, the correction is done by multiplying the field () by the scalar factor:
e9
Then, we define the structure error by
e10
where denotes
e11
which is the root-mean square of an arbitrary scalar field (). If the structure of the forecast feature matches the structure of the analyzed feature perfectly, . If there is a complete mismatch between the structure of the forecast and the analyzed precipitation, .

We note that in some cases where c is significantly smaller than 1 (the amplitude error is large), it is beneficial to make the correction for the amplitude error (the multiplication of the forecast field by c) before morphing. The correction for the amplitude error forces morphing to consider the entire forecast precipitation field. An example for such a case will be shown in section 4a.

e. An integrated measures of the forecast error

The desire to describe forecast quality by a single scalar score often motivates atmospheric scientists to define a scalar measure based on the components of the multivariate (vector) measure. In our case, the multivariate measure has three components, which could be combined into a single scalar measure, for instance, by the following formula:
e12
Because each term of the parenthetical expression of the right-hand side takes a value between zero and one, can take a value from zero to one, with zero indicating that no error of any of the three types is present in the forecast. We believe, however, that the right approach is to treat , , and as the three components of a vector-valued error measure.

4. Validation for ICP cases

The performance of the proposed approach is validated by applications to cases from ICP (Ahijevych et al. 2009). These cases include both idealized geometric examples and real forecasts by multiple models. Our verification results can be directly compared to those reported by KC07 and KC09.

a. Idealized geometric cases

In the geometric test cases, both the forecast and the analyzed (observed) precipitation features have elliptical shapes. Forecast errors are introduced by choosing the center, the area and the aspect ratio of the forecast features differently from those of the analysis features. These errors for the five verification cases are described in the second column of Table 1. This table can be directly compared to Table 1 of KC09.

Table 1.

Summary of the results for the geometric ICP cases (F = 8). The last column shows the rankings of KC07 and KC09 for comparison.

Table 1.

In two cases (cases 3 and 5) the amplitude error is nonzero, because the area of the forecast feature is much larger than the area of the verification (analysis) feature. These two cases are good examples for a situation in which correcting for the amplitude error before morphing can greatly improve the accuracy of the estimate of the location error. As an example, Fig. 8 shows the result of morphing with rescaling and Fig. 9 with no rescaling. When no rescaling is applied, part of the forecast field is left behind by morphing, leading to a spuriously low estimate (8.3 points) of the true location error, which is the 125-point distance between the center of the forecast feature and the center of the analyzed features. With rescaling, the entire forecast feature is shifted by the morphing and the relative error in the estimate of the location error (121.6 points) is a mere 3%.

Fig. 8.
Fig. 8.

Illustration of the result of morphing for geometric case 5. Gray shades indicate the silhouette of the verification feature.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

Fig. 9.
Fig. 9.

As in Fig. 8, but without correction for the amplitude error before performing morphing.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

The estimated location error provides a reasonable estimate of the true location error in all cases: the relative error of the estimate is the largest (11%) for cases 1 and 4, and the smallest (0.5%) for case 3. Because the amplitude error is computed rather than estimated, it shows accurately that the amplitude error is nonzero only for cases 3 and 5. However, because the estimate of the structure error has a dependence on the estimate of the location error, the error in the estimate of the location error also introduces an error into the estimate of the structure error. The nonzero value of the structure error in cases 1 and 2 reflects this error.

There are a number of differences between our results and the results of KC07 and KC09. First, the overall ranking of the forecasts is different, even though the two approaches agree that the forecast is the best in case 1. Second, and more importantly in our view, a comparison of our estimates of the location error and their estimates of the displacement error shows that the two error measures are not compatible. For instance, in their case, the estimated displacement error is the largest for case 1, the case in which the distance between the center of the forecast feature and the center of the verification feature is the smallest.

b. Real forecasts by multiple models

The set of real test forecasts consists of forecasts for nine different cases by three different models (Table 2). The ICP dataset includes both the model forecasts and stage II precipitation analyses (Lin and Mitchell 2005) of the National Centers for Environmental Prediction (NCEP) for the verification of the forecasts. It also includes an expert score for each forecast, which was obtained by averaging subjective scores from 24 experts, who were asked to rate each forecast on a scale from 1 (poor forecast) to 5 (excellent forecast).

Table 2.

Summary of the results for the real ICP cases. The last column shows the rankings of KC07 and KC09 for comparison.

Table 2.

Table 2 can be directly compared to Table 3 of KC09. The rankings of forecast quality based on our integrated score and the integrated score of KC07 and KC09 are obviously different: there is only 1 case (13 May) in which the rankings based on the two scores agree completely, and they agree on the best forecast only in 3 out of the total of 9 cases. Comparing the two sets of integrated scores to the expert scores, we found that our scores were correlated more strongly with the expert scores. To be precise, the correlation was −0.42 for our scores and −0.26 for the scores of KC07 and KC09. While these correlation values are based on small (27 member) samples of scores, and a stronger correlation with the expert scores does not necessarily indicate a better score performance, we can conclude that our scores tend to match the judgement of the experts better.

It is important to note that the results shown in Table 2 were obtained without correcting for the amplitude error before morphing. We found that in the real cases, the correction never had the dramatic effect observed for the geometric cases. In fact, the correction led to a slight drop (to −0.38) of the correlation with the expert scores.

5. Validation for TC forecasts

The performance of the proposed approach is further assessed by applications to forecasts that were produced in real time by the Coupled Regional Climate Model (CRCM) of Texas A&M University. The model domain is shown in Fig. 10. While the verification scores are computed for the entire model domain, the dominant precipitation feature at all verification times is the precipitation system of Tropical Cyclone Isaac. To assess the value of the information provided by the morphing-based scores, we also compute traditional precipitation verification scores, such as the equitable threat score (ETS), the root-mean-square (RMS) error, and the center of mass-based definition of the location error.

Fig. 10.
Fig. 10.

The CRCM model domain, which is also used for the computation of the verifications scores. The subdomain where stage IV analysis data are available is mark by a gray shade.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

a. Forecast data

The CRCM is a coupled version of the Advanced Research version of the Weather Research and Forecasting (WRF) Model (ARW) and the Regional Ocean Modeling System (ROMS) community ocean model. Earlier versions of the model have been used in a series of studies (e.g., Patricola et al. 2012, 2014). The version of the model that produced the 3-km horizontal resolution forecasts that we use in our examples is described in Ma et al. (2014, manuscript submitted to Nat. Geosci.). The initial and the boundary conditions of the model runs were provided by the operational North American Mesoscale Forecast System (NAM) analyses for the atmosphere component and the Real-Time Ocean Forecast System (RTOFS) analyses for the ocean component. A known model artifact of the CRCM simulations is that the model tends to produce a light rain over vast areas (P. Chang 2014, personal communication). To account for this systematic model error, we replace the values of the forecast precipitation that are smaller than 1 mm h−1 by 0 mm h−1 prior to the computation of the verification statistics.

b. Verification data

We use retrievals and analysis data for the verification of the precipitation forecasts. In particular, we combine the TRMM 3B42 (version 7) 3-hourly, 0.25° × 0.25° retrieval product of NASA and the stage IV precipitation analysis product of NCEP.

The TRMM 3B42 (version 7) retrievals are based on microwave (MW) observations at the locations where such observations are available, and on infrared (IR) observations at the location where MW observations are not available (Huffman et al. 2007). The retrievals are more reliable over ocean than land, because the retrieval algorithms can use radiance observations from more channels over the oceans, due to the larger difference between the emissivity of the surface and the emissivity of precipitating clouds (Chen et al. 2013).

The stage IV precipitation analysis product is based on radar and gauge observations of the precipitation over the United States. In particular, the stage IV precipitation analyses are produced by the assimilation of observations by Weather Surveillance Radar-1999 Dopplers (WSR-99Ds) and surface rain gauge observations (Lin and Mitchell 2005). The analyses are available as hourly rainfall accumulations for approximately 4 km × 4 km pixels. (The area covered by the stage IV precipitation analysis is shown in Fig. 10.)

To take advantage of the full coverage of the model domain by TRMM 3B42 retrievals and the higher accuracy of the stage IV analysis over land, we combine the two datasets into a single verification dataset by the following procedure:

  1. The data for the CRCM forecast domain are extracted from both the TRMM 3B42 and the stage IV datasets.

  2. The coarser-resolution forecast and the stage IV analyses are interpolated to the resolution of the TRMM 3B42 grid by using an area-mean interpolation to conserve total precipitation.

  3. Time averaging is applied to both the forecast and the stage IV and forecast datasets to match the 3-h intervals for which the TRMM 3B42 data are available.

  4. The TRMM 3B42 data are replaced by the regridded stage IV data wherever they are available.

c. Equitable threat score and root-mean-square error

For the computation of the ETS, both the forecast and the verification precipitation fields are replaced by fields of 1s and 0s: at a given location, a 1 indicates precipitation over a prescribed threshold, while a 0 indicates no precipitation over that threshold. Then, a 2 × 2 contingency table is prepared that includes counts of four mutually exclusive and collectively exhaustive events:

  • the number of hits, that is, the number of locations where the binary value is 1 in both the forecast and the verification field;

  • the number of misses, that is, the number of locations where the binary value is 0 in the forecast field, but 1 in the verification field;

  • the number of false alarms, that is, the number of locations where the binary value is 1 in the forecast field, but 0 in the verification field; and

  • the number of correct negatives, that is, the number of locations where the binary value is 0 in both the forecast and the verification fields.

The ETS is then defined by
e13
where is the number of hits that can be expected to occur by random chance. It is defined by
e14
where ETS takes a value between (complete mismatch) and 1 (perfect match). For a randomly generated forecast field, ETS is zero.
The RMS error is defined by
e15
where is defined by Eq. (11).

d. Tropical Cyclone Isaac

We verify forecasts from the time period from 0500 UTC 26 August to 0500 UTC 4 September 2012. Tropical Storm Isaac entered the Gulf of Mexico at the beginning of that period from the southeast direction. While crossing the Gulf, Tropical Storm Isaac gradually strengthened and became a category 1 hurricane around 1200 UTC 28 August near the mouth of the Mississippi River. Isaac made its first landfall along the coast of Louisiana on the mouth of the Mississippi River around 0000 UTC 29 August. It caused a relatively large storm surge and produced heavy precipitation. Slowly moving through Louisiana, Isaac weakend into a tropical storm at around 1800 UTC 29 August. It continued its northeastward movement over northern Louisiana, Arkansas, and Missouri and was degraded to a tropical depression around 0000 UTC 31 August just after crossing into southern Arkansas. On 1 September, Tropical Depression Isaac transitioned into an extratropical cyclone after interacting with an eastward-propagating upper-level trough. The main body of Isaac moved out of the CRCM forecast domain on 31 August. The forecast data we examine are from seven forecast runs, which were started at 0500 UTC each day from 26 August to 1 September (Berg 2013).

e. Parameters of the verification statistics

For the computation of the verification statistics, we remap the forecast data to the spatial and temporal resolution (0.25° × 0.25°) of the verification dataset. We impose the constraint introduced in section 2c on the pyramid matching algorithm only in the first iteration of the algorithm. In the constraint, we choose the parameter ε to be 0.5. This value was determined by a subjective evaluation of the morphing results for different values of ε. The subsampling parameter F is determined by the process described in section 3b. We choose the largest distance D over which morphing is allowed to match features to be Dmax = 1177 km, which corresponds to for . For the computation of the ETS, we use a threshold value of 1 mm h−1.

f. Verification results

We discuss the behavior of the verification metrics for four particular forecast examples. Table 3 summarizes the verification scores for these examples, while Figs. 1114 illustrate the individual examples.

Fig. 11.
Fig. 11.

Example 1: (a) the forecast, (b) the verifying data, (c) the morphed forecast, and (d) the forecast corrected for the location error (shifted by ). In (a),(c), and (d), gray shading indicates the silhouette of the precipitation feature in the verification data.

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

Table 3.

Summary of the verification scores for the examples of section 3f.

Table 3.

The four cases are the following:

  1. the 15-h forecast started at 0500 UTC 27 August: Isaac is in the middle of the Gulf of Mexico at verification time;

  2. the 57-h forecast started at 0500 UTC 28 August: making landfall, Isaac weakens into a tropical storm by verification time;

  3. the 36-h forecast started at 0500 UTC 29 August: same forecast situation as in example 3; and

  4. the 51-h forecast started at 0500 UTC 29 August: Isaac weakens into a tropical depression, and quickly moves in the northeast direction at verification time.

In what follows, we first provide a brief overview of the scores. Then, we discuss the most interesting specific aspects of the individual examples with the help of a four-panel figure for each example: the upper-left panel shows the forecast, the upper-right panel shows the verification data, the bottom-left panel shows the morphed forecast, and the bottom-right panel shows the forecast corrected for location error. The latter, corrected forecast is obtained by shifting the original precipitation field by . The purpose of showing this field is to demonstrate that the magnitude of is a valid measure of the location error. Notice that the structure error is computed by first correcting this shifted field for amplitude error.

g. Overview

The ETS, the RMS, and the integrated morphing-based measure, , agree on the ranking of the four forecasts in terms of accuracy (Table 3): the forecast of example 1 is the most accurate, which is followed by the forecasts of examples 2, 3, and 4 in decreasing order of accuracy. We consider the agreement between and the two conventional measures a positive results, because it gives us confidence that the three components of the morphing-based multivariate measure (, , and ) that contribute to provide realistic information about the forecast error.

The multivariate measure shows, for instance, that the forecast of example 4 is the lowest quality forecast by a large margin, primarily due to the large location and amplitude errors. In terms of structure, the quality of the forecast is similar to that of the other forecasts. The forecast of example 1 is the best forecast, because it has low location and amplitude errors, while its structure error is very similar to that of the other forecasts.

h. Example 1

This example (Fig. 11) shows a situation, in which the morphing-based technique indicates zero location error. Of course, it is highly unlikely that the true location error is exactly zero, but the finite resolution of the forecast and the verification datasets limits the smallest location error that a verification technique can detect. Because is a zero vector, Figs. 11a and 11d of the figure are identical. A comparison of Figs. 11a–c suggests that the morphing algorithm is highly efficient in mapping the complex forecast feature into an image that closely resembles the precipitation feature in the verification data. In addition, a comparison of Figs. 11a and 11b shows that the verification metrics correctly indicate small location and amplitude errors.

Figure 11 also shows that the structure of the main forecast precipitation feature over the Gulf of Mexico and the rainbands to the east (Fig. 11a) is somewhat different from that of the related features in the verification data (Fig. 11b). The value of the structure error (0.51) reflects these differences (recall that the value of the structure error is zero for a perfect forecast and one for a completely incorrect forecast of the structure).

i. Example 2

Similar to example 1, the location error is small (73.4 km) in this case. A comparison of Figs. 12a and 12d shows that the aforementioned value is a realistic measure of the location error, because when we correct for the related vector error, the match between the verified and the verification features clearly improves. There is also an obvious error in the prediction of the structure of the precipitation field (Figs. 12a,b), which is reflected by a structure error of 0.499. Finally, the forecast overpredicts the total amount of precipitation, which leads to an amplitude error of 0.426.

Fig. 12.
Fig. 12.

As in Fig. 11, but for example 2. The red arrow in (d) shows .

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

j. Example 3

In accordance with the larger value of in Table 3, the location error is visibly larger (Figs. 13a,b) than in the previous two examples. A comparison of Figs. 13a and 13d shows that provides a realistic measure of the location error. In addition, a larger value of (0.59) suggests that the structure error is also larger than in the previous two examples.

Fig. 13.
Fig. 13.

As in Fig. 11, but for example 3. The red arrow in (d) shows .

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

k. Example 4

As expected, based on the verification scores of Table 3, the location error and the amplitude error are clearly larger than those in the previous three examples, while the structure error is similar. This example shows that the morphing-based approach provides a reliable estimate of the location error, even if it is relatively large (Figs. 14a,d).

Fig. 14.
Fig. 14.

As in Fig. 11, but for example 4. The red arrow in (d) shows .

Citation: Monthly Weather Review 144, 1; 10.1175/MWR-D-15-0172.1

6. Conclusions

In this paper, we described a new morphing-based precipitation verification strategy. The strategy is based on the pyramid matching algorithm, which was first proposed for the verification of precipitation forecasts by KC07 and KC09. The novel aspects of our strategy are as follows:

  • it imposes a constraint on the pyramid matching algorithm to prevent overconvergence toward strong precipitation features during morphing;

  • it introduces an objective criterion for the selection of the subsampling parameter to avoid splitting or distorting features due to an arbitrary maximum displacement limit; and

  • it uses a new multivariate verification metrics to quantify the different aspects of the forecast error.

The components of the new verification metrics (structure, amplitude, and location error) are the same as those of the measure structure, amplitude, and location (SAL) introduced by Wernli et al. (2008). The main differences between our measure and SAL are that we use a different approach (the modified pyramid matching algorithm) to match the verified and verifying objects, and use different definitions of the location and structure error components.

We illustrated the properties of our verification approach by applications to both schematic and realistic examples to demonstrate that it could be used as a practical verification approach.

Acknowledgments

This study was supported by BP/The Gulf of Mexico Research Initiative and the National Science Foundation (Grant ATM-AGS-1237613). The critical comments made by George Craig and two anonymous reviewers on the originally submitted version of our manuscript helped greatly improve the presentation of our ideas. Christian Keil kindly provided us with the code of the morphing algorithm that was used by KC07 and KC09.

REFERENCES

  • Ahijevych, D., E. Gilleland, B. G. Brown, and E. E. Ebert, 2009: Application of spatial verification methods to idealized and NWP-gridded precipitation forecasts. Wea. Forecasting, 24, 14851497, doi:10.1175/2009WAF2222298.1.

    • Search Google Scholar
    • Export Citation
  • Berg, R., 2013: Tropical Cyclone Report Hurricane Isaac (AL092012) 21 August–1 September 2012. Tech. Rep. AL092012, National Hurricane Center, 78 pp. [Available online at http://www.nhc.noaa.gov/data/tcr/AL092012_Isaac.pdf.]

  • Chen, Y., E. E. Ebert, K. J. E. Walsh, and N. E. Davidson, 2013: Evaluation of TRMM 3B42 precipitation estimates of tropical cyclone rainfall using PACRAIN data. J. Geophys. Res. Atmos., 118, 21842196, doi:10.1002/jgrd.50250.

    • Search Google Scholar
    • Export Citation
  • Davis, C., B. Brown, and R. Bullock, 2006: Object-based verification of precipitation forecasts. Part I: Methodology and application to mesoscale rain areas. Mon. Wea. Rev., 134, 17721784, doi:10.1175/MWR3145.1.

    • Search Google Scholar
    • Export Citation
  • Done, J., C. A. Davis, and M. Weisman, 2004: The next generation of NWP: Explicit forecasts of convection using the Weather Research and Forecasting (WRF) model. Atmos. Sci. Lett., 5, 110117, doi:10.1002/asl.72.

    • Search Google Scholar
    • Export Citation
  • Ebert, E. E., 2009: Neighborhood verification: A strategy for rewarding close forecasts. Wea. Forecasting, 24, 14981510, doi:10.1175/2009WAF2222251.1.

    • Search Google Scholar
    • Export Citation
  • Ebert, E. E., and J. L. McBride, 2000: Verification of precipitation in weather systems: Determination of systematic errors. J. Hydrol., 239, 179202, doi:10.1016/S0022-1694(00)00343-7.

    • Search Google Scholar
    • Export Citation
  • Gilleland, E., D. Ahijevych, B. G. Brown, B. Casati, and E. E. Ebert, 2009: Intercomparison of spatial forecast verification methods. Wea. Forecasting, 24, 14161430, doi:10.1175/2009WAF2222269.1.

    • Search Google Scholar
    • Export Citation
  • Huffman, G. J., and Coauthors, 2007: The TRMM Multisatellite Precipitation Analysis (TMPA): Quasi-global, multiyear, combined-sensor precipitation estimates at fine scales. J. Hydrometeor., 8, 3855, doi:10.1175/JHM560.1.

    • Search Google Scholar
    • Export Citation
  • Keil, C., and G. C. Craig, 2007: A displacement-based error measure applied in a regional ensemble forecasting system. Mon. Wea. Rev., 135, 32483259, doi:10.1175/MWR3457.1.

    • Search Google Scholar
    • Export Citation
  • Keil, C., and G. C. Craig, 2009: A displacement and amplitude score employing an optical flow technique. Wea. Forecasting, 24, 12971308, doi:10.1175/2009WAF2222247.1.

    • Search Google Scholar
    • Export Citation
  • Lin, Y., and K. E. Mitchell, 2005: The NCEP Stage II/IV hourly precipitation analyses: Development and applications. 19th Conf. on Hydrology, San Diego, CA, Amer. Meteor. Soc., 1.2. [Available online at https://ams.confex.com/ams/pdfpapers/83847.pdf.]

  • Mass, C. F., D. Ovens, K. Westrick, and B. A. Colle, 2002: Does increasing horizontal resolution produce more skillful forecasts? Bull. Amer. Meteor. Soc., 83, 407430, doi:10.1175/1520-0477(2002)083<0407:DIHRPM>2.3.CO;2.

    • Search Google Scholar
    • Export Citation
  • Patricola, C. M., M. Li, Z. Xu, P. Chang, R. Saravanan, and J.-S. Hsieh, 2012: An investigation of tropical Atlantic bias in a high-resolution coupled regional climate model. Climate Dyn., 39, 24432463, doi:10.1007/s00382-012-1320-5.

    • Search Google Scholar
    • Export Citation
  • Patricola, C. M., R. Saravanan, and P. Chang, 2014: The impact of the El Niño–Southern Oscillation and Atlantic meridional mode on seasonal Atlantic tropical cyclone activity. J. Climate, 27, 5311–5328, doi:10.1175/JCLI-D-13-00687.1.

    • Search Google Scholar
    • Export Citation
  • Romero, R., C. Doswell III, and R. Riosalido, 2001: Observations and fine-grid simulations of a convective outbreak in northeastern Spain: Importance of diurnal forcing and convective cold pools. Mon. Wea. Rev., 129, 21572182, doi:10.1175/1520-0493(2001)129<2157:OAFGSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Speer, M. S., and L. M. Leslie, 2002: The prediction of two cases of severe convection: Implications for forecast guidance. Meteor. Atmos. Phys., 80, 165175, doi:10.1007/s007030200023.

    • Search Google Scholar
    • Export Citation
  • Szunyogh, I., 2014: Applicable Atmospheric Dynamics: Techniques for the Exploration of Atmospheric Dynamics. World Scientific, 608 pp.

  • Weisman, M. L., W. C. Skamarock, and J. B. Klemp, 1997: The resolution dependence of explicitly modeled convective systems. Mon. Wea. Rev., 125, 527548, doi:10.1175/1520-0493(1997)125<0527:TRDOEM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Wernli, H., M. Paulat, M. Hagen, and C. Frei, 2008: SAL—A novel quality measure for the verification of quantitative precipitation forecasts. Mon. Wea. Rev., 136, 44704487, doi:10.1175/2008MWR2415.1.

    • Search Google Scholar
    • Export Citation
  • Zinner, T., H. Mannstein, and A. Tafferner, 2008: Cb-TRAM: Tracking and monitoring severe convection from onset over rapid development to mature phase using multi-channel Meteosat-8 SEVIRI data. Meteor. Atmos. Phys., 101, 191210, doi:10.1007/s00703-008-0290-y.

    • Search Google Scholar
    • Export Citation
1

In the atmospheric sciences, the pyramid matching algorithm was first used for the computation of motion vectors from satellite images of clouds (Zinner et al. 2008). In that application, was defined by the intensity (brightness temperature) of the pixels of the satellite images.

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