Editorial Type:
Corrigendum
Article Type:
Correction

Corrigendum

Brandon R. Smith aCooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma
bNOAA/OAR National Severe Storms Laboratory, Norman, Oklahoma

Search for other papers by Brandon R. Smith in
Current site
Google Scholar
PubMed Close
,
Thea Sandmæl aCooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma
bNOAA/OAR National Severe Storms Laboratory, Norman, Oklahoma

Search for other papers by Thea Sandmæl in
Current site
Google Scholar
PubMed Close
,
Matthew C. Mahalik aCooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma
bNOAA/OAR National Severe Storms Laboratory, Norman, Oklahoma

Search for other papers by Matthew C. Mahalik in
Current site
Google Scholar
PubMed Close
,
Kimberly L. Elmore aCooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma
bNOAA/OAR National Severe Storms Laboratory, Norman, Oklahoma

Search for other papers by Kimberly L. Elmore in
Current site
Google Scholar
PubMed Close
,
Darrel M. Kingfield aCooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma
bNOAA/OAR National Severe Storms Laboratory, Norman, Oklahoma

Search for other papers by Darrel M. Kingfield in
Current site
Google Scholar
PubMed Close
,
Kiel L. Ortega aCooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma
bNOAA/OAR National Severe Storms Laboratory, Norman, Oklahoma

Search for other papers by Kiel L. Ortega in
Current site
Google Scholar
PubMed Close
, and
Travis M. Smith aCooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma
bNOAA/OAR National Severe Storms Laboratory, Norman, Oklahoma

Search for other papers by Travis M. Smith in
Current site
Google Scholar
PubMed Close
Online Publication:
04 Jun 2021
Print Publication:
01 Jun 2021
DOI:
https://doi.org/10.1175/WAF-D-20-0125.1
Page(s):
1131–1133
A related article is available
Full access

Mahalik’s current affiliation: Cherokee Nation Strategic Programs and NOAA/OAR Weather Program Office, Silver Spring, Maryland.

Kingfield’s current affiliation: Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, and NOAA/OAR/ESRL/Global Systems Laboratory, Boulder, Colorado.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Brandon R. Smith, brandon.r.smith@noaa.gov

Mahalik’s current affiliation: Cherokee Nation Strategic Programs and NOAA/OAR Weather Program Office, Silver Spring, Maryland.

Kingfield’s current affiliation: Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, and NOAA/OAR/ESRL/Global Systems Laboratory, Boulder, Colorado.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Brandon R. Smith, brandon.r.smith@noaa.gov

The purpose of this corrigendum is to address errors in several equations that were discovered in Mahalik et al. (2019, hereafter M19), which presented the derivations for both the azimuthal and divergent shear linear least squares derivative (LLSD) equations that serve as the foundation for the publication. While two of the errors are minor and typographical in nature, additional errors in two of the matrices, the coefficient matrix [M19’s Eq. (8)] and the adjugate matrix [M19’s Eq. (10)], introduce incorrect components within the final versions of the LLSD horizontal shear equations provided in M19’s Eqs. (12a) and (12b), in addition to their fully expanded forms in M19’s appendixes A and B. The errors and their subsequent corrections are addressed below in the order that they are encountered within M19.

The first two equation errors in M19 are both typographical in nature. The Δ (delta) modifier was incorrectly excluded from the range component of the two-dimensional radar variable uij, shown in M19’s Eq. (5). This delta is needed as it describes the offset in range from the center of the kernel in which the derivatives are being calculated. The updated M19’s Eq. (5) is as follows:
R=∑k=0m×nwk[u⁡(Δrk,Δθk)−(u0+urΔrk+uθΔθk)]2.
Similarly, the Δ (delta) modifier was incorrectly excluded from the azimuth offset in M19’s Eq. (6c):
∂R∂u0=0=∑k=0m×n2⁡(−uk+u0+urΔrk+uθΔθk)wk.
The remainder of the equation errors are attributed to an incorrect setup of both the coefficient matrix used to express the complete system of LLSD equations and its associated adjugate matrix. M19’s Eq. (8), which outlines the complete system of LLSD equations expressed in matrix form, contains three matrices, the details of which are outlined in M19. Out of the three matrices, only the coefficient matrix contains errors. These errors relate only to the cross-diagonal coefficients, which are incorrectly switched across the diagonal, not allowing the original LLSD equations in M19’s Eqs. (7a)–(7c) to be generated when the coefficient and variable matrices are multiplied. To correct this, these cross-diagonal coefficients are held in their original element positions, producing the corrected M19’s Eq. (8) below:
[ΣwkΔrkΔθkΣwkΔrk2ΣwkΔrkΣwkΔθk2ΣwkΔrkΔθkΣwkΔθkΣwkΔθkΣwkΔrkΣ wk][uθuru0]=[ΣwkΔrkukΣwkΔθkukΣwkuk].
As discussed in M19 and shown by M19’s Eq. (9), the derivatives of the individual LLSD azimuthal and divergent shear components are found by multiplying the answer matrix by the inverse of the coefficient matrix. This inverse is found by calculating both the determinate and adjugate of the coefficient matrix. The adjugate matrix shown in M19’s Eq. (10), however, is not set up correctly. Part of the process of creating the adjugate matrix involves switching the cross-diagonal elements. In the case of M19’s Eq. (10), only the a23 and a32 elements were switched. To correctly create the adjugate matrix, the a12 element must be switched with the a21 and the a13 element switched with the a31. Adjusting M19’s Eq. (10) to account for these errors produces the following corrected equation:
M−1=1D adj⁡(M)=[a11a21a31a12a22a32a13a23a33]⁡(1D).
Since the adjugate matrix shown in M19’s Eq. (10) had to be corrected, all downstream equations have been adjusted using the corrected adjugate matrix. The corrected versions of these downstream equations are displayed below where Eq. (5) is the corrected form of M19’s Eq. (11), Eq. (6) is the corrected form of M19’s Eq. (12a), and Eq. (7) is the corrected form of M19’s Eq. (12b):
[uθuru0]=[a11/Da21/Da31/Da12/Da22/Da32/Da13/Da23/Da33/D]⁡[ΣwkΔrkukΣwkΔθkukΣwkuk],
uθ=∑k=0m×nwkΔrkuk⁡(a11D)+∑k=0m×nwkΔθkuk⁡(a21D)+∑k=0m×nwkuk⁡(a31D),
ur=∑k=0m×nwkΔrkuk⁡(a12D)+∑k=0m×nwkΔθkuk⁡(a22D)+∑k=0m×nwkuk⁡(a32D).
The corrected versions of M19’s appendixes A and B, the expanded forms of M19’s Eqs. (12a) and (12b) [Eqs. (6) and (7) above, respectively] are shown, respectively, in Eqs. (8) and (9) below. Using Eq. (3), the original determinate in M19’s appendixes A and B was also corrected and is also shown in Eqs. (8) and (9):
uθ=(∑k=0m×nΔrkΔθk∑k=0m×nΔrkΔθk∑k=0m×nwk−2∑k=0m×nΔrkΔθk∑k=0m×nΔθk∑k=0m×nΔrk+∑k=0m×nΔrk2∑k=0m×nΔθk∑k=0m×nΔθk−∑k=0m×nΔrk2∑k=0m×nΔθk2∑k=0m×nwk+∑k=0m×nΔrk∑k=0m×nΔθk2∑k=0m×nΔrk)−1[∑k=0m×nΔrkuk⁡(∑k=0m×nΔrkΔθk∑k=0m×nwk−∑k=0m×nΔθk∑k=0m×nΔrk)+∑k=0m×nΔθkuk⁡(∑k=0m×nΔrk∑k=0m×nΔrk−∑k=0m×nrk2∑k=0m×nwk)+∑k=0m×nuk⁡(∑k=0m×nΔrk2∑k=0m×nΔθk−∑k=0m×nΔrk∑k=0m×nΔrkΔθk)],
ur=⁡(∑k=0m×nΔrkΔθk∑k=0m×nΔrkΔθk∑k=0m×nwk−2∑k=0m×nΔrkΔθk∑k=0m×nΔθk∑k=0m×nΔrk+∑k=0m×nΔrk2∑k=0m×nΔθk∑k=0m×nΔθk−∑k=0m×nΔrk2∑k=0m×nΔθk2∑k=0m×nwk+∑k=0m×nΔrk∑k=0m×nΔθk2∑k=0m×nΔrk)−1[∑k=0m×nΔrkuk⁡(∑k=0m×nΔθk∑k=0m×nΔθk−∑k=0m×nθk2∑k=0m×nwk)+∑k=0m×nΔθkuk⁡(∑k=0m×nΔrkΔθk∑k=0m×nwk−∑k=0m×nΔrk∑k=0m×nΔθk)+∑k=0m×nuk⁡(∑k=0m×nΔrk∑k=0m×nθk2−∑k=0m×nΔrkΔθk∑k=0m×nΔθk)].

It should be noted that while M19 did not include the correct final version of the LLSD azimuthal shear (AzShear) equation, the correct version is employed in the operational code set that is utilized in the workflow to generate the rotation track products within the current Multi-Radar Multi-Sensor (MRMS) system. In addition, the figures in M19 are unaffected by these equation errors.

Acknowledgments

We thank Monika Freeman, with the Environmental Remote Sensing Laboratory at École Polytechnique Férédale de Lausanne and the Radar, Satellite, and Nowcasting Division at MeteoSwiss, who discovered the errors and notified the authors while providing documentation of the work that had been done to uncover the errors. Monika’s documentation helped immensely in the diagnosis and subsequent correction of the incorrect equations. Funding was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA16OAR4320115, U.S. Department of Commerce.

REFERENCE

Mahalik, M. C., B. R. Smith, K. L. Elmore, D. M. Kingfield, K. L. Ortega, and T. M. Smith, 2019: Estimates of gradients in radar moments using a linear least squares derivative technique. Wea. Forecasting, 34, 415–434, https://doi.org/10.1175/WAF-D-18-0095.1.

  • Crossref
  • Search Google Scholar
  • Export Citation
Save
Cover Weather and Forecasting
Weather and Forecasting

  • Mahalik, M. C., B. R. Smith, K. L. Elmore, D. M. Kingfield, K. L. Ortega, and T. M. Smith, 2019: Estimates of gradients in radar moments using a linear least squares derivative technique. Wea. Forecasting, 34, 415–434, https://doi.org/10.1175/WAF-D-18-0095.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 672 0 0
Full Text Views 529 290 88
PDF Downloads 238 83 3