jonathan e. martinmarrella.aos.wisc.edu/mclay_and_martin2005i.pdf · 1 using filtering to mitigate...

47
1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of a Prototype Filtering Scheme. Justin G. McLay 1 NRC/Naval Research Laboratory, Monterey, California Jonathan E. Martin University of Wisconsin, Madison, Wisconsin (Submitted to Monthly Weather Review, November 29, 2004) 1 Corresponding Author: Justin G. McLay, Naval Research Laboratory, 7 Grace Hopper Ave., Stop 2, Monterey, CA, 93943-5502. E-mail: [email protected].

Upload: others

Post on 17-Aug-2020

1 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

1

Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble

Covariance. Part I: Evaluation of a Prototype Filtering Scheme.

Justin G. McLay1

NRC/Naval Research Laboratory, Monterey, California

Jonathan E. Martin

University of Wisconsin, Madison, Wisconsin

(Submitted to Monthly Weather Review, November 29, 2004)

1 Corresponding Author: Justin G. McLay, Naval Research Laboratory, 7 Grace Hopper Ave., Stop 2, Monterey, CA, 93943-5502. E-mail: [email protected].

Page 2: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

�����������

�� �������������������� ��"!��#�$�%��&'�(�*),+.-*�����/���$���#�����%�# ��0����&21��3�$�#��42�0�#���5�768�9� �+;:<+�=>��&��$��6�?�@����+��$�/-*���$+BA3���#�$��&�4C:D���#��4�&��3:E�#+F6G��� ��42�0�#�H�# ��I�3JK��A9�H+�=L�$�#+BAM ����$�#��AI6G+B:��3@��3�#�$+��#��+�&N��&��$��6�?�@��A�+O10���$�P��&�A3��Q��� ��56G�3�# �+B:C�����/=R+��#6G�S�T�$���$�����U+0=!�@�� �3�#��&�4"�WVD�����$��6G��&2� �S)L�X� F�# ���+������#�0� ��+�&���@��&D�#��6%?�@��868�36�?K�3�#�3Y�+�?D� ����&���&�4(�Z�#�9��+�=�=R+��$��A����$�%�$�M�.� ����)L ���AM <���%@����#��A�+��$�#[��B� ��:N?2\]���$�#+��$�+�=,�$� +;AM ����/� ��A"+��$��4���&^Q8_`�a� D��&<[��$�����#+�6G�8&b[�6%?K�3��+�=,� ��H!�@��#���$��:'�/�M�0�#���a� +7A�+�6G��@��368�3&2��+���#[D����@P�0&b�L�# ���+��K�3� �0�#��+�&���@c6G��6�?����$��Y�=R+��$68��&�45�5�#+.-*Ad�0@�@��3:I 2\B?D�#��:7��&D�#��6%?�@��0Q

_*&�� D�������0�K�3�e� D�>68�9� �+;:�������&2� �$+B:D[�A��3:5��&�:H�f�D�#+��#+��*\;�K�>!�@X� �3�#��&D4f�#AM D��6G�����g�3�$� ��?�@����# ��3:=R+��h[��$�i��&5� ��j68�9� �+;:^kl���310��@�[��0� ��+�&^Qm�� ��i�9nGAd��A9\H+�=c�# ���!�@��#���#��&�4U�#AM D��6G�f���h4���[�4��3:G�# ��#+�[D4� A�+�68��+��#�X� �"��&�:<�3&��#�36�?�@��W-*?2\2-*�3&��#�36�?�@��IA3+�6G�����#���#+�&D�%?��3�*)�����&E� D�T�#+;+��$-`68����&B-`�#ob[����$�8�3�#�#+��AM ����#��A3�#���$���$�#��A3�i+�=g� ��a�#�9�S+�=�!�@�� �3�#�3:Z�$� �0� �3�S��&D:(� D+��#��+�=g�# ��%+������#�0� ��+�&���@p�3&��#�36�?�@���Q��� ��3�#�A�+�68���0�#���$+�&��%���#�8?����#�3:E[��K+�&N�(\�������kl��)�+��$�# N+�=L4�@�+�?���@���&��$��6�?D@��T:��.�M�DYh�0&�:N�#[�4�4����/��� ��.�� D��!�@��#���#��&�4"�$AM ���6G�HAd��&C��&F=q��A3�SA�+�&��$���/� ��&2�#@�\F�D�#+;:�[�A����T�#�3��+0=��/�M�0�#���S)L ���AM r���#��4���&��3� ��@�@�\@��3�#�iA�+��$�#[��D�#��:C?2\(�$� +;AM ����/� ��A%���$�#+��$�i�# ���&Z�# ��%+������#�0� ��+�&���@p6G��6%?K�3�#�S�0�#��QfsB+�6G�a[�&�:��3�#@�\;��&�4���$+��K�3�$�#�����a+�=�� ��G���#+0� +��*\;���G!�@X� �3�#��&D4(�$AM ���6G�T���$�"�0@��#+t ���4� �@���4� 2� �3:^Q"�� D���a���a��A�A3+�6G��@����# ��3:?2\N��&21����$�#��42�.� ��&D4F� ��I!�@X� ���$��:E�$� �0� �3��kh�����/=R+��#68��&�A3�T=R�#+�6u� ��t�����#�$�K�3A3�#��1��3�8+0=f���$+�?���?D��@����$�#��A6G+B:��3@�@���&�4%��&�:7�#+0-`Ad��@�@���:t�0&�+�68��@�\8���#+wv$��A3�#��+�&cQ

Page 3: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

x

y�z {�|�}O~w�p�e�m��}w�R��|

_*&I&b[�6G���#��Ad��@�)��d�0�# ������D�#��:D��A3�#��+�&G� ��f��&��X� �P�0@��$� �0� �SA���&T+�&D@�\G?K�j=R[�@�@X\H:����$A��$��?���:I��&G� �3�#6G�+�=��#+�6G�5���$+�?���?���@��X�*\(:D��&��$���*\C=R[�&�A9� ��+�&������U�h�WQ^�f@��#+�Y����$?����#� ���$��@X\(�$6"�0@�@g�3�#�$+��#�U��&��# ��H��&��X� ����@�$� �0� ��68�d\T�9VB ���?����L@����#4��S4��#+O)��# ^Qh�� ����$���*)�+8+�?��#�3�$10�0� ��+�&����9Jc�3A3� �X1��3@�\I�3@���6G��&��0� �i��&2\IAM ���&DA���0�j�#[DA�A��3�#�/=R[�@�:��9� ���$68��&����/� ��AS),���0� ��3�i=R+��$��A����$�#�f�9VBA3���D�f+�&(�# �+��#��� ���$��+BA3Ad���$��+�&��L)L ��3�#����&Z� D�=R+��$��Ad�0�$�T�91�+�@�[D� ��+�&����8�$��@��0� �X1���@X\N��&D�#��&D�#���#��1��7�#+<��&��X� ����@,�/�M�0�#�Z[�&�A3���/�M����&b�*\�Q�_*&�1B���3)�+�=f� �����#����@��X�*\�Y�),���0� D����=R+��#��A����$�#�%�$ �+�[�@�:'?��8���$+�?���?D��@����$�#��A���&'&��0�#[��#�0Qt�h&��#�36�?�@��8���#�3:���A9� ��+�&]+��#��4��X-&��0�#��:C�0�S�"6G�d��&D�i� +8���/� ��68�0�#�a�# ���=R+��$��A����$�S��U��4���1���&F�$+�68�a��&D��� ����@p���U�����$�#+;A����0� �3:()L���# � D�%[�&�A3���/�M����&b�*\(��&7� ����#����A3��!�A��0� ��+�&Z+�=h�# ��%��&����#�P��@c�$�M�.� �I�q�p�����# '�O���0��YK�(+�@X� ��&D�e�9�U��@�Qa�d�����DY�m+�� r��&�:r����@�&��d\<�O�������9Q�_*&r�D� ��A9� ��A3��Yp� D�568�9� �+;:���&21�+�@X1��3�U�K�3�$=R+��$68��&�4I�I�#�3�#���3��+�=,6G+;:���@��&2�#��4��#�0� ��+�&��S=R�#+�6��#+�6G��!�&��X� �H�#�9�%+0=,��&����#�P��@eA�+�&�:D��� ��+�&��U+�?B�M����&���:C� D�#+�[�4� ]�����/� [��$?��0� ��+�&�+�=� D��?����/�i�3�$�#��68�0� ��+�=m� ��a��&D��� ����@��$�M�.� ��Q��� ��a�#�3�i+�=m=R+��#�3Ad���/�f�/�M�0�#���j���#+O1;��:D��:Z?2\I�# �����&2� �34�� �.-� ��+�&��e:��3!�&D���h� ��,���$�#��68�0� �,+�=�� ���=R+��#�3Ad���/���U��Q��h�#+�?���?���@����*\U=R+��#��A����$�#��=R+��h���$?����#� ���/\�),���0� ��3��31���&2� �iAd��&7?��a:D���#�X1��3:7=R�$+�6��# ������$�#��68�0� �U+0=e�# ��U=R+��#�3Ad���/�f���U�,Q

�� ��T�9Jc�3A3�#��1��3&����$�5+�=��0&<��&��$��6�?D@��"�D�#��:D��A3�#��+�&r�$\;�/� ��6u���a:��3�K�3&�:���&2�5[D�K+�&]�*),+Z=q�0A3� +��#��� �+O)�)���@�@L� D�(��&����#�P��@����U� ���G:��3!�&D��:¡��&�:�� ��6G��@���:^Y��0&�:� �+O)�)���@�@j�# ��Z&b[�6G���$��Ad�0@j6G+;:���@�#��6�[�@��0� �3�S� ��H:D\;&���6G��A3�a��&�:r�� 2\B�$��A3��+�=�� ��H�$�d��@��0�#68+��#�� ��3�#�0Q8�� ��G4��$�d�0�#�������#+��K+��/� ��+�&C+�=�d�0�#@�\��9Jc+��$� ����&2� +7��&D�#��6%?�@��8���#�3:���A9� ��+�&]),���%:��91�+��#��:<�#+Z:��9!�&���&D4F��&�:'� ��6G��@���&�4I� ��8��&��X� ����@���U�,Q^�¢@����#4���?K+;:D\Z+�=,�$��A3��&2��),+��$£F ������$[�4�4��3�$� �3:^Y� �+O),�91��3��Ye�# ��0��6G+B:D��@g�3�#�$+��#�UA���&] ��O1��A�+�&��#��:����#��?�@��7:��9� �#��68�3&2�M��@L��6G����A9�"+�&�� D�Z�#£b��@�@L+�=U�3&��#�36�?�@��W-*:��3�#�X1���:¤�D�#+�?��0?���@����*\'=R+��#��A����$�#��¦¥,+�@�[�A3A��h��&D:�§>��[�6G ��3=R&D���5�O����¨DYp§,[���©�©��T�3�a��@�QG�d�����DYcªf���$�#���$+�&C�3�a�0@¦QH�O�����BYp�g��@�68�3�f�9�a��@¦Q�O�����DYps;�#��&��$�#[�:r�3�a�0@¦Q5�O�����DYp��10��&��U�3�a�0@¦Q ��«�«�« Yp�g�0@�6G��� �0«�«�« Y�¬S�$�#��@�@e�9�a��@¦Q �0«�« ��Yp�e��@�6G�����«�« ��Y;­'��&�:D���# D��&T�9�L��@¦Q ��«�« ��Y;�f@� ���6G��:T�9�L��@�Q ��«�«2� YD§,�$��4� 2�>��&�:7�([�@�@��3& ��«�«2� Y��7\;@�&��S�3����@¦Q��«�«�� Y�§>���#£b6G���®v$���L�3�S��@¦Q ��«�«�x YK¬S�#�#�3@�@ ��«�«�x �9Q��� b[��3Yc�X�f ��0�f?���A�+�68�aA�@��d���L� ��.�f�X=g� D�%[D�#��@��X�*\+�=��3&��#�36�?�@��W-*:��3�#�X1���:C�D�#+�?��0?���@����*\I=R+��$��Ad�0�$� �f���i�#+T?���=R[�@�@�\7�#����@���©���:cY��# ���&F�# ���AM ����#��A3�#���$���$�#��A3���&�:��$+�[��#A3����+�=�6G+B:D��@g�3�#�$+��#�U6�[D�$��?�����:��3&2� ��!���:^Y���&D:C68����&��U+�=6G��� ��42�0�#��&�4H+��S�3@���68��&��0� ��&�4

Page 4: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� D���#�a���$�#+��$�L6�[��$�L?���:��31;���#��:Z��&�:7��6G��@��368�3&2� ��:cQ��+����3@P��?�+�� �.� ��+�&'+�&<�# ��I68+;:��3@,���$�#+��%���#+�?D@���6¯A�+�&D�#��:D���%�*)�+�:���JK���$��&2�5=R+��#�3Ad���/�5� �#�wv$��AW-

� +��#���3��Ym+�&��G�31�+�@�1���:'=R�$+�6°�(4���1��3&'��&����#�P��@e�$�M�.� �G[��#��&�4Z�7&b[�68�3�#��Ad��@h6G+B:D��@���&�:]+�&��G�31�+�@�1���:=R�#+�6±�# ��5� ��6G����&����#�P��@p�/�M�0�#�5[��#��&�4I�#����@��0�#68+��#�� ��3�#��A�:D\;&���6G��A3�U��&�:C�� 2\;�#��A��8���e��4�Q5�w�WQ��� ��:��XJc�3�#��&DA����0���$+�6G�S� ��6G�i²�?��3�*)�����&t� D�U�$�M�.� �U+�&I� D�U&;[D68�3�#��A���@c6G+;:���@�k³�=R+��#�3Ad���/��� � �Ov$��A3�#+��$\��&�:T�# ��S�/�M�0�#�S+�&"�# ��S�#����@^�.� 6G+��#�� D���#��Aj=R+��#��A����$�,� � �Ov$��A3�#+��$\8�#�3���#�3�#�3&b�#��� ��S�3�#�$+��>��&2�#�#+;:�[�A��3:?2\"�# ��S6G+B:��3@c�0�>+�=p� ��68�L²f���g��4�Q � �WQ��� ����>���$�#+��>68�d\G ��d1��S�*)�+GA3+�6G�K+�&D��&2� �3YD+�&��f+�=��$\;�$�#��65-�0�#��Aj+��#��4���&G��&�:"+�&��f+0=p�$�#+BAM ��0�$� ��Af+��$��4���&^Qm�� ��S���$�#+��A�+�68��+�&��3&b�>���#�$+BA3�P�0�#��:")L�X� 8�$\;�$�#��68�0� ��A���$+BA3���#�$���8 ����5� ��t�#��68�t:����#�3A3� ��+�&���&�:�6"�04�&����#[�:��T=R+��5����AM ��3&��#�36�?�@��76G��6%?K�3�7���g��4�Q x �9Q�� ����%[�&��X=R+��#6G���*\F�#���$[�@��#�%��&]� D�"6G�d��&]+�=�� ��8��&��$��6�?D@��":D���$�#�#��?�[D� ��+�&r?K�3��&�4(:����$��@P��A3��:r=R�#+�6� ��0�a+�=>� D�8�$�d��@��0�#68+��#�� ��3�#��AG:D���$�#�#��?�[D� ��+�&^QG�� ��"�3&��#�36�?�@��8:����$�#�#��?D[D� ��+�&������ ����:r�#+Z?��H´�?��X-���$��:�´C��&]� D���%A3���$A�[�6G�$� ��&�A��0QF�� ��T�3�#�$+��%A3+�68��+�&��3&2�5���$�#+;A��P�.� ��:])L��� '�$� +;AM ����/� ��AT���#+;A��3�#�$���:�+;���i&�+��S&D��A��3�#�#���#��@�\( ��d1��%�# ���� �068��:����$��A3�#��+�&Z��&�:F68��4�&D��� [D:��a=R+��i����AM C��&��$��6�?D@���6G��6�?�������e��4�Q��2�9Qh�� ����3JK��A9�j+�=e�# ������3�#�$+���A�+�6G��+�&���&2�L���,� +HA�+��#�#[D�D�L� ��S10���$�P��&�A3����&�:tA�+O10���$�P��&�A3�U+�=� D����&��$��6�?D@��5:����$� �$��?�[B� ��+�&^QU��+��f��@�@�[��$�#� �0�#��+�&^Y�A3+�&��$��:��3�S� ��0�i� ��5A3+O1��0�#�P�0&�A���+0=h�*),+t� �0&�:�+�610���#����?�@����,µ¶��&�:7·¢�����WVD�D�#���$�#�3:����

¸�¹wº �RµF» ·��½¼ ¾¤¿ µ�·�À�Á°¾¤¿ µFÀ,¾¡¿l·�ÀS» �Â�w�

)L ��3�#�S¾¡¿ À������ ��j�9VB����A9�M�0�#��+�&8+��K�3� �0�#+���Qg_`=p�$�#+BAM ��0�$� ��Ad��@�@�\2-*��&�:�[�A3��:G���$�#+��$�,ÃH��&�:"Ä����#�i��:�:D��:� +Iµ ��&�:]·8Y��#�3�#����A9� ��1���@X\�Y�� ��3&]� D�GA3+O1��0�#�P�0&�A��5+�=�� ��H�$���#[D@�� ��&�4tA�+��$�#[��B� ��:�1��3�#�$��+�&��a+�=µ��&�:(·��q�$�3=R�3�#�#�3:(�#+8�0�LµCÅi��&D:(·%ÅM�,���

¸�¹wº �Rµ Å » · Å � ¼ ¸a¹.º �RµZ»M·��"Æu¾¤¿®µ�Ä À�Æ°¾¡¿Ç·%Ã�À�ÆȾ¡¿lÃbÄ ÀS» � � �

)L ��3�#�j����������$�#[�6G��:G� ��.��# ��j�9VB�K�3A3�#��:"10��@�[���+�=^?�+��# "Ã��0&�:"Ä>����©����$+�Q��� b[��3YD�$�#+BAM ����$�#��Ad�0@�@�\2-��&D:�[�A��3:I���$�#+��$�,)L��@�@c��@X� �3��# ��UA�+O10���#����&�A3�f+�=m� �0&�:�+�6¢10���#����?�@������# ��#+�[�4� "�# ��U@P���/�,�# ��#�3�f� �3�#6G�+�&I� D�a�$��4� 2�/-* ���&D:7�$��:���+�=g�hoÉQ � Q

�(�9� �+;:��T+�=f:��d�0@���&D4r)L�X� �6G+;:���@L�$\;�/� ��68�0�#��A(��&�:��/� +;AM ����/� ��A(���$�#+��$�"��&���&D�#��6%?�@��(���#�W-:���A3� ��+�&FA���&�?K��A3@P���$�#�X!���:r���S�#+T)L ��3�# ����U� D�3\F+��K�3� �0�#��:�[��$��&�4T�# ��5A�+�[D�#�#�H+�=��# ���&b[�6G���#��Ad��@

Page 5: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Ê

��&2�#��4��#�0� ��+�&��+��>�0=Ë�#���� ��i��&2� �34�� �0�#��+�&D�,���$�fA3+�68�D@��3�#��:^Q�7�3� D+B:��,+�=^� D�f=R+��#6G���A�@����#�,��&�A3@�[�:��� D+��#�L+�=c§,[���©�©d�S�9����@¦Q,�/�O�����2�g��&D:"§,�$��4� 2���0&�:8�([�@�@���&(� �0«�«2� �WY�)L ���AM G��&DA�+��$�K+��#�0� �>��:�:��X� ��+�&���@� �3�#6G����&r�# ��G&;[D68�3�#��A���@h6G+B:��3@�� +(68�X� ��4��0� �%�# ��G�3JK��A3��+�=��/� +;AM ����/� ��AG���$�#+��$����&r�# ��G68+;:���@�k³������#��6G�3� �3�#��©���:7A�+�[���@���&�4G?K�9�*),�3��&F[�&D�#���$+�@�1���:B-���&�:(�#���$+�@�1���:B-`�#A���@���Ì�+O)�Q��f@��#+G��&t� ���=R+��#6G���A�@����#�L���L�# ���´�=R+��$A���&D4G�$��&�4�[�@P����1���A9� +��$´"6G�3�# �+B:7+�=g§>���$£;6G���®v$�����9�S��@�Qj� ��«�«�x �9Y�)L D��AM (���#�$���#�$���� D�a�9Jc�3A3�j+�=g�$\;�$�#��68�0� ��AU�K�3�$� [D�#?��0�#��+�&D�>� +G68+;:���@Í-`10���$�P��?�@��L�� 2\B�$��A���@p� ��&D:���&�A3���3��Q

�(�9� �+;:��>� ��0�>A3+�68����&D� �0�#�f=R+��,6G+;:���@����$�#+��$���0=Ë� �3�,� D�S&b[�68�3�#��Ad��@���&2� �34�� �.� ��+�&��,�0�#�SA3+�6H-��@��3� �3:G���#�>�$�3=R�3�#�#�3:G�#+����h´���+��$�/-*���$+BA3���$�#��&D4�´U68�9� �+;:��3Q�¬S&���+�=É� ��3�#�0Yb�#���$68�3:H�# ���Ad�0@���?D� �0�#��+�&6G�3� D+B:^YK��:.v$[D�$� �i��&��$��6�?D@��9-`:����$��1���:F�91���&2�����$+�?���?D��@��X� �����>��&F��A3A�+��$:���&�A3�%)L���# Z�#+�68�a:����/� �$��?�[B-� ��+�&]�q���0� ��6G�3�#�#��AU+��j&D+�&������#��6G�3� �$��Ad�� ��.�i:D���#A3�#��?K�3�i� D�a D���$�#+��#��Ad��@c@���£0��@��� �+;+B:8� ��0�L�# ��U1����Â-��=Ë\;��&�4"�91���&2�U)L��@�@e���#�$[�68���"A��3�$�M�0��&F@�+BA��0� ��+�&(��&Z� D�� ����3� ���$AM 2\Z+�=��3&��#�36�?�@��9-*:D������A3� �3:��31��3&2� ���ªi��6G��@�@c��&�:F¥,+�@�[�A3A����O����¨DYÉ�hAM£���@e��&�:7­<�0@�� �3�#���O����¨DY�ªf��6G��@�@ ��«�« �w�9Q,¥>��@���?�� �.� ��+�&I���j��?�@����#+6G��� ��42�0�#�G�# ��t�3JK��A3�#�H+�&E��&��$��6�?�@��9-`:����$��1��3:��D�#+�?��0?���@���� �����a+�=j?K+0� E68+;:��3@,�$\;�/� ��68�0�#��At���$�#+��$���&�:8��&�A3+��#�$��A9�,�3&��#�36�?�@��L10���#����&�A3���Ë)L ���AM 8��������$�#@�\��# ��j���#+;:�[�A9�>+�=c68+;:���@��/� +;AM ����/� ��Ai���$�#+��$�M�9Q�f&D+�� ��3����+��/�$-*�D�#+;A����$�#��&�4(6G�3�# �+;:r���U� ��H=R+��#68�0�#��+�&F+0=��t6�[�@��#�X-`68+;:���@m��&D�#��6%?�@��Z��ªi���#�$���#+�&�3�i��@�Q>�O�����DYÉs;�#��&��$�#[�:(�3�i��@¦Q,�O�����DY��h10��&D�j�9�i��@¦Q �0«�«�« YD­'��&�:D���# D��&I�3�i��@�Q ��«�« ��Y��7\;@�&D�a�9�i��@¦Q��«�«�� Y^�f@� ���6G��:C�9�a��@¦Q ��«�«2� �9Qa�� ����U��&b1�+�@�1����S�# ��5A�+�6%?���&��.� ��+�&F+�=��&��$��6�?D@��H68�36�?����#�i=R�#+�6�*)�+T+��j68+��$��:���JK���$��&2�f�3&��#�36�?�@��%���$��:���A3� ��+�&7�$\;�$�#��6G��Yc+�&(� D�%?����$���L� ��.�i�# ���6G��6%?K�3�#�L=R�#+�6+�&��j�$\;�/� ��6Î68�O\�&�+���?K�j�#[�?bv$��A3��� +�� ��i� ��6G�L68+;:��3@��$\;�$�#��68�0� ��Af��&D:"�$�#+BAM ����$�#��Aj���#�$+���3JK��A3�#����S�# ��568�36�?����#�S=R�$+�6�� ��H+��# ����U�/\B�/� �36F�R�M�9Q�­¡ ��0��6G��4� 2�S?���� �3�#6G��:r�I:����$��10�0�#��1���=R+��#6±+�=� D�,6�[�@X� �Í-*6G+B:��3@���&D�#��6%?�@���68�9� �+;:������ D�,�����$�#[��#?���:;-*6G+B:��3@B�3&��#�36�?�@��,6G�3� D+B:T��ªi+�[D� �3£0�068�3��3�U��@¦Q5�O�����DYpsb� ��&D�#�#[D: �0«�«�« �WQU_*&F� ����U�������$+2��AM ^YK� ��5�3&��#�36�?�@��%68�36�?K�3�#�U�0�#�5��@�@e4���&��3� �0�#��:[��$��&�4r� ��7� ��6G�t68+;:��3@¦Y,?D[D�"����AM �6G��6%?K�3�8���G4���&��3� �0�#��:�[��$��&�4r:��XJc�3�#��&2�8A�+�6%?���&��.� ��+�&��5+�=� ��0�I6G+B:��3@¦kl�T�d10����@���?�@��(�� 2\;�#��A��I���0AM£0�04����3YfA�+�&b1���A9� ��1����#AM ��368�3��Yf�9� A0QÏ�i&�+�� D����Yj��6G���$4���&D4��+��$�/-*���$+BA3���#�$��&�4%68�9� �+;:T���� ��.�>+�=��3&��#�36�?�@��f68�36�?K�3��:D�#���$�#��&�47��ÐL+�[�@��$� +�&T��&�:7s;68�X� �0«�«�x Y­'��&�48��&�:(§,���$ �+�� ��«�« �b�WQ��� D���j�������$+2��AM (��&21�+�@X1��3�j=R+��$68��&�4H�H�$�M�.� ���/� ��Ad��@^�3&��#�36�?�@���=R+��j����AM 6G��6�?����L+�=��# ��%:B\B&���68��Ad��@p�3&��#�36�?�@��T�R�¦Ql��Q>� D���3&��#�36�?�@���+�?D� ����&��3:Z?2\7��&2� �34�� �.� ��+�&���+�=��G&;[;-

Page 6: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

6G���#��Ad��@c68+;:���@P��?2\t� ��6G��@���&�4��# ����$����A3�%���$+�[�&�:t� D��4��X1��3&(6G��6�?�������&Z�0A�A�+��#:���&DA���)L��� t� D�6G��6�?����3k³�S D���$�#+��#��Ad��@��3�#�#+��S�$� �0� ���$� ��A��3QU�� ��5�$� �0� ���$� ��Ad��@��3&��#�36�?�@��3�G�q:��3�#A3�#��?���:r���f´�:��#�3�#�$��&�4�´2����$�T� ��3&�A3+�6�?���&���:<)L�X� N� D�I:D\;&���6G��Ad�0@,��&D�#��6%?�@��T�#+F=R+��#6¶�F�#+0-`Ad��@�@���:' b\;?��$��:E��&D�#��6%?�@��0Y=R�#+�6Ñ)L D��AM I���$+�?���?���@����$� ��A>=R+��#�3Ad���/� ���0�#�f D��&�A3�3=R+��/� I:��3�#��1���:^Q��� ��f:��#�3�#�#��&�456G�3� D+B:8���#+O1;��:��3��#+�68�tA�+�68����&��#�0� ��+�&N=R+���� D�7�9Jc�3A3�8+�=i�$� +;AM ����/� ��A(6G+B:D��@>�3�#�#+��#��Y,��&�:����#�5:��31���@�+���68�3&2�" ��0����/�M��?D@����$ ���:t� D��=R�����#��?���@����*\"��&�:7��A3+�&�+�6%\t+0=g[��$��&�4G�$� �0� ���$�#��Ad�0@¦YD+���+�� ��3�$)L���#�a&�+�&B-`:D\;&���6G��Ad�0@¦Y� �068��@�������&7�3&��#�36�?�@��a�D�#��:D��A3�#��+�&cQ

_`�����,��&2�#���#�3�$�#��&�4%�#+�&�+��#�f� ��0�>&�+�&��S+�=p�# ��S�9VB���$� ��&�4�6G�3�# �+B:D��=R+���68�X� ��4��0� ��&�4U68+;:��3@K�3�#�#+���3JK��A9� ����6G��@�+O\F!�@X� �3�#��&D4�QH�f��� ��G+�?;v$�3A3� �X1��G+�=,!�@�� �3�#��&�4t���U� +7�#�9� �#���31��G�# ��8[D&�:����$@�\;��&D4(4�+;+;:��&B=R+��#68�0� ��+�&Z=R�$+�6��IA�+��#�#[D�D� �3:'�#��4�&���@�Y^�������U&�+��a[�&��$�d���$+�&���?D@��5� +I� D��&�£F�X��68��4� 2�U?��G[D�#�3=R[D@��&C�$+�6G��Ad������A��X�*\Z=R+��U:�����@���&�4")L�X� F=R+��$��Ad�0�$�U� �#�wv$��A9� +��$���3�S� ��0�U���#�5A3+�&2�M��6G��&��.� ��:C?2\C6G+;:���@���$�#+��$��QE_`�G���H��@��#+C��&b�#���$���$�#��&�4C�# ��0�8��@�@�?�[D�G+�&��t+�=j� D�7�WVB���$�#��&�4�6G�3�# �+B:D�H���#�t��&2�#��&�:��3:��#+��6G���$+w1��%� D�568����&���&�:ÉÒ.+��f10���#����&�A3��+�=�� ��5�3&��#�36�?�@��G:D���$�#�#��?�[D� ��+�&^Y�?�[D��&�+0�U&���A3���$� ���$��@�\Z���#�A�+O10���$�P��&�A3��QH�� ��G��&��$��6�?D@��H:��#�3�#�$��&�4768�9� �+;:�+�=,­'��&�47��&D:'§,���# �+��]���S� ��H�9VBA3���D�#��+�&^Ye��������$�#�#�X1����j=R+��j��6G���#+O1��368�3&2�S��&7��&D�#��6%?�@���A�+O10���#����&�A���?2\(�3&��#[D�#��&D4"� ��.�i�# ��%�/�M�0�#���$�#��A���@�@X\T4��3&����Â-�0�#��:768�36�?K�3�#�3k�A�+O10���#����&�A��3�j���#����:D��&2� ��Ad��@c�#+5� ��a:D\;&���6G��A���@^6G��6�?����$��k��$�d���$+�&���@�@�\t�d1��3� ��4���:���$�#+���A3+w10���$�P��&DA����3QTs;��&�A3�G�# ��H���#+�?D@���6G�S)L���# ]�3&��#�36�?�@��8:����$�#�#��?D[D� ��+�&���kK�$� �0� ���$� ��Ad��@g68+�6G��&2�#� ��d1��j���$+w1���&T�$���$��@�����&2�h� +a��&D:���1;��:D[���@D�$+�@�[B� ��+�&���Y��# ��i�WVB���$�#��&�A3�f+�=K?�[D��+�&��L6G�3�# �+B:H=R+��h��68���$+O1b-��&D4r�3&��#�36�?�@��7A3+O1��0�#�P�0&�A��I���5��6G�K�9� [���=R+��5��&21����/� ��42�0� ��&�4r��@��#���$&��0� �X1��T�$[�AM �6G�3�# �+B:D��Q�ÓU��1��3&� D���#�C+�?��#�3�$10�0� ��+�&��3Yf�<&��.� [��#��@fob[��3�$� ��+�&¤���8)L ��3�# ����t�<��+��$�/-*���$+BA3���$�#��&D4<� ��AM D&���ob[��C� ��0�T���?����$��:8[��K+�&8!�@X� ���$��&�4UA���&"?��L:����$��4�&D��:8=R+����# ��j��[��#��+��$�i+�=c68�X� ��42�0� ��&�4i�# ��j�3JK��A3��+�=c�$� +;AM ����/� ��A6G+B:��3@p���$�#+��$�j+�&(�3&��#�36�?�@��aA�+O10���$�P��&�A3��Q�� ��aA�[��$�#��&2�i���������i��&�:Z�(Ad���d\7��&�:F�Z���/� ��&]� ��«�«2Ê ��M�0£��%�# ���+�����+��/� [�&��X�*\t� +"�$�3�S=R+��$� F��&�:����$�#���$�U+�&��a� ��AM D&���ob[����# ��0�S���f?�����&�4"�WVD�D@�+��$��:C��&(�#�W-�#��+�&��$�T� +Z�# �����ob[����/� ��+�&^Q��f@�+�&�4()L�X� <��&b�#�#+;:�[�A3��&�4Z�# ��T&��9)��#��AM �&D��ob[���Yh�# ����$�I���������#����@��$+ ��d1��S�# ��S?��$+2��:��3�>+�?;v$�3A3�#��1��U+0=�+��K�3&���&D4H�%&��3)Ô@���&��j+�=m��&�ob[����$\G��&2� +�� ��SAM ���@�@���&�4���&�4%�D�#+�?�@���6+�=e�$�#+BAM ��0�$� ��A�68+;:���@c���#�$+��#�3Q

�� ��j+��#4���&���©��0� ��+�&�+�=�� D�jA3[��#�$��&2�����������h�����0��=R+�@�@�+O)L�3Qg�¤�$\;&�+����#����+�=�� D�j��+��/�$-*�D�#+;A����$�#��&�4

Page 7: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� �3AM �&���o;[D�j���h���#+O1;��:D��:H��&GsB�3A3�#��+�& � Qe_*&"s;��A3�#��+�& x Yb�U���$+�� +0�*\B���>!�@X� ���$��&�4f�#AM ��368�j����:D�3!�&��3:G�#+=q��A3��@��X�M�0�#����&21����$�#��42�.� ��+�&�+�=D� ��� ��AM D&���ob[��0Q�sB��A9� ��+�&%�i���#�3�#��&2�#���iA�+�68���$�� ��3&��#�X1��,���#�$���#�$68�3&2�h+�=� D�j�D�#+��#+��*\;�K�L!�@��#���$��&�4U�$AM ���6G��kl�>��?D��@��X�*\�� +a���#+;:�[�A3�i���#�9�+�=^�/�M�0�#���h)L���# 8�$��:�[�A3��:B-*��6G��@��X� [�:���$�#+BAM ����$�#��Af�3�#�$+��#�3YB?��0�#��:"[��K+�&T��@����#4��j�#��6G��@��j+�=p�3&��#�36�?�@��3��Qh_*&ts;��A3�#��+�& Ê Yb�#+�6G�j[�&�:��3�#@�\;��&�4���$+��K�3�$�#�����I+0=a�# ��C���$+�� +��*\;���F!�@��#���$��&�4N�$AM ���6G�r���#�F�$�31��d��@���:¡� ��$+�[�4� ¡�9VD��6G��&��0� ��+�&�+�=a� D�ob[����/� ��+�&¡+�=U)L ��3�# ����IA3���/�M����&�!�@��#���#�3:¤�$� �0� �3�I�$\;�$�#��68�0� ��Ad��@�@�\��K�3�$=R+��$6 ?��3�$� �3�"� ��0&¤+�� ��3�#�3Q¥,+�&�A3@�[��$��+�&D�i�0�#�U���$���#�3&2� ��:Z��&(sB��A9� ��+�&7�DQ

Õ^z ÖI×BØ���~0�RÙ�}w�q��|���ÚL}wÛ�×FÜÝ×2}wÛ��p�

�gVB���#�3�#�#�3:]68�0�# ���68�0� ��Ad��@�@�\�Y�� ���?����#��A�+�?;v$�3A3�#��1��%���i�#+"!�&�:(!�@��#���#�3:F1��3�#�#��+�&��S+0=h� ���A�+��Â-�#[D�D� �3:(�#��&�:�+�6Î10���#����?�@��3�,µ�Åj��&�:7·%Å��q�$�3=R�3�#�#�3:7�#+G���>µ8Þ Å ��&D:Z·�Þ Å Y�� D�U�#[�?��$A��$���D�fßF��:��3&2� �X-=Ë\;��&�4G��!�@X� ���$��:t10���#����?�@��d��$[�AM Z� ��0�

à ¸�¹wº �Rµ8Þ Å »M·�Þ Å �7Á ¸a¹wº �˵F»M·%� à�á à ¸a¹wº �˵ Å »M· Å �(Á ¸a¹.º �RµZ»M·�� à » � x �

)L ��3�#�]�# ��C1��3�$� ��Ad��@�?��0�#�7:���&�+0� �]��&��0?��#+�@�[D� �C10��@�[���QÎ�e+E=R[��/� ��3�(�WV��068��&��C� ����7�$��@��0� ��+�&^Y�#�3A�+�4�&���©3���# ��0� ¸a¹.º �Rµ8Þ Å »M·�Þ Å �>Ad�0&7?����#�3���#�3�#��&2�#��:F��&(�0&���@�+�4�\8�#+"�hoÉQ � ���

¸a¹.º �˵8Þ Å »M·�Þ Å � ¼ ¸a¹.º �Rµ8Þ�»M·�Þw�"ÆȾ¡¿ µ"ÞOÄWÞdÀ�ÆȾ¤¿Ç·KÞOÃbÞdÀ�Æ°¾¡¿ÇÃbÞ.ÄWÞdÀi» �R�b�

)L ��3�#�S�X�,���,���$�#[�6G��:8� ��0��� ��j!�@��#���$��&�4a�#AM ��368�f���:����/� �#��?�[D�#��1��i��&�:8� ��0�>¾¤¿ÇÃ;ÞdÀK¼â¾¡¿lÄWÞdÀK¼ « Q��+��U�d�0�#�H+�=����#�3�#��&2� �0� ��+�&r�# ��5� ��$���H� �3�#6G�U��&r�ho;�3Q � ��&D:]�I� ��0�a��&21�+�@X1��H�$�#+BAM ����$�#��AH���$�#+��$�A�+�@�@��3A3�#��1��3@�\G)L��@�@^?K�U�$�3=R�3�#�#�3:(�#+8�0�iãa��&�:(ãOÞ2Y��#�3�#����A9� ��1���@X\�YÉ�#[�AM 7� ��.�

¸a¹wº �˵ Å »M· Å � ¼ ¸�¹wº �RµF» ·5�"ÆÈãf» ��&�: � Ê �

¸�¹wº �˵8Þ Å »M·KÞ Å � ¼ ¸�¹wº �Rµ8Þ2»M·KÞd�TÆuãOÞiä �q�2�

sB[�?D�$� �X� [D�#��&�4E� D�]�$��4� 2� ���&�:â�#��:D���7+�=5�hob��Q Ê ��&�:Ô����&2�#+��hoÉQ x =R+�� ¸a¹wº �˵�Å�»M·�ÅM�7�0&�:¸a¹.º �Rµ8Þ Å »M·�Þ Å �9Yg�#���$�K�3A3�#��1��3@�\�Y�� ��3&'�M��£b��&�4C��:D10��&2� ��4��"+0=L�Z�#[D@��G+�=j��?��$+�@�[D�#�810��@�[��"�0&�:N���Â-�#[D68��&�4G�# ��0�i¾¡¿ µ"ÞdÀc¼Ý¾¡¿®µZÀg�0&�:Z¾¡¿Ç·�Þ�Àp¼Ý¾¤¿l·aÀm���L���j=R+�[D&�:(�# ��0�i�ho�Q x Ad��&(?K�a�9VB���$���$�#��:

Page 8: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

¨

���

à ¾¡¿ µ8ÞO·KÞ�À%Á�¾¡¿ µ�·aÀ à�á à ã à Á à ãOÞ à ä �¦���

�hoÉQ"�t��@�@�+O)L�f=R+��U�$+�68�H��&��#��4� 2�U��&2� +I�# ��HA�+�&�:D��� ��+�&��S�# ��0��6�[��/�a?K�H6G�3����=�!�@X� �3�#��&D4I���S� +7?��[��$��:(�#+H��6G���#+O1��U�3&��#�36�?�@��aA3+O1��0�#�P�0&�A��0Q�>)�+G+�?��#�3�$10�0�#��+�&���=R+�@�@�+O)â[���+�&t��� ����&��#����A9� ��+�&^���Qh�� ��a�#��4� 2� ��0&�:t�#��:D�U+�=��hoÉQ,�56%[��$�L?���4��#���0� �3�L�# ���&t©��3�#+�Q��� ����L68����&��>� ��0��� ��U!�@X� �3�#��&D46�[��/�,�D�#+;:�[�A��f��:����$�#�#��?D[D� ��+�&H+�=p�$� �0� �3��)L�X� "�/� +;AM ����/� ��Aj���$�#+��$���# ��0�����#�i4���&D��� �0@�@�\�+�=^�$��:�[�A3��:��6G��@��X� [�:D��YD�$+G�# ��0� à ãOÞ àbáåà ã à Q� QG�� ��G@��3=Ë�# ���&�:r�#��:��H+�=��hoÉQI�T��:��d��@�@�\(),+�[D@�:r?K�H©��3�#+�Q8�� �����A3���#A3[�6G�$�M�0&�A��H����6G+��$�a�����#��@�\�#����@���©��3:'��=,� ��G!�@��#���#��&�4ZA���[��$����&�+Z�#��:D[�A3�#��+�&<��&<��6G��@���� [D:��H+�=�� ��G� �$[��.æa�/�M�0�#���%µ �0&�:N·8Qªi+O)��31�����Yc?K�3Ad��[��$��?�+�� (�#��:D���f+0=h�hoÉQS�H��&21�+�@X1��a:���JK���$��&�A3���S�X�j���i���������$��&2�f�# ��0�i�#+�6G��!�@��#���/-��&D47+0=,� D�G�#�#[��H�$� �0� �3�������K�3�#6G���#�$��?�@���Yp���$+O1B��:���:r� ��.���$+�6G�G�$��:�[DA3� ��+�&r��&r��6G��@��X� [�:��5+0=,� D��$�#+BAM ����$�#��AU���$�#+��$�������0AM ����91���:7�#��6�[�@X�M��&��3+�[��$@�\�Qg�� �����6G�d��&D�,� ��.��� ��S!�@X� �3�#��&D4�:�+;����&�+��L ��O1��� +8�9VD��A9� @�\7:��XJc�3�#�3&b�#�P�0�#��?��3�*)����3&C�$�#+BAM ��0�$� ��A%�3�#�#+��#�S�0&�:(�#�#[����$�M�.� ���i��&(+��#:��3�i�#+"���$+w1;��:��a=R+����6G���$+w1���:HA�+O10���$�P��&�A3��Qg­¡ ��0��� ���!�@X� ���$��&�4U:D+B�3� ��O1��j�#+�:�+����h�#����AM "�0&8�9Jc�3A3� �X1��LA�+�68���$+�6G���#�?��3�*)�����&(� �����@���6G��&��.� ��+�&8+�=e���$�#+��$�j��&�:t� ����#6G+;+�� ���&�45+�=m� �$[��UÌ�+O)ç�$� �0� �3��Q

_`�m6�[��/��?K��?�+��#&��h��&�68��&�:S� ��.�p� �����+���&2� �p68��:��h�$��42�0�#:���&�4>� ����$��A�+�&�:�+�?��#�3�$10�0� ��+�&a��?�+w1���#�3�$�i��&7�����/�f+�&t� D�%�0=R+��#��6G��&2�#��+�&��3:Z�0�#�#[D68�D�#��+�&t� ��0�S¾�¿Çµ8Þ�Àp¼ç¾¡¿ µFÀm��&D:F¾¡¿Ç·KÞ�Àc¼Ý¾¤¿³·�À¦Q�� ����8�0�#�#[D68�D�#��+�&E�3�#�#�3&2� �P�0@�@�\<�#���D�#���$��&2� �"��&�+��# ����HA3+�&�:��X� ��+�&E�# ��0�H6�[D�$�G?K�t�#�0� ���$!��3:���=j� D�!�@X� ���$��&�48���j� +T���$+O1��5?���&��9!�A�����@¦Qi_*&F� ���A����#��� ��0�S¾¡¿lÃbÞdÀm¼å¾�¿lÄWÞdÀe¼ « �q�"&�+��S[�&��#�����#+�&���?�@���#A3��&��0�#��+2��� D��&5� ����gA3+�&�:��X� ��+�&�)L��@�@b?K�>6G�3�g�#��68��@X\���=�¾¤¿ µ8Þ Å ÀK¼�¾¤¿ µ Å ÀD��&�:H¾¡¿Ç·KÞ Å ÀK¼â¾¡¿l· Å À¦Q

�è!�@��#���#��&�4G�#AM ��368�a� ��0�i� �.� ���/!����j� ��aA�+�&�:����#��+�&D�j:D���#A3[��#�$��:���?�+O1���Ad��&7=q��A���@���� �0� �S��&Z��6H-���$+w1���6G��&2�i��&I��&D�#��6%?�@���A�+O10���$�P��&�A3��Q��� ���68����&I��68����:���68�3&2�,� +H[��$��&�45!�@X� �3�#��&D4�=R+��>� �������[��Â-��+��#�U���,� ��f�#��:D[�A3�#��+�&t��&T��68�D@����#[�:��j+�=�� �$[��SÌ�+O)��/�M�0�#���>�# ��0�������0�#�#��&�:���&2��)L��� T�# ��f!�@X� �3�#��&D4���$+BA3���#�3QC�� ����%�#��:D[�A3�#��+�&<��&N��6G��@��X� [�:D�H)L��@�@�+������#�0� �"�#+F��&DA��#�����#�8� ��T@��3=Ë�# ���&�:<�$��:��8+�=j�hoÉQ

é*êËëHì¦íwîfï/ð9ë�ì�î/ñdì�ðWòeë�ó.ô,î$õ�ö÷ï/ø9ùKúcî$øWì�íwî$õ�òÍð3õ�î#ï/ø9ûqì¦û$üÉøaýWì�õ¦ówî$ýaûRì`øMì�îfö÷ûð3ëwîLì¦í.øMì>úKð3ówù÷þHÿ�îiõ¦î$ø9ù³ö��/î#þ�ölòeø9ëö ëwölì¦ö ø9ù;ûqì¦øMì¦î�ö ûpî��9ð3ù��3î$þLúpö³ì�ífì¦íwî�þ��dë.øWô>ö÷ï/û^ð9òBì¦íwîhõ�î#øWù2øMì¦ô>ð3û��wíwî/õ¦î

Page 9: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

�T��&�:C�# ����$��?2\C:��34�� ��:D�5���$�#��68�0� �3�S+�=A�+O10���$�P��&�A3��Q%sB��6G��@����#@X\�YÉ���i)L��@�@e�$���$1��5� +t:���6G��&D���# (���/� �X-68�0� �3�i+�=g1��0�#�P�0&�A��0Qj_*&(=q��A3��YK�91���&C�X=g!�@X� �3�#��&D48A���&F��&D4���&�:D���U��&(��6G���#+O1���6G��&2�f��&ZA3+w10���$�P��&DA���Y� D������6G���#+O1���6G��&2�6"�d\5&�+0��#� ��&D�#@P�.� ��� +���&G��6G���#+O1��3:86%[�@��#�X-*:D��6G��&��$��+�&��0@;�D�#+�?��0?���@����$�#��A,=R+��#�9-Ad�0�$�j?K�3Ad��[D�#��+�=m� ���:��9� �#��68�3&2�L+�=m:���6G��&����# ��3:T10���#����&�A��0Qh�� ��� 2\B��+��# ����$����� ��.�L[�&�:��3�#@�������� D����$���#�3&2�j��&���@X\B�$������� ��0�,+�&��i68��4� 2��?��U��?D@��L� +�6G���#��42�0�#�>� ��f���#+�?D@���6¢+0=p� ��j!�@��#���#��&�4%+0=p� �#[D��$� �0� �3�j)L�X� �+�[D�L��@���6G��&��0�#��&�4a� �����+��#�$��?���@��X�*\8+0=���6G���$+w1���:tA�+O10���$�P��&�A3��?b\t[D�#��&D4G�$+�6G�a&b[�6%?K�3�+�=��# ���!�@�� �3�#�3:F�$� �0� �3�S��&(A�+�&�A3���/�f)L���# Z�$+�68��&b[�6�?����f+0=�� ���[�&D!�@X� �3�#��:Z�$�M�.� ���f��&F�G�#+0-`Ad��@�@��3:´� 2\;?��#��:�´G��&��$��6�?�@���:����/� �#��?�[D�#��+�&^Qh�i:�+��D�#��+�&7+�=m� ����L b\;?��$��:7��&��$��6�?D@����������$+2��AM 768����&���� ��.��(�*)�+0-*�����$�5��+��/�$-*�D�#+;A����$�#��&�4C68�9� �+;:�+�@�+�4�\C)L��@�@�?K�"�9VB��@�+��#��:N��&'� D�T���#�3�#�3&b�G�0&���@�\;�$����QZ�� ��6G�3� D+B:�+�@�+�4�\8�#����A3��!�A���@�@X\T��&b1�+�@�1����3�

�w���� ��T���$+w1;���#��+�&'=R+����Z4���1���&N+������#�0� ��+�&���@��3&��#�36�?�@��8+�=j�Z�#��68�D@��"+0=>!�@��#���$��:N�$� �0� �3�%)L �+��$��$�#+BAM ����$�#��A����$�#+��$�i�0�#�U4��3&����#��@�@X\T+�=e�#�3:�[�A��3:F��6G��@���� [D:���Q� Q5�� ��G�#�3@��3A3� ��+�&�+�=,�7�$[�?��#�9�%+0=��# ��5!�@��#���#�3:]�/�M�0�#���U� +7A�+�68��@���6G��&2�U+����$[�����@���&2�U� ��G+������/-�0�#��+�&��0@���&��$��6�?D@��G68�36�?K�3�#�3Ye[�&D:����a� ��GA�+�&D�$� �#����&2��� ��0���# ��G�#���$[�@��#��&�4t:����$�#�#��?D[D� ��+�&^kl�U68����&��&�:t10���#����&�A3�U����A�+�68���0� ��?�@��f� +5�# ��0�j+�=m� ���+������#�0� ��+�&���@K��&��$��6�?�@���Q

�� ��5 2\;?��#��:C��&��$��6�?D@��%�# ��0�S���i�# ���A�[D@�6G��&��0�#��+�&7+�=h�# ����$���*),+t�����/� �i ����S6G�d��&��0&�:Z10���#����&�A3�A�+�68���0� ��?�@��S� +H� ��.�L+�=g�# ���+��K�3� �0�#��+�&��0@c��&D�#��6%?�@��a?2\t:����$��4�&^Q�� b[���YK���$�#[�6G��&�45�# ��0�L� ��a 2\2-?��$��:H��@��#+��0JK+��#:D�h��6G���#+O1��3:5A�+O10���#����&�A3��Y2�����# �+�[D@�:H���#+O1;��:D�>=R+���6�[�@X� �Í-*:���68�3&��#��+�&���@2���$+�?���?D��@��X�*\=R+��$��Ad�0�$� ���# ��0�i���#�U?��3�$� �3�L�# ���&t� �+��#��?����$��:Z[D�K+�&I�# ���+��K�3� �0�#��+�&���@c�3&��#�36�?�@��0Q

�^z ��~w��}O��}� pÙh×����Ë}O×b~0�q|��'Ød�.Û�×��]×

_*&21��3�$� ��42�0�#��+�&�+�=� ��H���$+���+��#�3:]6G�3�# �+B:D+�@�+�40\(�$��ob[����#����:��9!�&��X� ��+�&C+�=>�T!�@X� �3�#��&D4I�#AM ��368�0Q�� ��]���#+��#+��*\;�K��!�@X� ���$��&�4E�$AM ���6G�N��:�+��D� �3:Ô=R+��I� D�]���$���$��&2����&��0@�\;�#���7��&21�+�@X1����t=R+��$68��&�4���@�@��+��#�$��?�@��a�������$�j+0=h+������ �.� ��+�&���@^6G��6�?����$�f�0&�:(�# ���&Z�O1����#��4���&D48� D��6G��6�?����$�i��&(����AM Z�������3Q>_*&� D�5Ad���$�G+0=�# ����f�.� ��+�&���@h¥,�3&2� ���U=R+��S��&21;���$+�&�6G��&2�M�0@h�h�#��:D��A3�#��+�&����S¥>�����Ó�@�+�?���@e��+��#�3Ad���/�s;\;�$�#��6 ��Óa��s�� «�«�«�«�� �j¥ç��&����#�P��@���©d�.� ��+�&C���3-*6G��6%?K�3�i�3&��#�36�?�@��0Y^� ��3�#�H���$� Ê�Ê ��+��$�#��?D@������0���#�

Page 10: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� «

��&�:7 ��3&�A�� Ê�Ê �#+0-`Ad��@�@��3:I´��������/-`)L���#�3´�!�@��#���#�3:t�$�M�.� ���3Q�>)�+%+�?��$���/1��.� ��+�&���� +�4��3� D�����#[�4�4����/���# ��0�h� D�j�������Â-`)L���$�,!�@X� ���$��&�4�6G��4� 2���0A3� [���@�@X\%�$���$1��j�#+

�#�3:�[�A3�%� ��%��6G��@��X� [�:���+�=��$� +;AM ����/� ��A%���$�#+��$�f:��3�#���X� ����� �j�#��68�D@���A3���*\�QL¬S&��%���j�# ��0�f�# ����d1���� �04��+�=É�*)�+a:D���#�$��6G��@����p!���@�:���)L��@�@B4��3&����#��@�@X\% ��O1��L@����$����6G��@��X� [�:���� ���&H�����# ����g+�&���+�=É� D�>�*),+�!���@�:���Q��1;��:��3&�A���+�=�� D���>4��3&����#��@^�$[�@��SA���&I?��U�#���3&(��&"� D�U�#��6G��@��f�#AM D��68�0� ��A���+�=e�g��4�Q Ê Qh�� D�a�$��A3+�&�:+�?��$���/1��.� ��+�&����m� ��0�e� ����$�#+BAM ��0�$� ��A9-`���#�$+��e!���@�:D����&5�0&b\��*)�+U+������ �.� ��+�&���@b68�36�?����#�g���#�,[�&�@���£��3@�\� +"?K����:���&2�#��Ad�0@¦Y�4���1���&F� D��+��#��4���&(+�=h� D�����$�#+��$�S��&F�#��&�:�+�6 ���$+BA3���$�#���3Q��� D���#�H�0�#��YK �+O),�91��3��Y&�+N�$��6G��@��(�'���$��+��$�i+�?D�#���/10�0� ��+�&��8�#+N�#[�4�4����/�T� ��.�T� ��Z�������Â-`)L���#�t!�@��#���#��&�4'A���&¡��AM ����31�����&�3JK��A9� �X1��"A3+�6G���#+�6G���$�5?K�9�*),�3��&<�# ��8�3@���68��&��0� ��+�&F+�=,���$�#+��$�%��&D:]� D�8�$68+;+�� D��&�4t+�=�� �#[D�GÌ�+O)A�+�68��+�&��3&b�#��Q

�� ��HAM �+���A��5+�=���$+�� +0�*\B����!�@X� ���$��&�48�#AM ��368�H���$+�6G�D� �S�$+�6G�H��:�:��X� ��+�&���@��#�36"���$£b��Q���+��i��&B-�$� ��&�A3��Yh���������d���$��@X\��# �+O)L&N� ��.���# ��T6G�d��&<+0=>� ��"��&2� ���#�"�$�3�5+0=��������/-`)L���#�H!�@��#���$��:<�/�M�0�#���������:D��&2� ��Ad��@É� +��# ��0��+�=��# ��S+������#�0� ��+�&���@���&��$��6�?D@���Q�sB��68��@P���$@�\�Y����,���>�����#��@�\G�# �+O)L&t� ��.�>� ��f68����&+�=��# ��"���0���/-¦)L���$�8!�@X� ���$��:<�/�M�0�#���3ke� �$[��TA3+�68��+�&��3&2� ��������:���&2� ��Ad��@�� +(� ��0�%+�=�� D�"+������#�0� ��+�&���@��&D�#��6%?�@��0k³��� �$[��HA�+�6G�K+�&���&2� �3Ym��&�:C� ��.�a�# ��H68����&�+�=� ��5!�@X� ���$��:��$� �0� �3��k��$�#+BAM ����$�#��AH���$�#+��$����L©3���#+DQ��� b[��3Y��# ��U!�@��#���#�3:(�/�M�0�#���j:����$��@P�d\T�# ���:����$���$��:Z6G�d�0&7�D�#+������/� ���3�L+�[D� @���&��3:t��&(sB��A9� ��+�&� Qr�� ����$�I�����/� �$+�&�4C�#�����#+�&E�#+C?K�3@����31��T�# ��0���# ��I6G��6�?����$�# ����<+�=i�F4���1���&E 2\;?��#��:N��&��$��6�?�@��)L��@�@��9VB ���?�����A�@�+��#�j� +��# ��S�#��68�j6G�d��&8���#+������/� ������Q¥,+�&��#��:����� ��0��� ��i68�36�?����#�+�=^�# ��S 2\;?��#��:)L��@�@¦Ym?2\':��3�#��4�&^Yh?��"�#�3@��3A3� �3:N� +F+��B� ��6G��©3�5� ��T 2\;?��$��:^kl�%10���#����&�A3��Yh��&D:<�#+����$�T���B�%� +Z?K�I�0�:����#�#��68��@P���%���G�K+��$�#��?�@��0Q¡sB��&DA��(��&��0�����#�3A�����?�@��tAM ��0&�4��Z��&�6G�d�0&����$+������$�#���3�G)�+�[�@�:��#�3ob[����$���&D ����$��&2��A�+�6G6G+�&���@����*\�Y�&�+���:D���#�$��6G��@����#�X�*\�Y���6G+�&�4"�# ��5�#�3@���A9� �3:]6G��6�?����$��Y^�$[�AM <�TAM ���&�4��5���[�&�@���£0��@�\8� +H?��a�3&�4��3&�:����$��:(?b\I�# ���A�+�&D�$� �$[�A3�#��+�&t+�=��5 b\;?��$��:7��&��$��6�?D@���Q

�f&D+�� ��3�%�$��68���#£������ ��0���d1���� �04���&�4Z+�=���&D�#��6%?�@��8+�[D� �D[D�� ��0�%���$+w1���&N6G+��$���3JK��A9� �X1��T�.����$+B:�[DA���&D4I�#��:D[�A��3:B-*�3�#�#+���4�@�+�?���@m���0�$� �3�#&�����&��/\B&D+��D� ��A9-¦�*\B���5!���@�:��U�#[DAM <��� Ê�«�« ��g�t4���+��K+0-� �3&2� �P�0@p �����4� 2�dQh­¡���# t�# ����L��+���&2�L��&76G��&D:^YD� D�a�������Â-`)L���$�S!�@��#���$��&�45�$AM ���6G�%���L������@�����:t+�&D@�\"�#+!��3@�:���+�= Ê�«�« ��g�54���+��K+��#��&2� ����@c D����4� b�L��&T� ������$���$��&2�i��&���@X\B�$���3Qh�f@��$+�YB�# ��U!�@��#���$��&�4��$AM ���6G��kl������D@���A��0� ��+�&N���5=R+;A�[��$��:�+�&�=R+��$��A����$�#�8��&��# ��(6G��:D:�@��T�#+]@��0� �t�$� ��4��3�8+�=i� ��76G��:���[�6 � �0&�4���Y

Page 11: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

���

)L ��3�#�%=R+��$��A����$�#�f+�=�10���$�P��?D@����L�$[�AM C���i���#�3A����D���M�.� ��+�&(�0&�:Z�$[��$=q��A3�%� �368�����#�0� [D�#�a ��d1���@����#�#@��S+��&�+H�#£b��@�@¦YD��&�:7 ���&DA���=R+��#�3Ad���/� �L+�=��/\;&�+��D�#��AUÌ�+O)ç���.�#� �3�#&��L�$��68����&I+�=eA�+�&��#��:����#��?�@����$��@��910��&�A��0Q

ªf�d1;��&D4G:D�3!�&��3:Z�5���$+�� +0�*\B���U!�@��#���$��&�4��$AM ���6G��YÉ��&�ob[����/\"A���&t&�+O)ç?���4���&I��&2� +H�#�91��3� ��@^=R[�&;-:���6G��&2� ��@�o;[D���$�#��+�&D���%_*�a� ��G���$+�� +��*\;���G!�@�� �3�#��&�4(�$AM ���6G�T�0?�@��5� +7���$+B:�[DA��"+�&'�7�$\;�$�#��68�0� ��A?����$���i&��3)èÌ�+O) �$� �0� �3�f� ��.�U���$�%@��3�#�S�3&�A�[D6�?K�3�#�3:C?b\(���$�#+��$�f� ���&Z� D�%+������#�0� ��+�&���@�6G��6�?����$����$��� ¥>��&' 2\;?��#��:]�3&��#�36�?�@������0A3� [���@�@X\F+�JK������6G���$+w1���:]A�+O10���#����&�A����è¥>��&' 2\;?��#��:'�3&��#�36�?�@��3�\;���3@�:76�[�@��#�X-`:���6G��&��$��+�&���@É���#+�?���?���@����/� ��Ai=R+��#�3Ad���/� �L� ��0�f���$�a?��3�$� �3�j�# ���&(�# �+��$�a\;���3@�:��3:(?2\t� D�+������#�0� ��+�&���@���&D�#��6%?�@����E�� ��3�#��ob[��3�$�#��+�&��S���#����&21�+�@X1��3:C��&�+�[D4� Z� +"?K�%��:�:��$���#�$��:��#�3�����#�0� �3@�\�Y)L���# a�# ��A�[��$�#��&2�g����������kl�g&D�9V;�g�$��A9� ��+�&����$�#��4�&���:a� +i�# ���!��#�/�gob[��3�$� ��+�&��0&�:5�(A����O\���&D:5�Z���/� ��&� �0«�«2Ê ��:D��:���Ad�0�#��:t� +H� ���@P�.�#� �3�,�*)�+�Q

��z �! #"#����"D}w�R��|��ÉÚL}OÛm×FÙm~w��}O��} ^Ùh×����R}O×b~.�q|$�rØd�.Ûm×%�'×'&XØH×)(<��"��* +-, ./+*01+

�f&���@�\;�#�������U?����$��:r[���+�& x �D�H:���JK���$��&2�/�f�0�#��+�&���@h¥,�3&b�#����=R+����h&21B���#+�&D68�3&b� ��@��h�#�3:���A9� ��+�&���f¥>�����7ÓU@�+�?��0@U��+��#�3Ad���/�Fsb\B�/� �36 ��Óa��s�� «�«�«�«*� �j¥±��&����#�P��@���©��0� ��+�&Ô���3-*6G��6�?����"��&��$��6�?�@��=R+��$��Ad�0�$� �3QI�� ����$�"�3&��#�36�?�@��3��)����$�"4��3&����#�0� �3:<:�[��$��&�4t� ��G+�&��W-`\��d���a�K�3�#��+B:r?K�9�*),�3��& � �T�S�9-A��36�?���� ��«�«2� ��&D: � �G�S�3A���6%?K�3� �0«�«�x Q5�� ��H:��.�M�T),�3�#�G+�?D� ����&��3:]+�& � ä Ê*2 -*?2\2- � ä Ê*2 @P�0�#��� [D:��9-@�+�&�4����#[�:��84��#��:��%��&E�ZA3\;@���&�:��$��Ad�0@��3o;[D��:����$�M�0&b�t�¦¥>���������$+.v$��A9� ��+�&^Q��f�5�$� �0� �3:E��&EsB�3A3�#��+�& x Y��&��0@�\;�#���a�����#���/� �$��A3�#��:r� +t=R+��#�3Ad���/� �a+�= Ê�«�« ��e�(4��3+���+�� �3&b�#�P��@g �����4� 2�%�0&�:]�# ��(�d� � �=R+��#�3Ad���/�@�����:D�#��6G��QF�� ��t�0�����#+����#���0� � « N@�����:D� ��68�8A�+�&2�#�#+�@=R+��#�3Ad���/�5)>���5[D�#��:�����1����$��!�A��0� ��+�&N��&<��@�@=R+��$��Ad�0�$����$�#+���Ad�0@�A�[D@P�0�#��+�&��3Q�s;�31��3� ��@É�3&��#�36�?�@��3�,�0�#�L68���#�$��&�4U+��h��&�A�+�68��@��3�#��:�[��#��&�4U� ��j�0=R+��#�9-6G��&2� ��+�&���:H+�&D�9-`\��d���������$��+;:8�0&�:8 D��&�A3�i�0�#�L[�&21��3�#��!���?�@��0�e�# ���� �43 [�@�\ �0«�«�x Y � �S�i[�4�[��/� �0«�«�x Y��&�:¡�O�5�j+O1���6%?K�3� �0«�«�x ��&D�#��6%?�@��3��QN�f�H�F=R[��/� ��3�5A�+�&D�#��ob[��3&�A��0YU�O� � N@��d��:B� ��6G�8=R+��#��A����$�#���&D��� ����@���©��3: «�«�«�«�� �j¥¤� 3 [D@�\ �0«�«�x ��&�:��O�H�f[�4�[��$� ��«�«�x ���#��[�&21��3�#�X!É��?�@���Q

6 , .87:9;=<*>?+*0�;1@A>B@DC8@:EF7G�+*0�;1@A>?+*HI7>J9�7K 6 HL7MKN7K 6 7GO9/+*>?PMEJ+*;QGRTSU;L9�7WVXHY0D7GO7ZP�901+*0D7:9�� ��N���Z68�36�?K�3�#�T+0=a�'4���1���&¤�3&��#�36�?�@��Z��&�A3@�[�:��(� ��FA3+�&2� �$+�@¦Y>!�1��C68�36�?����#�8+�?D� ����&��3:

Page 12: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� �

� D�#+�[�4� Ô� D�'��:D:����#��+�&¤� +E� D�'A�+�&b�#�#+�@a��&���@�\;�#���t+�=%!�1��':��XJc�3�#�3&b�(�����$�#[��#?��0� ��+�&���Y��0&�:Ô!�1��6G��6�?����$�>+�?D� ����&��3:"� D�#+�[�4� T� ��U�$[�?D� �#��A3�#��+�&T=R�$+�6¢� ��UA3+�&2� �$+�@^��&���@�\;�#���,+�=��# ��f!�1��a:D��JK���#�3&2������$�#[��#?��0� ��+�&���Qg�� ��f+������#�0� ��+�&���@D:����$��4�&��.� ��+�&��=R+����# ����$�f6G��6�?����$�,�0�#� ¸ «�«�� Y%[ «�« ��Y�QXQXQ�Y�[ «�«2Ê Y��&�:]\ «�« ��YXQ�QXQ�Y�\ «�«2Ê Y�#���$�K�3A3�#��1��3@�\�Q���+����# ��t���#�3�#�3&b�"��&��0@�\;�#����Y�� ���� « �����$�#[��#?���:�6G��6�?����$���[ «�« ��Y^[ «�«2� YiQ�QXQXY_\ «�« ��Y`\ «�«2� YiQ�QXQ÷� �0�#�7�#�3=R�3�#�$��:��#+r)L�X� ���@��#���#&��0� �X1��t:����$��4�&��0� ��+�&��5� ��.��#�9Ì���A9�g�# ���6G��6%?K�3�#��kw�$68�m:����/�M��&DA�����=R�#+�6ç� ����&��$��6�?D@��,6G�d�0&^Q�s;�K�3A���!�Ad��@�@�\�YO�# ��,:��3�#��4�&��0�#��+�&��\ æ Y?\badY�QXQXQ�Y?\ æ�c ���#�����#�$��4�&D��:(�#+G�# ��a68�36�?K�3��=R[��$�# ����/�i=R�$+�6 � ��a6G�d��&^Y��# ���6G��6%?K�3�j�$��A3+�&�:=R[��/� ��3�$�j=R�#+�6 �# ���6G�d�0&^Y�QXQXQ�QXYl�# ��a68�36�?K�3�LA�@�+��#���/�i�#+8�# ���6G�d�0&^Y��#�3�#����A3�#��1���@�\�Qd�i+��#�a�# ��0�e\ æ Y\/aOYXQXQ�QXYF\ æ�c )L��@�@K&D+��L&���A3���$� ���$��@�\"A�+��$�#�3�#��+�&�:t� +�� D�a�#��6G�S+������ �.� ��+�&���@�@X\H:����$��4�&��0� �3:(6G��6�?����=R�#+�6½��&D�#��6%?�@��U� +G��&D�#��6%?�@��0Q

�� ��a:����$��4�&��0� ��+�&I=R+��L��&2\I�#����A3��!�Aa�������/-¦)L���#�i!�@��#���#�3:(�/�M�0�#�������#��68�D@�\"�# ���A�+�6%?���&��.� ��+�&T+�=���#���*),+TA3+�6G�K+�&D��&2�f+��K�3� �0�#��+�&��0@^68�36�?����#�3kÉ:����$��4�&��.� ��+�&���Q���+��L�WV��068��@���Y?\ æ \baf�$�3=R���$�i�#+8� D��������/-¦)L���#�i!�@��#���#�3:(�/�M�0�#��:����$��1��3:7=R�$+�6 +������#�0� ��+�&���@K68�36�?����#�d\ æ �0&�:f\baOQgh, .87iVX>F;Q0�;1@A>B@DCkjOlm+A9;1g)jb+*>?PBjAn_o*7G�+*HQHpjeqr>J9�7K 6 HL7:9

���#�3o;[D��&2�7�#�3=R�3�#�3&�A������I68��:��Z��&�� D����&���@X\B�$���"�#+<� ��C�$+0-*A���@�@���:¤´�?����#��A�´E��&�:¤´�+w1����#��@�@�´+������#�0� ��+�&���@p�3&��#�36�?�@��3��Yp�0&�:Z� +8� ����#+0-`Ad��@�@���:(´�?����$��A3´T��&�:Z´�+O1����#��@�@�´"+������#�0� ��+�&���@p6G��6%?K�3�#��Q�� ��>´�?����#��A�´i+��K�3� �0�#��+�&���@2��&��$��6�?D@��,���e:D�3!�&��3:�� +iA�+�&��$���/��+�=��# ��,A3+�&2� �$+�@D�0&�:%� D�f� « �K�3�$� [D�#?���:6G��6�?����$��Yb��&�:H�U´�?����#��A�´S6G��6�?����g���h�U6G��6�?�����+0=��# ����g��&��$��6�?�@���Q��� D��´�+O1����#��@�@�´S+������#�0� ��+�&���@��&D�#��6%?�@��i���,:D�3!�&��3:T� +�A3+�&��$���$�,+�=p� D�f?����$��Ai+��K�3� �0�#��+�&���@���&��$��6�?D@��S��@�[���� ��S�3&��#�36�?�@��f6G�d��&^Y��&�:(��&t´�+O1��3� ��@�@�´56G��6�?��������L�G6G��6%?K�3��+�=e�# ���+O1����#��@�@c��&��$��6�?�@���Q

P-, sB70itF@hP%@AHL@Z<*u�� ����$�d��@����*\t���i� ��0�f=R+��i��&2\(4���1���&F�3&��#�36�?�@���=R+��#�3Ad���/�S+�&�@X\7+�&����#�#[��%�$�M�.� �%���f[�@X� ��6"�0�#��@X\

£b&�+O)L&<� +Z�(4�+;+B:]�����D�#+dVB��68�0� ��+�&^Yp� �����?�����&�4t� ��G=R+��#�3Ad���/�%1��3�#�X!�Ad�0�#��+�&��q��Q³�0QT� ��H1��3�#��=Ë\;��&�4��&��0@�\;�#���M�9Qg�f&I���$�#[��S�#+�?��U�#���$+�@�1���:I���> �+O)��X�,A���&t?K�S���#A3���$� ����&D��:T� ��0�>� D�S�������/-`)L���#�j!�@��#���#�3:�$� �0� �3�I���$�(4��3&����#��@�@X\E@��3�#�"A3+��#�$[��D�#��:�?2\����#�$+��#�8+�=U�/� +;AM ����$�#��A(+��#��4���&E� ��0&�� D�Z+������#�0� ��+�&���@

Page 13: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� x

6G��6�?����$�>���$��YD4���1��3&�v$[��$��� ����,+�&D�f� �$[��f�/�M�0�#�f=R+��,�$�3=R�3�#��&DA���Q��m+�=q��A���@��X�M�0�#�>� ����,���$�#[��0k³���$���$+�@�[B-� ��+�&5�������h&�+��#��:5� ��.��# ����������/-¦)L���#�>!�@�� �3�#�3:G�/�M�0�#���� ��d1��L� D�j�#��6G��68����&^Yb��&�:G ��3&�A���� D�j�#��6G��$\;�/� ��68�0�#��AL���$�#+���A�+�68��+�&��3&b��Y;���h� ��L+������#�0� ��+�&���@�6G��6%?K�3�#��Qg�� ;[D��Y;��=K��&5� �3�#6G�+0=c�$+�68�L�3�#�#+��6G�d���$[��#�f�# ��f!�@X� �3�#��:"�$�M�.� ���>���#�f=R+�[�&�:"�#+�?K�f?K�9�#� �3�������D�#+dVB��68�0� ��+�&���+�=p� �$[��S�/�M�0�#���,�# ���&T� D�+������#�0� ��+�&���@h6G��6�?����$�����$��Y��X�%����?���A���[��$�"� ��8!�@��#���#�3:<�$� �0� �3��km�$�#+BAM ��0�$� ��AT���$�#+��aA�+�68��+�&��3&b�#� ��d1��G?�����&]�#�3:�[�A��3:^Q"�i@��$+�Yp� +t=q��A���@���� �0� ���#�3�#+�@�[D� ��+�&����������0�#�#[D68�3:]� ��0���7�$[���� ��?�@��56G�d�0�#[��$�+�=e� ������$�#+��L��&Z�54��X1���&Z�������$+dVD��6"�.� ��+�&I+�=��5� �$[��a�$�M�.� �����L�#+;+��/-*6G�d��&B-`�#ob[����$�G�q�$68� ��:D���$� ��&�A3�=R�#+�6Ï� ��0�h� �$[��L�$�M�.� ��Qg�f:�:��X� ��+�&���@�@�\�Yw=R+;A�[��h���h��@P��A3��:H+�&5� D�j@�+O),�3�?�+�[�&D:G+0=K�# ���� ��&D4��L+�=��#6G����$�#+����q�$68�$�O��+�=�� D��+O1����#��@�@�+������#�0� ��+�&���@��3&��#�36�?�@��%��&�:Z�# ��0�i+�=h� D�%�#�9�S+�=��������Â-`)L���#��!�@��#���#�3:�$� �0� �3��Q�� ��a@�+O),�3��?K+�[�&�:7+�=g�# ���� ��&�4��U+�=��$68�$�S=R+��j�H4���1��3&(+O1���� �0@�@p��&D�#��6%?�@�������:D�3� �3�#6G��&��3:?2\E!�&�:���&�4]�# ��Z6G��&���6�[�6¶10��@�[��t+�=U�$68�$�F���$�#+;A��P�.� ��:�)L���# ���&2\�68�36�?K�3�H+�=S� D�Z��&D�#��6%?�@��0Q�� ���@�+O)�����?�+�[�&�:t+�=��# ��U� ��&D4��U+�=m�#6G�#�S=R+���� ������$�#+;A����0� �3:7�$�3�j+�=�!�@X� ���$��:t�$� �0� �3�������$��6G��@P�0�#@�\:��9� ���$68��&���:cQZ­¡���# '� ��3�#�I:��9�M����@��a=R+��%�$�3=R���$��&�A3��Y����%Ad��&'?K�I�0�#4�[��3:<� ��.�%� D�"!�@��#���$��:N�$� �0� �3����$�i4��3&����#��@�@X\H?K�9�#�#�����������$+OVB��6"�0�#��+�&D�+0=^� �$[��i�/�M�0�#�����0&�:" ��3&�A��f@����$��A3+��#�$[��D�#��:T?2\8�$� +;AM ����/� ��A���$�#+��$�U��=�� D�5@�+O),�3�U?K+�[�&�:�+�=� ��3���S�#��&�4��5+�=��#6G�#�%���U�#68��@�@����j� ��0&��# ��5@�+O)����U?�+�[�&�:�+�=� D�� �0&�4��L+�=^�$68�$�L+�=K�# ��j+w1����#��@�@��3&��#�36�?�@��L+�&8���$\;�$�#��68�0� ��Aj?����#����Q��m+�[D&�:����$�$� ��&�:"�# ����3Y;A3+�&��$��:��3��H�#A��3&����$��+5)L ����$����&Z+�&��� ��0�i�5@����#4��U�#��6G��@��U+�=m=R+��$��Ad�0�$� �i��&�:cY�=R+��L� «-v +0=e�# ��a=R+��#��A����$�#��Y�� D��31���&2�(�q D���#���0=Ë� �3���#�9=R���$�#��:<�#+C���%¾ æ ��+BA3A�[��$�%� ��.�%� D�"!�@��#���$��:<�/�M�0�#����ke� ��&D4��T+�=��$68�$�T ����%��#68��@�@����L@�+w)����j?�+�[�&�:Z�# ���&Z�# ��0�i+�=��# ��%+O1��3� ��@�@e�3&��#�36�?�@���kl�f� �0&�4��%+0=h�#6G�#�0Qfs;��&�A3��¾ æ +BA3A�[��$�)L���# � ���4� C=R�$��ob[���&DA3\]��A��$+��#�a�t@P���$4����#��68�D@���Y��X�S=R+�@�@�+O)L�S=R�#+�6±� ��5=R�#�3ob[���&�A9\r��&2� �3�#���$�3�M�.� ��+�&+�=����$+�?���?D��@��X�*\T� ��0�S¾ æ ���j1��3�$\(@���£0��@X\I� +T+;A�A3[��i=R+��i��&2\(4���1���&Z=R+��$��A����$�G�qÐj+��#�%�d����¨2�WQi_`�S��@��$+=R+�@�@�+O)L�>=R�#+�6��3@��368�3&b� ���$\T���#+�?���?���@����*\"A�+�&�A3���D�#�i�# ��0�i¾ æ Ad��&(?K�U1��3�$\7@���£��3@�\T�#+8+;A�A3[��i+�&�@�\I�X=� D��=R+�@�@�+w)L��&�4"A3+�&�:��X� ��+�&F���U�#�0� ���$!��3:^�U����AM r+�=��T@P���$4���68�wv$+��$���*\7+�=� ��5��+��#�$��?�@��%� �$[��5�$� �0� �3�²Tw`»Zx�¼ ��»däXä�äX»�y ���$�#+;A��P�.� ��:C)L�X� ��I4���1���&C=R+��$��Ad�0�$�����f?K�9�#� �3���������$+dVD��6"�.� ��:�?2\F�$+�6G��!�@��#���#�3:�$� �0� ��� ���&%?b\��#+�68��+w1����#��@�@;+������ �.� ��+�&���@b��&��$��6�?D@��,6G��6%?K�3��QW�j+O)�Y�� ��,o;[D���$�#��+�&����#���#���3Y�k³�S+B�3�� D���LA�+�&D:����#��+�&^kl��� �.� ���/=q��A3�#��+�&t��68��@X\8�# ��0�j6G+��$�j+�=m� ���!�@X� ���$��:7�$� �0� �3�f���$��?K�9�#� �3�f�0�����#+dVB��68�.-� ��+�&��L+�=g�K+��$�#��?�@��S�#�#[��a�$� �0� �3�:��k�Q�¬U�/� �3&��#��?�@�\�YÉ�# �����&��$)����j� +H� ����L����k³&D+�k�Q>¬S&���A���&7��68��4���&D�U�

Page 14: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

�d�

�#A3��&��0�#��+G)L ����$����&Z� ����K+��#�#��?�@���� �$[��a�$�M�.� ���S���#����@�@p4��#+�[�����:Z&��d���f�8�$��@P�.� ��1���@X\I�#68��@�@p�D�#+���+��/-� ��+�&�+�=,�# ��H!�@��#���#�3:'�/�M�0�#����Q"_*&r�# ����a�#A3��&����$��+�Yp�# ��"�0?K+O1��8A�+�&D:����#��+�&�A�+�[�@�:r?K�H�#�0� ���$!��3:^Yg?�[B�+�&�@X\G�%6G��&�+��$���*\5+0=^� ��j!�@��#���$��:T�/�M�0�#���>6G��4� 2����A9� [���@�@�\H?��f?��3�$� ���,�������$+dVD��6"�.� ��+�&���+�=��K+��#�#��?�@��� �$[��5�$� �0� �3��Q5ªj+O),�91����3Ye�$[�AM ]�t�$A��3&����#��+t���SA3+�&2� �#�X1��3:^Ym��&�:CA�+�&�A��3�D� [��0@���©��0� ��+�&��S+0=��X�U�#[D4�4����/�� ��0�U���i),+�[�@�:���&21�+�@�1��%� ��5+��K�3� �0�#��+�&��0@e�3&��#�36�?�@��5?�����&�4I�#[D�K�3�/-`10���$�P��&2�f+��S�9V;�#�#��6G��@X\F?������#�3:���e��4�Q��2�WQz�j���X� ��3�S+�=�� ��3�#�HA����#A3[�68�/�M��&DA����S=R�$��ob[��3&b�#@�\CAM ���� ��A9� �3�#��©3�5+������ �.� ��+�&���@m��&��$��6�?�@����3Y��A3A�+��$:���&�4C�#+]:�+;A�[D68�3&b�#��:��$�#[�:����3��Q�sB����A��X!�Ad�0@�@�\�Y��$� [D:����3�8��&�:���A��0� �T�# ��0�H+�&D�7+0=f� D�768�wv$+�����$+�?�@��368�S)L�X� �+������#�0� ��+�&���@e��&��$��6�?�@����a���S�# ��0�U� D�3\r���$��� +;+7+0=Ë� ��&r�#[D?B-`10���$�P��&2�dYp&D+��a�#[������Â-10���#����&2�i�q�0Q³4�Q�t\B@�&��j�3����@¦Q ��«�«2� �9Qg�i@��$+�Yb+�?��#�3�$10�0�#��+�&���#�342���$:���&�4U�# ���kl���$+��K+��$� ��+�&H+�=^+�[D� @������$��k�q��Q³�0Q��# ��5���#+��K+��/� ��+�&C+�=�� ��68����&F)L ���AM C�# ��51����$��=Ë\;��&D4I�$�M�.� �5@������S+�[D� �$��:����# ��H��&21��3@�+����H+�=���&��&D�#��6%?�@��a:����/� �$��?�[D�#��+�&��=R+��L+������ �.� ��+�&���@^�3&��#�36�?�@��3�f+�= Ê�«�« ��g�G4���+��K+��#��&2� ����@^ ��3��4� 2�f���#��&�+0�A�+�&��#���$� �3&2�i)L�X� t� ��3�#��=R�#��ob[��3&2� @�\t?�����&�4G�9V;� �$��6G��?���������&t� D���#�a:����$� �$��?�[B� ��+�&���Q��+��L��&��/�M��&DA���Y� D���#�3�#[�@X� �f+0=��(+�@��#��&��m�3�S��@¦Q��/�O�����2�L��&�:F�>�#4����H�Â�O�������L?�+�� Z��&�:���Ad�0�#�a�# ��0�i� ������#+���+��/� ��+�&+�=�+�[D�#@����3�#�j=R+��S� ��H�h[��$+����d��&'¥,��&2� �$�G=R+����(��:D��[�65-*�#��&�4��%­]�d�.� ��3����+��$��A����$�#��&�4r�q��¥��t­â�h���&D�#��6%?�@��L+�= Ê�«�« ��e�a ��3��4� 2����h@����$���# ���& ��«-v =R+����*\;����A���@�68�3:���[D6H-`� ��&�4���=R+��$��Ad�0�$�@�����:D� ��68�3��Q­]���#���# ����$�H:����/� �#��?�[D�#��+�&��j� +I?���=R�#�3o;[D��&2� @X\CAM ����#��A3�#���#��©��3:r?2\C�WV;� �#�368�5?D�P���3Y^+�&���)�+�[�@�:r�WVb-����A3���# ��H���#+���+��/� ��+�&C+�=,+�[D� @������$�f� +t?��HA�+�&��$��:��3� ��?D@�\F4��#�d�.� ���3Q5_*&r1;���3) +0=�# ����$�8��+���&b�#��Yp� D��#A3��&��0�#��+G)L ����$����&Z� ����K+��#�#��?�@���� �$[��a�$�M�.� ���S���#����@�@p4��#+�[�����:Z&��d���f�8�$��@P�.� ��1���@X\I�#68��@�@p�D�#+���+��/-� ��+�&r+�=,�# ��G!�@�� �3�#�3:'�$� �0� �3���$ �+�[�@�:'&�+0��?��8�7A3+�686G+�&r+BA3A�[��$�#�3&�A��0QI�� ����a68����&��U� ��0��)L D��&� D�%��?�+O1��aA�+�&D:����#��+�&]�R�# ��0�j��&D�#[��$���U¾ æ ���L1����/\(@���£0��@�\D�,���j�#�0� ���$!��3:^YÉ���j���L4��3&����#��@�@X\T?���A���[��#�%�@P�0�#4��i���#+���+��/� ��+�&C�q?D[D��&�+��>&���A3���#�#���#��@�\T��@�@P��+�=p�# ��S!�@X� ���$��:I�/�M�0�#�������$�S?��3�#�#���L���D���#+dVB��68�0� ��+�&��+�=e� �#[D���/�M�0�#���i�# ���&Z���#�a��&2\768�36�?����#�L+�=g� ��a+O1����#��@�@p+��K�3� �0�#��+�&��0@p��&��$��6�?�@���Q>_`�f��@��#+G68����&���Y?2\7�9V;� �3&��#��+�&^YK�# ��0�i��=�¾ æ ����=R+�[�&�:Z�#+"?��a1����/\Z@���£��3@�\I�X�j���j?K�3Ad��[��$�%6G+��$�i+�=��# ���� ��6G�a�8@����#4�����$+��K+��$� ��+�&T+�=p�# ��S!�@X� ���$��:I�/�M�0�#���L���$�U?K�9�#� �3�������D�#+dVB��68�0� ��+�&��+�=�� �#[D�U�$�M�.� ���3Qh­¡��� 8� ����>A�+�&;-A�@�[��#��+�&^Y��0�#�#�3�#�$68�3&b�i+�=e�31���&2�S¾ æ kl�j@���£0��@��� �+;+B:F�q�¦Ql��Qh�X� �>=R�$��ob[���&DA3\t+�=e+BA3A�[��$�#�3&�A��d�L���>� ��S=R+;Ad��@��+���&2��+�=��# ��U�#�3@P�0�#��1��fA�+�6G�����$���#+�&T?K�9�*),�3��&7� ��U:����$�#�#��?D[D� ��+�&T+�=m�������/-¦)L���#�i!�@��#���#�3:I�$� �0� �3�i�0&�:� D�a+��K�3� �0�#��+�&��0@c��&D�#��6%?�@��0Q

Page 15: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� Ê

7h, {r@AK!EF@�9;Q0T7}|d7H=+*0�;Qo*7�{r@AK!EJ+*G;L9�@A>�¢�#�3@P�0�#��1��5A�+�6G�����$���#+�&F+�=� ��5�$68�$�5AM ���� �0A3� �3�#���$� ��A���+�=�� D�G�������Â-`)L���$��!�@X� �3�#��:��/�M�0�#�G:D���/-

� �$��?�[B� ��+�&8�0&�:"� D+��#�j+�=^�# ��i+��K�3� �0�#��+�&��0@���&��$��6�?�@��f:����$�#�#��?D[D� ��+�&HAd��&8?K�jAd���$�#����:"+�[D�,[D�#��&D4�� D�@P�0�#4��L��&D�#��6%?�@��i:��0�M�0�#�3��:��3�#A��$��?���:T��&IsB�3A3�#��+�&8�2�DQ��� ��j!��#�/�>���0�$��+�=c�# ��fA3+�68�����#���#+�&G��&21�+�@�1��3��d�0AM 5:����/� �#��?�[D�#��+�&^kl�mA�+�6G�K+��#���#���#��&�4���+�=��$68�$��Qh¥,+�68��[B�M�0�#��+�&�+�=�� ���A�+�6G��+��#�X� �� ��&�4��,+�=É�$68�$�=R+���� ��f?����#��Ai�3&��#�36�?�@��f!��#�/�>��&b1�+�@�1�����+�?D�M�0��&���&�4a�# ��f�#��&�4��i+�=��#6G�#�L=R+����d��AM T+0=p� �� x �D�f?����#��A��&D�#��6%?�@��3���&5� ���:��.�M���$�3�dQg�� ��L� ��&�4���+�=c�$68�$�>=R+�����&2\5+�&D��+�=K�# ����$�L��&��$��6�?�@��������h:D�3� �3�#6G��&��3:?2\�!�&�:���&�47�# ��G68��&���6�[D6���&�:]6"�wVD��6�[�6±1��0@�[��3��+�=>�#6G�#�G���#�$+BA3�P�0�#��:r)L��� '��&2\�68�36�?�����+�=� D�G4���1��3&���&��$��6�?�@���QG¬S&�A3�H�# ����$�G�*)�+7o;[���&2� ���#���3�����#�5:��9� �3�#6G��&��3:�=R+�����@�@ x �D�5��&D�#��6%?�@��3����&� D�(:��.�M���$�3�dY,�d��AM �o;[���&2� ���*\E���5� ��3&��d1��3� ��4��3:�+O1��3�G�# ��(:��0�M���$�3�G� +r+�?D� ����&N� D�(A3+�68��+��$��� �� �0&�4��G+�=��#6G�#�H=R+��U� D�"?����$��AG��&��$��6�?D@���QI�i&���@�+�4�+�[����/� ���D��:��3�#���$68��&��G�# ��GA�+�6G�K+��#���#�H� ��&�4��+�=�#6G�#��=R+��S� ��5�$�3�a+�=�������/-`)L���#�%!�@�� �3�#�3:r�/�M�0�#���3Q��e+tA3+�68�D@���6G��&2�S�# ��5A�+�68��+��#�X� ���#��&�4��3�U+�=�#6G�#��=R+��L� ��a?����$��Aa��&��$��6�?D@����0&�:(�# ����$�3�f+�=�!�@X� ���$��:(�$�M�.� ���3Y��# ��%A3+�6G�K+��$���#�a@�+O),�3�i?�+�[�&�:(+�=� D�5+w1����#��@�@��3&��#�36�?�@���kl�U� ��&�4��5+�=�#6G�#�����U��@��$+I+�?D� ����&��3:^Qa�� ��5@�+O)����U?�+�[�&�:�+�=�� D�5� ��&�4���+�=�#6G�#��=R+�����&2\C+�&��H+�=� ��H+O1��3� ��@�@���&D�#��6%?�@��3�����U:��3�#���#6G��&D��:�?2\F!�&�:���&�4t� ��56G��&���6�[�6�10��@�[��+�=e�#6G�#�����$�#+;A����0� �3:t)L�X� 7��&2\I6G��6�?�����+0=e�# ���4���1���&t��&��$��6�?D@���Q�¬S&�A3���# �����ob[��0&b�#���*\I����:��3�#���Â-6G��&��3:t=R+��i�0@�@ x �D���3&��#�36�?�@��3�f��&t�# ��a:��0�M�0�#�3��YÉ���j���j�d1��3� ��4��3:Z+O1��3�i�# ���:��0�M�0�#�3�j� +G+�?D� ����&t� D�A�+�68��+��#�X� �j@�+O)����>?�+�[�&�:I+0=p� ��S+O1���� �0@�@c�3&��#�36�?�@���kl�>� ��&D4��S+�=��$68�$��Q���+��>�0:�:����#��+�&���@��#�9=R���#�3&�A��0Y� D�aA3+�6G�K+��$���#�U68�3:��P��&T10��@�[��U+�=g�#6G�#�U���L+�?B�M����&���:I=R+���?�+�� I�# ���?����#��A���&��$��6�?�@��a��&�:t� ����#�9�+�=��������Â-`)L���$��!�@��#���$��:'�/�M�0�#���3Q7�� D�"A�+�68��+��#�X� �H�#�3�#[�@X� �����#�8:����$��@P�d\���:'��&'�e��4�QT�BY���&D:'�#����:���@�\�0JK+��#:(�#�91����#��@��$[�4�4��3�$� ��+�&��3QS��+��j��&��/�M��&�A3��Y�� ����#��@��0� �X1����K+��$���#��+�&7+�=��# ����#+����S+�=g� ��a�*),+I?��0�:�����4�� �068���&�:D��Ad�.� ����� ��0�,� ��S68�.VB��6%[�6¢1��0@�[��i+�=m�#6G�#�S��&I��&2\T�$�3�L+�=p!�@X� �3�#��:I�/�M�0�#���>�#��&�:��>�#+?��fA�+�&��#��:����#��?�@�\G@����#�� ��0&8� ��0�>��&8� D�fA�+��#�#�3�#��+�&�:���&�4�?��0�#��Aj��&D�#��6%?�@��0Qh�f@��$+�Yb� ��i�K+��$���#��+�&G+�=� D�%?�+��#�#+�68�L+�=g� ����*),+"?����j:��P��4�� ��6G�L�#��@��0� �X1���� +G�d��AM C+0� ��3�f��&D:(� +G� ���@���&����#���D�#���$��&2� ��&�4� D�7A3+�6G�K+��$���#�I@�+O),�3�H?K+�[D&�:�+0=i�# ��t+w1����#��@�@��3&��#�36�?�@��0k³�G� �0&�4��t+�=i�#6G�#�t��&�:D��Ad�.� �����# ��0�H� D�6G��&���6�[�6��#6G�#����&C�0&b\Z�#�9�U+�=�!�@��#���$��:C�$� �0� �3�f� �3&�:��S�#+I?K��@����$�f� ��0&F� ��0�f��&F?�+��# F� ���?����#��A

Page 16: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

�O�

��&�:H+O1��3� ��@�@���&D�#��6%?�@��3��Q��g��&���@�@X\�Yw� ��L�$��@P�.� ��1��>��+��#�X� ��+�&�+�=�� ��>�*)�+�@���&��,�#��4�68�3&b�#���#���D�#���$��&2� ��&�4� D�"A�+�68��+��#�X� �H6G��:�����&r10��@�[����a+�=��$68�$�8�$[�4�4��3�$�#�%� ��.���# ��86G��:D�P��&r�#6G�#�H)L���# ���&'��&2\��#�3�%+�=!�@X� ���$��:7�$� �0� �3�j�#��&�:��j� +G?K��A3+�&��#��:����#��?�@X\I@��3�#�L� ���&t� ��0�j��&I� ��a?����$��A���&D�#��6%?�@��0Q,�� b[���Y�+�&(�A�+�68��+��#�X� �L?����$���3Y;�# ��f�$�3�>+0=^�������/-¦)L���#��!�@X� �3�#��:"�$�M�.� �����9VB ���?����#��@�+w)�����10��@�[D���+�=p�#6G�#�L� ���&"� D�+������#�0� ��+�&���@��3&��#�36�?�@��8:�+;���a=R+��a�0�a@������$��� D�#���8ob[���&2� �X� ���3���a6"�wVD��6�[�67Yp68��&���6%[�6(Y^��&�:'6G�9-:�����&H�#6G�#��Qg�� ��j=q�0A3�h� ��0��� ��LA�+�68��+��#�X� ��6G��&���6�[�6è�#6G�#��=R+��h�# ��L�#�3�+�=K!�@��#���$��:G�$�M�.� �������@����$�� ���&G�# ��LA�+�6G��+��#�X� �>6G��&���6�[�6 �#6G�#�>)L�X� ���&H�# ��L+O1����#��@�@��3&��#�36�?�@��j���h�����/� ��A3[�@P�0�#@�\���6G��+��$� ��&2�dY?���Ad�0[��#�I�X�%�������#�3@���68��&����$\Z��&�:D��Ad�.� ��+�&]� ��0�%�# ��TA��$[�A�����@�31��3&2�"¾ æ +�=isB�3A3�#��+�&<��:<=R�#��ob[��3&2� @�\+;A�A�[D�#�U��&D:^YK?2\(�9V;� �3&��#��+�&^Y�� ��0�i� ����������/-`)L���#��!�@��#���$��&�4G���#+;A��3�#�SA���&CA�+�&��#���$� �3&2� @�\7��:���&2� �X=Ë\7��#�9�a+�=,�/�M�0�#���U� ��0�a���#�5@����$�UA�+��$�#[��D�#��:r?2\C�$�#+BAM ��0�$� ��AH���#�$+��#�U�# ���&C� ��5+��K�3� �0�#��+�&��0@g6G��6�?����$����$��Q

C~, ./+*;QHYu8|d7H=+*0�;Qo*7�{r@AK!EJ+*G;L9�@A>�f:D:����#��+�&��0@i��&��#��4� 2�IAd��&¤?��C42����&���:¤?2\��310��@�[��0� ��&�4]�# ��C�������/-`)L���#�Z!�@�� �3�#�3:¤�$�M�.� �C:����$� �$�X-

?�[D�#��+�&ck³�%���#�$+BA3�P�0�#��:<�$68�$�810��@�[D���%��&'�$��@P�.� ��+�&r�#+Z� �+��#�"+0=�� ��8+��K�3� �0�#��+�&���@���&��$��6�?�@��"+�&<��&��&D�#��6%?�@��W-*?2\2-*��&D�#��6%?�@��(?����#����Q¤��+��5�# ��Z�0�#�#�3�#�$68�3&b�"� ��0�H=R+�@�@�+O)L��Y,��@�@ Ê�Ê �������Â-`)L���$�I!�@��#���#�3:�$� �0� �3�f)����#�%4���&��3� �0�#��:F=R+��i�d��AM C+0=�� �� x �D�%+��K�3� �0�#��+�&��0@p��&��$��6�?�@����U��&(� D�%:��0� ���#�9�dYc��&�:F� D��#6G�#�G+�=,�d��AM '!�@�� �3�#�3:]�/�M�0�#�F�q���a)���@�@��������AM <+������#�0� ��+�&���@g68�36�?K�3�M�f),���%�$[�?��#�3ob[���&2� @X\'A���@X-A�[D@P�0�#��:^QÝ�i:�:����#��+�&���@�@X\�Y�=R+��"�d�0AM â+0=U� �� x �D�F��&D�#��6%?�@��3�T� ��F+O1���� �0@�@f+��K�3� �0�#��+�&��0@i6G��6�?�������$�#+;A��P�.� ��:N)L�X� N� ��I�$6"�0@�@��3�$��10��@�[��T+�=i�#6G�#�r�R ����$�d�0=Ë�#���5�$�3=R���$�#�3:��#+]�0��� ��t´�?����/� ´�+O1����#��@�@+������#�0� ��+�&���@�6G��6�?���� ��),���S��:���&2� �X!���:cYc��&D:F���#�i10��@�[D��+�=��$68�$��),���S[��$��:����f�"?���&�AM �68���$£t=R+��A�+�68���0�#���$+�&^Q��m+�?���4���&")L���# 8�# ��U���$�#���$�#6G��&2�dYÉ�e��4�Qg¨%:����$��@P�d\;�,=R+����d��AM I+0=p� �� x �D�S�3&��#�36�?�@��3�� D�f&b[�6�?�����+�=^!�@X� ���$��:8�$�M�.� ����)L��� 8@��3�#�,�$68�$�j�# ���&"�# ��j?K�3�$�>+O1���� �0@�@�+��K�3� �0�#��+�&���@�68�36�?�����Qg_`����e��&D=R�3�#�$��:5�#��4� 2�h�d),�O\�� ��.�g=R+��h�i4��#���0��68�wv$+��$���*\a+�=�� D�>��&��$��6�?D@������f!�@�� �3�#�3:��$� �0� ���WVD���$�#�g� ��.�+�[D�#�K�3�$=R+��$68�U�# ��8?����/�%+O1���� �0@�@�68�36�?�����Q8_*&�:��3��:^Ym� ����a���U� �#[D�G=R+�� x�x�� �R+��a� ��v ��+0=,� D� x �B���&D�#��6%?�@��3��Qh­¡ ��0�L���>68+��#��YB[��#[���@�@X\8�# ����$�U���>6G+��#�i� ���&t+�&D�S!�@��#���#�3:I�$� �0� �S�# ��0�L+�[D�#�K�3�$=R+��$68�� D�7?����/�"+O1���� �0@�@�6G��6�?����3Q�s;�K�3A���!�Ad��@�@�\�Y+�&��d1����#��4��t� ��3�#�7���$�f����Q³�F!�@��#���$��:��$�M�.� ���H)L���#

Page 17: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

�w�

@��3�#�U�$68�$��� ���&F�# ��5?����$��+O1���� �0@�@�6G��6�?����3Q�_*&�+��# ����S)�+��$:���Y^�# ����U6G�d��&D�f� ��.�S=R+�����&2\F4���1��3&��&D�#��6%?�@���+�&���A���&F�9VB����A9�S� +G!�&�:F=R+�[��j+��j!�1��%!�@�� �3�#�3:F�$� �0� �3�f)L���# Z@��3�#�i�#6G�#���# ���&Z�# ��%?����/�+O1��3� ��@�@p68�36�?K�3��Q

_`�j�#�36"����&��,�#+G?����#�3��&�v$[��$�j �+O)ç6%[�AM (��68�D�#+O1��368�3&b�L��&I�#6G�#��6G��4� 2��?������#�$+BA3�P�0�#��:I)L���# ��&2\E+�&��t+�=i� ��3�#�7!�@��#���#�3:��/�M�0�#���3Q��m+<��:�:D�#���$�G�# ����3Y>�g��4�Q��r:D���#�D@P�d\;�H=R+��G����AM �+0=f� D� x �B���&D�#��6%?�@��3�L�# ��S�����$A���&2� ��4�����6G���#+O1���6G��&2�dY���&"� �3�#6G��+�=m�#6G�#�U�0&�:t�#��@��0� �X1��f�#+�� ��U?����/�L+O1����#��@�@+������#�0� ��+�&���@h6G��6�?����3Y����$�#+;A����0� �3:')L���# ]� D�"!�@��#���$��:'�/�M�0�#�")L���# ]�# ��"@�+w)����/�5�#6G�#�Z�q D���#���0=Ë� �3��#�9=R���$�#��:C�#+t���S� ��%´�?K�3�$�#´t!�@�� �3�#�3:C�$�M�.� �O�WQ%ÐL�d�0:���@X\F���D�����#�3&2�U��&F�# ���!�4�[��$�5���j� ��0�U�# ��������Â-A��3&2�M��4��L��68���$+O1���6G��&2�h=R�$��ob[���&2�#@�\5�����0��@������$� Ê�v Yb��&D:G�$+�6G�3� ��68�3�h�9VBA��3��:��j� «-v Q��� D�i�d1���� �04�������#A3��&2�a��6G���#+O1���6G��&2�U��� Ê Q �*v Y^�# ��56G��&���6�[�6������#A3��&2�a��6G���#+O1���6G��&2�U���j-Â�DQ³� v ���0�$�M����&���:C��&���$�#+;A��P�.� ��+�&%)L���# �� ��U� « ¬SA9� +�?���� ��«�«�x ��&��$��6�?D@��O�WYb��&�:5� D��6"�.VB��6�[�6è�K�3�#A3��&2���6G���#+O1��368�3&2���� ��« Q «-v ���0�$�M����&���:���&����$�#+;A��P�.� ��+�&�)L�X� �� �� « �j����?��$[����/\ ��«�«�x ��&D�#��6%?�@��d�9Q��e��4�[��$��� « ���#+O1;��:��3����D�#��A3���$�S?��#����£b:�+O)L&I+�=^�# ��f&b[�6�?�����+�=p?����/�,!�@X� ���$��:T�/�M�0�#���,)L D+��#�i�K�3�#A��3&2�M��4��f��6G���#+O1��368�3&2���&r�#6G�#�H=q��@�@��S)L���# ���&r�74���1���&r� ��&�4��8+�=�10��@�[��3��QT_`�a),���%�/�M�0�#��:'�D�#�31;��+�[��#@X\C� ��.�a=R[D@�@�\ x�x2� +�=x �D�U?����/�L!�@��#���#�3:t�$�M�.� �����R+��m� � ��v �,�#�34����/� �3���#+�6G�S!�&��X� �S��+��#�X� �X1��U�����#A3��&2�M�04�����6G���#+O1��368�3&2��#�3@P�0�#��1��a�#+"� ���?K�3�$�f+w1����#��@�@m��&��$��6�?D@���6G��6�?����3Qm�i+O) ���jAd��&��0@��#+G?��%��&B=R���#�$��:Z=R�$+�6±�g��4DQU� «� ��0�a�O¨ « +�= x �D��?K�3�$�j!�@��#���$��:7�$�M�.� ���%�q+��m� Ê�«-v �>�#��4����$�#���4� Ê�v ��68�D�#+O1��368�3&b��YÉ��&�:t� ��.� x �+�= x �D�H?����$��!�@X� �3�#��:r�$� �0� �3�8�q+��/� � «-v �f�$��4����$� �3����� «-v ��68�D�#+O1��368�3&b��QGs;�M�.� ��:r:��XJc�3�#��&2�#@�\�Y� D���#�f�#���$[�@��#�,��&�:���A��0� �L� ��0�>�0?K+�[B�>+�&��i+�=p�91����/\8�*)�+5��&D�#��6%?�@��3�,)L��@�@� ��d1��U��&T���#�$+BA3�P�0�#��:"?����/�!�@X� ���$��:8�$�M�.� �i�# ��0�,�0JK+��#:��e� Ê�v ��6G���#+O1��368�3&2�,��&8�$68�$�f�#�3@P�0�#��1��L�#+��# ��f?����/�>+O1��3� ��@�@��$+�@�[B� ��+�&+�=�� D�U4���1���&t��&��$��6�?D@���Y���&�:I� ��.�j��?K+�[D��+�&��U+0=e�91��3�$\C� « �3&��#�36�?�@��3�>)L��@�@c ��d1��a��&t�0�#�#+;A����0� �3:?����$�L!�@X� �3�#��:7�$� �0� �U�# ��0�j�0JK+��#:D�b�¢� «-v �$��@P�.� ��1��U��6G���#+O1���6G��&2�dQ

�m+S�#�3��� �3� �0�#��Y.� ���A�+�@�@���A9� �X1���#���$[�@��#�����$���#�3&2� ��:H��&��g��4���Qm¨.- � « �$ �+O)��# ��0�g�i10���$�g6"�Ov$+��#�X�*\U+�=� D�L�#��6G�,� D�i:����$�#�#��?D[D� ��+�&�+�=c�������Â-`)L���#�,!�@X� �3�#��:G�$� �0� �3�)L��@�@BA�+�&2�M�0��&H6�[�@X� ���D@��,�/�M�0�#����� ��.�� ��O1��@��3�#�>�$68�$�f� ���&T� ��f?K�3�$��+O1��3� ��@�@c+������#�0� ��+�&���@É6G��6�?����3Y���&D:"� ��.�>=R�#�3o;[D��&2� @X\"� D�S:����/� �$��?�[D�#��+�&A�+�&b� ����&D�U�0�S@��d���/�S+�&D���$� �0� �%)L D+��#�5�$68�$�%���f�T&�+�� ��?�@�����6G���#+O1���6G��&2�S�$��@P�.� ��1����#+"� ��.�S+�=�� D�?����$�a+O1��3� ��@�@�+������#�0� ��+�&���@e68�36�?K�3��Q��� ����$�8�$���#[D@�� �a���$�5���$�#��&2�#�P��@h?���A���[��#�H� D�3\C:��368+�&��$�#� �0�#�

Page 18: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

�O¨

� ��0��� D�T�31���&2�8¾ æ +�=isB�3A3�#��+�&N��:N+;A�A3[��#�%+�&E�Z1��3�$\<A3+�&��$���$�#��&2�H?����$���3Qr_*&N+0� ��3�%),+��#:��3Y���&��A3A�+��$:���&�A3�G)L�X� C� ��G���$4�[�6G��&2�����$���#�3&2� ��:'��&]sB��A9� ��+�&���:^Yp� ��3�#�G�#�3�#[�@X� �����/�M��?�@����$ ��# ��0��� D�:����$� �$��?�[B� ��+�&G+0=^�������/-¦)L���#�>!�@�� �3�#�3:8�/�M�0�#���,:D+B�3��Y;��&H=q��A3��Y;A3+�&2�M����&8�/�M�0�#����� ��.�>���$�L4���&��3� ��@�@�\5@����$�A�+��#�#[D�D� �3:(?2\t�$�#+BAM ����$�#��A����$�#+��$�L�# ���&7��&2\t+������#�0� ��+�&���@K68�36�?K�3�#�L���$��Q

�^z �S×b~wÚ`�É~���"�|e��×Z�e�Q��×b~w×B|�}w��"#��Øz�8�Ë}wÛe�q|�}OÛm×Z�e�qØ3}O~.�i�g�p}w�R�K|E�ÉÚm���Ë}O×b~.×;��Ø3}�"D}O×;Ø�m+��# ����e�K+���&2��� D�>A�+�&DA����$&8 �����?�����&5)L�X� H�S�#�3@P�0�#��1��>A�+�6G�����$���$+�&�+�=��# ����������/-`)L���#�!�@��#���#�3:

�$� �0� �3��k�����$=R+��#68��&�A����0&�:7�# ��0�j+�=e� ���+������#�0� ��+�&���@K68�36�?K�3�#�3Q,��[��/� ��3�j[D&�:����$�$� ��&�:���&�48+0=g� D�!�@X� ���$��&�4G���$+BA3���#�i[�@��#��68�0� �3@�\"Ad��&(?K��42����&D��:7?b\7�WV��068��&���&�4H� D��ob[��3�$� ��+�&F+�=g)L ��3�# ����iA����/�M����&!�@X� ���$��:<�/�M�0�#�����/\;�$� �36"�.� ��A���@�@X\C���$+w1��I68+��#�G�3JK��A3�#��1��"� ��0&<+�� D���#�3QF��1;��:��3&�A��T�# ��0�%�#[�4�4����$�#���&7�0nG�#68�0�#��1��S��&��$)������#+�� �����o;[D���$�#��+�&t����YD��&T=q��A3��YD=R+��/� �A3+�68��&�4%=R�$+�6 �0�>@��d�0�$�>�*)�+8:D��JK���#�3&2������#�$�K�3A3�#��1��3��Q

+-, �rGO@ 6 + 6 ;QHY;L90�;1g!�m7GO9TEF7:g0�;Qo*7�u���#+�?���?���@����/� ��AZ68+;:���@�@���&�4<�����#�$�K�3A3�#��1��]���#+O1;��:D���7��&��$��4� 2�t��&2�#+��*)�+�10���$�P�0�#��+�&��I+�&� D�

��?�+O1��Lob[����/� ��+�&^�,�w���S+UA3���/�M����&5�#[�?��$�3�#��+�=�� D�>!�@��#���#�3:H�$�M�.� �����A3A�+�[D&b��=R+�����:D���#�D�#+���+��$�#��+�&��0� �&b[�6�?����+�=c�# ��f�/�M�0�#����� ��.��\;���3@�:H� ��j6G+��$�&�+�� ��?�@��%�R�¦Ql�/�è� «�v ���$��@P�.� ��1��L��6G���#+O1��368�3&2����YD�0&�:� �G�S+<A��3�$�M�0��&��$[�?��$�3� �t+�=U�# ��(!�@�� �3�#�3:¤�$�M�.� ���T��A�A3+�[�&2�T=R+��"�':����$���#+��K+��/� ��+�&��0�#�(&b[�6�?����8+�=� D�T´�?����$�#´Z!�@X� ���$��:<�/�M�0�#���7�R�¦Ql�G�# ��"!�@X� ���$��:<�/�M�0�#���%)L�X� '�# ��T6G��&���6�[�6��$68�$�8=R+��%��&2\'4���1��3&��&D�#��6%?�@��d�Z�

ÐL��42���$:���&D4L�# ��h!��#�/�g+0=B�# ����$��ob[��3�$� ��+�&��3Y�A�+�&��#��:����p� ��0� x ��+�=B�# �� x �D��?K�3�$��!�@X� ���$��:a�$� �0� �3�e��&� D�jA3[��#�$��&2�),+��$£H�0JK+��#:��3:��Ý� «-v ��6G���$+w1���6G��&2��+O1��3��# ��i?����/�+w1����#��@�@�+������#�0� ��+�&���@B68�36�?�����Y��&�:5�# ��0� x�x +0=��# ����$� x �f!�@��#���$��:H�$� �0� �3��),�3�#�L��:��3&2� ��!���:�)L���# 5� ����$[�?��#�9�A�+�6G���#���#�3:5+�=É!�@��#���#�3:�$� �0� �3�T?����$��:¤[���+�&¤+�&D�Z+��868+��$�Z+�=U�# ��Z�#+���!�1��F?����$��AZ6G��6%?K�3�#�T6G+��$�8:����$�M�0&b�8=R�#+�6 � D���&D�#��6%?�@���6G�d��& �R�¦Ql��Q�\ æ Y!\baOY}\b�dY}\4�OY}\b�W�9Y}���������OQ �gVD��68�D@����I+0=%�/�M�0�#���t��&��# ����T�$[�?��$�3���&DA�@�[D:��k\ æ ¸ aOYX\/a�\b�dYr\4��\/�OYX\4�\ æ�c YX\b��\z��Y��3�#A�Q]�gV��068��@����5+�=L�/�M�0�#���H&�+�����&'�# ������$[�?��$�3���&DA�@�[D:��m\/�:\/�dY�\b� ¸ aOY%\/�:\ æ�c Yb�3� A0Qh�� ��jo;[D���$�#��+�&8����)L ��9� ��3�,+�&D�jA3+�[�@�:8�$��6G��@�\��0�#�#�#��?D[D� ��� D�

Page 19: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

�O�

��:D��&2� �X!�Ad�0�#��+�&r+�=>�# �+��$� x�x !�@�� �3�#�3:'�$� �0� �3�%)L���# ����������%�#+ZAM ��0&�A��0Yh4���1���&'�# ��0�%�7� +0�M��@+�= � � «+�=��# �� x �B�T?K�3�$��!�@�� �3�#�3:<�$� �0� �3�%),�3�#�I��:���&2� �X!���:])L��� ����������dQZ�� ��I�D�#+;A���:D[��#�"=R+��%��&D�$)����#��&�4� D����ob[����/� ��+�&<���a����=R+�@�@�+O)L��QT�e���#�/�dY�� ��H10���#����?�@��������a:D�3!�&��3:<� +7?K�H� D�"&b[�6�?����a+�=,�# �� x �!�@X� ���$��:T�/�M�0�#���,���#�$+BA3�P�0�#��:")L�X� f� � «-v ��68�D�#+O1��368�3&b�� ��0�����$�i��:���&2�#��!��3:")L���# k���������,�# ��#+�[D4� AM ���&DA���QÏ�� ���&cYj�# ��C���#+�?���?���@����*\E68���$�"=R[�&�A9� ��+�&¤+�=4�Ý���8Ad��@�A�[�@��0� �3:^Yj��&�:�� D���"=R[�&DA3� ��+�&����[��$��:��#+'�����$=R+��#6 ��&���&D=R�3�#��&DA��(�#���/�T+�&�)L ��3�# ����G� ��(�31��3&2�T� ��.� x�x +�=f�# ��(!�@�� �3�#�3:��$� �0� �3����$�%��:D��&2� �X!���:F)L�X� ����������UA���&F?��5�0�#�#�#��?�[D� �3:Z� +"AM ���&�A3�5+��f&�+0�dQS�� ��%&;[D@�@m b\;��+�� D���#���U+�=h� D���&B=R���#�3&�A��a���L� ��0�L� ����31��3&2�S���i�H6"��&D��=R���/�M�0�#��+�&t+�=gAM ���&�A3��Yc��&�:(�# ��a�#��4�&���!�A���&�A3��@��31��3@�+�=�� D���&B=R���#�3&�A��t���5:��3�#���#6G��&D��:E:����$��A9� @�\r=R�#+�6¯� D�7�D�#+�?��0?���@����*\r6"�0�#��=R[�&�A9� ��+�&������ ��t���$+�?���?D��@��X�*\+�=j��:D��&2� �X=Ë\B��&�4f�Ô¼8� x�x !�@��#���#�3:N�$� �0� �3�H)L�X� ������L��OQ�_`=L� D�������$+�?���?D��@��X�*\r���%@����$��� ���&N�$+�6G����$?����#� ���/\��# ��#�3�# �+�@�:^Y2� �d\I� v Y��# ���&5�# ��L&;[D@�@D 2\;��+�� ��3�#������h�#�*v$��A3�#��:8��&�:5� D���31��3&2�,���h���#�$[�6G��:&�+��e� +U?����f68��&��X=R���$� �0� ��+�&a+�=ÉAM ���&�A3��Qh�>)�+��#68��@�@b�$�#������=q�0A���@���� �0� ���&�[D&�:����$�$� ��&�:���&�4�+�=�)L ����$�� D�S�9VB���$���#�$��+�&I=R+���� D�S���#+�?���?���@����*\H68���$��=R[�&�A9� ��+�&TA�+�68�3��=R�#+�6(Q��� D�f!��#�/������ +��$�3=R�3�,� +�����AM +�=j� �� x �C!�@�� �3�#�3:��/�M�0�#���5� ��0�G�0JK+��#:��3:��°� «-v ��68�D�#+O1��368�3&b�G�0�H��´��#[�A3A����$�#´�Y�=R+����d�0�#�t+�=:����#A�[D�#�#��+�&^Q��� ����#�3A�+�&D:Z����� +H�#�3�$�M�.� �U� ���ob[��3�$� ��+�&7�K+��$��:(��?�+w1��a����Y�´�_`=e+�&��U ����j�H�$��ob[��3&�A��+�= x �D�5�3@���6G��&2�#�aA3+�6G���#���#��:C+0= x �I�$[�A�A3���#�$���%��&�: x2��Ê &�+�&;-*�#[DA�A��3�#�$���%��&r��&r���#?��X� �#���$\(+��#:D����Y)L ��0�����U�# ��8�D�#+�?��0?���@����*\(+�=,!�&D:���&�4��E�#[�A3A����$�#�3�5��&]��&<���$?����#� ���/\F�#��68�D@��H+�= � � « +0=,� D� x �B���@���6G��&2� �O�d´8�� ��������#+�?���?���@����*\8�����$�d��:D��@�\"A�+�6G��[D�#��:7���

["�i�p�h¼

��  x ��¡£¢¤

��  x �D�LÁ x �

� � « Á��¡£¢¤

��  x �D�

� � «

¡£¢¤

)L ��3�#� ��  Ã

Ä

¡ ¢¤ ¼ ÃF¥

�qÃ�ÁEÄ9�¥³Ä�¥ ä�� ��G@��9=Ë� ���&D:r�#���#6°��&C� ��G&b[�68�3� �0�#+������S�# ��8&b[�6%?K�3��+�=�)>�d\;���#+t� ��6G��@����E+�=�� �� x �I�#[�AW-A��3�#�$���5)L ���&N+��#:��3�%+�=L�$��@��3A3�#��+�&]:�+B�3�#&^kÇ�568�0�#�#���3Ye� D�T�#��4� 2� ���&D:]� �3�#6u��&r� ��T&b[�6G���#�0� +������

Page 20: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

��«

� D�I&;[D6�?K�3��+�=L),�O\;���#+C� ��6G��@�� � � « -p��+�=��# �� x2��Ê &D+�&B-*�$[�A�A3���$�#���H)L ��3&�+��$:����%+�=L�#�3@���A9� ��+�&:�+;���$&^kl�,6"�.�#� �3��YB��&�:G� ��j� ���$6Ñ��&8�# ��f:D��&�+�6G��&��0� +�������# ��f&b[�6�?�����+�=^),�d\B��� +%�#��6G��@�� � � « +�=� D��� +0�M��@m+�= x �B����@���6G��&2� �j)L ���&C+��$:����i+�=��#�3@��3A3� ��+�&Z:D+B�3�#&^kÇ�U68�0�#�#����QfÓU��1��3&Z� D���f�WVB���#�3�#�#��+�&=R+��z["�i�p�9Ye� ��TA3+��#�$���$�K+�&D:���&�4Z���#+�?���?���@����*\�68���#�a=R[�&�A9� ��+�&<����+�?D�M�0��&��3:<?2\'A3+�68�D[D� ��&�4¦[8���^�=R+��§�(¼ « »d��»�ä�äXä�» x �DQh¥>�0�#�$\;��&�4%+�[B���# ��f���$+�?���?D��@��X�*\568���#�=R[�&�A3�#��+�&8Ad��@�A�[�@��0� ��+�&����&�:8�K�3�$=R+��$6H-��&D4T� ��H��&B=R���#�3&�A����#���/�dY����U���S=R+�[�&�:C�# ��0�U� D�5���#+�?���?���@����*\(+�=��:��3&2� ��=Ë\;��&�4T)L���# B���������a�# ��#+�[D4� 6G���#�GAM ���&DA��¨��¼�� x�x !�@��#���#�3:'�/�M�0�#�����0�#�#+;A����0� �3:')L���# ©�±� «-v ��68���$+O1���6G��&2����� � ä �Aª�� «%« � Q��[��$�# ����$68+��#��Ye� ��T&b[�6%?K�3�%+�=�!�@X� �3�#��:N�/�M�0�#�����WVB�K�3A3� �3:N� +F?��T��:D��&2� �X!���:')L�X� ������L��5����+�&�@�\��Ê Q³�DQ�� b[���YÉ� ��3�#�a���L�$�#�#+�&D48�91B��:���&DA����# ��0�!���������j����)L ��0�i68��4� 2�L?K��A3+�&��#��:����$��:Z�G´��D�#�3=R�3�#�$��:�#+�[��#A3��´�+�=c� ��L!�@��#���$��:"�/�M�0�#���,���#�$+BA3�P�0�#��:8)L�X� G� ��j68+��$��&�+��M�0?�@����q��Q³�0Q_�è� «-v �h��6G���#+O1��368�3&2�+O1��3�j�# ���?K�3�$�j+O1����#��@�@c68�36�?K�3��Q

���$��6G��@����h��&��0@�\;�#����A���&"?��f�����/=R+��#6G��:G=R+���# ��jAd���$�S+�=^�$[�?��$�3�m�������:�wY;A�+�6G���$���#�3:"+�=c!�@��#���#�3:�$� �0� �3�"?��0�#��:�[���+�&�+�&D�(+��H6G+��#�t+�=i� ��7� +��E=R+�[��H?��0�#��A76G��6�?����$�G6G+��/�":����$� ��&2�G=R�$+�6 � D���&D�#��6%?�@��i68����&C�R�¦Ql��QU\ æ YJ\badY#\b�dYJ\4�3�9Q�¥,+�&��#��:����� ��0� x�« +0=^� �� x ��?����/�,!�@X� �3�#��:"�$�M�.� ���� ��.��0JK+��#:D��:©��� «�v ��6G���#+O1���6G��&2�a+w1����a� ��G?K�3�$��+O1����#��@�@h+��K�3� �0�#��+�&��0@�68�36�?����U),�3�#�8��:��3&b�#��!��3:)L���# ��������:�wY2��&D:�� ��0� ����x +�=��# �� x �B�>?K�3�$�h!�@X� �3�#��:5�/�M�0�#���g)����$�L��:��3&b�#��!��3:%)L���# ¬�������:�OQh¥>�0�#�$\;��&�4+�[D�j� D�%��&���@�\;�#����)L��� t� D���#��&b[�6�?����#�3YK�X�j����=R+�[�&�:7� ��.�j�# ��a���#+�?���?���@����*\"+�=���:���&2� �X=Ë\;��&�45)L���# �������:�S�# ��#+�[D4� F6G���#�aAM ���&DA��8�r¼8� x�« !�@X� �3�#��:(�$� �0� �3�U���#�$+BA3�P�0�#��:7)L��� ­��� «�v ��6G���#+O1��368�3&2���� x äX�hª�� « « � QZ��[��/� ��3�#6G+��#�0Ye� D�"&b[�6�?�����+�=�!�@��#���$��:'�/�M�0�#�����WVB�K�3A3� �3:<� +F?��"��:���&2�#��!��3:])L���# �������:�U���j+�&�@�\ ��� Q � Q��� b[��3Y^���j)L���# 5���������dYÉ� ��3�#�����j�$� �$+�&�48�31;��:��3&�A��a� ��.�M�������:�U���j)L ��0�f6G��4� b�?��iA3+�&��#��:����$��:I�a´����$�3=R���$�#�3:"�#+�[��#A3��´�+�=K� ��j!�@��#���$��:"�/�M�0�#���,���#�$+BA3�P�0�#��:H)L��� G� ��j6G+��$��&�+�� ��?�@���q��Q³�0Q`�¢� «-v �,��6G���$+w1���6G��&2�j+O1������# ���?K�3�$�j+O1����#��@�@p+��K�3� �0�#��+�&��0@c6G��6�?����3Q

¬S&�����:�:D��� ��+�&���@É+�?D�#���/10�0� ��+�&I6"�0£����>�# ��f!�&D:���&�4��>�#�3@P�0�#��:T� +¬�����L���i��&D:®����������68+��$�S6G�d��&B-��&D4�=R[�@¦Qf�� ����S���i�# ��0�S�# ����������/-¦)L���#�a!�@��#���$��:C�$� �0� �3�S� ��.�UA�+�6G���$���#�8�������:����&�:B�������������$��?����$��:[���+�&F+������#�0� ��+�&���@p6G��6%?K�3�#�f�$��@��0� �X1���@X\I:����$�M�0&b�i=R�#+�6 �# ����3&��#�36�?�@���68����&^YK��&�:C�0�f�#[DAM C�#���;-�#�3�#�3&b�fA�+�6%?���&��.� ��+�&���+�=�6G��6%?K�3�#��� ��.�f�0�#�U� ��a6G+��$�j:��XJc�3�#��&2�L=R�#+�6�+�&��a��&�+��# ����j��&7�H�#+;+��$-6G�d��&B-`�#ob[����$�i�$��&��$��Q�ÓU��1��3&8�# �����+�?��$���/1��.� ��+�&^Y2� ��i��?�+w1��i!�&D:���&�4��A����#�/\H� ��j��6G��@���Ad�0�#��+�&a� ��.�

Page 21: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� �

�������/-¦)L���#�5!�@X� �3�#��&D4(����6G+��$���3JK��A9� �X1��T�.�%�$�d��@���©���&�4t�#�3@P�0�#��1��G��6G���$+w1���6G��&2����&'�$68�$�G)L D��&<� D�?����$��AU6G��6�?����$�L+�=��54��X1��3&(���0���/-¦)L���$�f!�@��#���$��:7�$�M�.� �a���#��&�+0��1����/\t�#��68��@P�����&t���0�$� �3�#&^Q

ÐL��42���$:���&D4f� ��,�#��A3+�&�:Hob[����/� ��+�&H��?�+w1���Y2�X�g�����#����:���@X\�:��3�#���$68��&���:��# ��0�h�.��@������$�g+�&��>�$[�?��$�3�+�=�� ��H!�@��#���$��:'�/�M�0�#���a:�+;������A�A3+�[�&2�%=R+��a�t:����$���#+��K+��/� ��+�&��0�#�5&;[D6�?K�3��+0=,� D�8´�?����/� ´7!�@��#���#�3:�$� �0� �3��Q�¥,+�&��#��:����e�#+�6G���$[�?��$�3�§�Z+�=D� �� Ê�Ê !�@X� �3�#��:��$� �0� �3��Y�+�=��#��©3�^¯CQe_`=��d��AM %+�=�� D� Ê�Ê !�@��#���#�3:�$� �0� �3�,=R+��,�%4���1���&T�3&��#�36�?�@��S�����ob[���@�@�\G@���£��3@�\5� +%?K�j� ��S?����/�$-`�K�3�$=R+��$68��&�4a!�@�� �3�#�3:T�$� �0� �0YB� D��&� D�����$+�?���?D��@��X�*\"�# ��0�L� ��a?����$�j!�@X� ���$��:(�$�M�.� ��)L��@�@c?K�a��:��3&2� ��!���:t)L��� (�#[�?D�#�3�M�����U°C¼±¯7Ò Ê�Ê Q�f�$�#[�6G��&D4a=R+���#��68�D@���A3���*\a� ��.���# ��i��:���&2�#���*\H+�=K� ��j?K�3�$��!�@��#���#�3:"�/�M�0�#�j=R+���a4���1���&"�3&��#�36�?�@��i�����&D:�������&�:D��&2��+�=�� D����:��3&b�#���#�����m+�=É� D�>?K�3�$��!�@�� �3�#�3:5�$�M�.� ���h��&�+��# ����L�R�����/�W�e�3&��#�36�?�@��3��Y��# ���&5� D�&b[�6�?����>+0=m�#��6G������&I���#��68�D@��f+0=I�Z��&��$��6�?�@����>�# ��0�>�# ��U?K�3�$�>!�@�� �3�#�3:I�$�M�.� �U������:��3&2� ��!���:T)L���# �#[D?��#�9�²�ÏA���&�?��t68+;:��3@�@��3:����G�r?���&�+�6G�P��@�@�\�:����$�#�#��?D[D� �3:��#��&�:�+�6¯1��0�#�P�0?�@��")L�X� ��9VB����A3�#��:10��@�[D�t¾¯¼³��°�¼´��¯7Ò Ê�Ê Qµ�i+O)�YA�+�&D�#��:D���H��&E�����/� ��A�[�@����%� D�7�$[�?��$�3�8A�+�6G���#���#�3:�+0=i�# ���� «!�@X� ���$��:T�/�M�0�#���,�# ��0�>���#�i?����#�3:t[��K+�&"� ��iA�+�&2�#�#+�@���\ æ ¸ aOYJ\ba ¸ adY#\b� ¸ aOY�QXQXQ�QXY�\ æ�c ¸ aW�9Y#�·¶iadQ�­¡���# � D�j�#��6G��@��>+0=·�(¼ x �D�L��&��$��6�?�@�����d10����@���?�@��=R+���� D��A�[��$�#��&2�),+��#£ÉY2� ��L&b[�6�?�����+�=�� ��68�3�g� ��.�� D�f?����$�,!�@��#���#�3:T�$� �0� �f�����:��3&b�#��!��3:")L���# ²�·¶ia,���,�9VB����A9� ��:I�#+�?K�f¾ ¼ x �D�*¸�� « Ò Ê�Ê �Ï���DQ � ��+�&��&D�#����A3�#��+�&cY�� D�Z?����$�8!�@��#���$��:��/�M�0�#�(���G=R+�[�&�:���&¹�·¶ia"�D�(�#��6G����Y�+��8��?K+�[D�"� «-v 68+��#�t+�=Ë� �3&� ���&N68��4� 2�%?��I�9VB����A9� ��:cQ'��[��/� ��3�#6G+��#�0Y�� ��T�D�#+�?��0?���@����*\�+�=��# ��I?K�3�$��!�@�� �3�#�3:N�$�M�.� �I?�����&D4��:D��&2� �X!���:�)L���# ��·¶iaa�# ��#+�[�4� '6G���$�HAM ���&�A3�T�D�G+��a68+��$��� ��68�3�a����+�&�@�\ Ê ä Ê ª�� «%« � QT�� b[��3Yg� D���&B=R���#�3&�A��j�����# ��0�� �0&�:�+�6 �D�#+;A����$�#�3�>���$�j D��4� �@X\%[D&�@���£���@X\%�#+%��A3A�+�[D&b�=R+����# ����$�i?�����&�4��#+a6"��&2\?����$�a!�@X� ���$��:r�$� �0� �3�a��:���&2�#��!��3:r)L�X� ­�·¶�a�Q8_`��=R+�@�@�+O)L��Y^�# ���&^Y�� ��0�z�·¶�aaA�+�[�@�:�?��GA3+�&��$��:��3�#��:'��#+.-*Ad�0@�@��3:8´����#�9=R���#�$��:T+��#��4���&�´U+�=c� ��i?K�3�$�!�@��#���#�3:T�$� �0� �0Q � ��+�&8�#�3Ì���A3�#��+�&cYB� D���h!�&�:���&�4%�$ �+�[�@�:&�+��f?K��A�+�&D�#��:D���#�3:��"�#[D�#���$���#�0Yp���i���i���#+�?���?�@�\7���S�"68��&��X=R���$� �0� ��+�&t+�=h� D�%=q��A3�i� ��.�S��&7� �3�#6G�+�=��$68�$�i� D�fA�+�&b�#�#+�@�=R+��#�3Ad���/�>���,?��3�#�#���� ���&8� ��S�����/� [��$?K�3:I68�36�?����#��+�&I�%�/\;�$� �36"�.� ��Ai?����$����Q�i+�&��3�# ���@����#�3YB�X�h���h6G+��$���31;��:��3&�A��L� ��0��� ��3�#�j���#��:D���#A3���$&���?�@����K�3�$=R+��$6"�0&�A���:��XJc�3�#�3&�A��3�)L�X� ���&� D�a:D���$�#�#��?�[D� ��+�&I+�=e�������/-`)L���#�f!�@�� �3�#�3:(�/�M�0�#���L���#�#��@X=ÂQ

6 , º!>·@AK¬+*HYu}E?GO@D»�7:g0�;1@A>¬EF7G:9TEF7:g0�;Qo*7

Page 22: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

���

�f&t�0@�� �3�#&��.� ��1��f�0&�:T��&2�#���#�3@�\G:��XJc�3�#��&2�,�K�3�#�#����A9� �X1��f=R�$+�6¢� ��0�,+�=p���$+�?���?D��@����$�#��A�6G+B:��3@�@���&�4��@��#+j�#[�4�4����$�#�e�# ��0�mA��3�$�M�0��&%�������Â-`)L���$��!�@��#���$��:%�/�M�0�#���e�$\;�$�#��68�0� ��Ad��@�@�\S�D�#+O1���68+��$���3JK��A3�#��1���� ���&+��# ����$��QI�m+F[�&�:��3�#�/�M��&�:]� ��G?����$���a+�=>� D���a�K�3�#�#����A9� �X1���YgA�+�&��$��:��3�a!��#�$�%�d��AM '?����#��AH��&��$��6�?�@��6G��6�?����3k³�8��&�+�68��@X\N�#�3@P�0�#��1��t� +r� ��(��&��$��6�?D@��Z6G�d�0&^Qâ�É+��8�0&b\�?��0�#��A(68�36�?����G+�=��]4���1��3&��&D�#��6%?�@��0Yb�# �������&�+�68��@�\a�����#��68��@X\a:��3!�&D��:8�#+U?K�>�# ��>!��3@�:5+�?D� ����&D��:H?2\%�M�0£;��&�4S� D��4��#��:��K+���&2�$-?2\2-*4��$��:���+���&2��:��XJc�3�#�3&�A��i?K�9�*),�3��&T� D�i?��0�#��Aj6G��6�?������&�:G� ��j��&D�#��6%?�@��j68����&^Q��� D�S��&�+�6"��@X\���$+.v$��A9� ��+�&7+0=���&2\I�*)�+8:D��JK���#�3&2�f�0&�+�68��@�\G!���@�:��LAd��&7��@��#+H?��a:D�3!�&��3:F���

~½¼�¾ ¼ ¿�À »:Á ÂÃ�Ä�Ã!ÃŮà »

)L ��3�#�f�# ��S��&D+�6"�0@�\�!��3@�:��,���#�i���$�#[�6G��:T�#+%?��f��&G1��3A3� +���=R+��$6(Y ¿ Â�:���&D+�� �3����&"��&�&��3�,���$+B:�[DA3�dY��&�: à :��3&�+��#���T1��3A3�#+��t6"��4�&����#[�:���Q��� ��r��&�+�68��@�\E�D�#+.v$�3A3� ��+�&¤68�O\����$�#[�6G�C��&2\�10��@�[��F��&� D�5�#����A3�#�#[�6�-M���#+C��Yp��&�:C6G�d�0�#[��$���S� D���*),+7��&�+�6"��@X\I!���@�:���kc�#��6G��@����#�X�*\I��&(� ���$68�f+�=?K+��# ���0�$� �3�#&E��&�:N�$��4�&cQr�� D�t��?��#+�@�[D�#�"10��@�[��8+�=L� ��"���#+.v$�3A3�#��+�&N6G�d���$[��#�3��� ��"�#��6G��@����#�X�*\C+�=L� D��*)�+C��&�+�68��@X\Z!��3@�:��3ke���.�#� �3�#&��3���� ��8@P���$4����U� D����1��0@�[��H���3Ye�# ��86G+��$�G�$��6G��@����f� D�8�*)�+Z!���@�:���k���0�$� �3�#&������$��Q��� ��H�#��4�&F+�=��# ��5���#+wv$��A3�#��+�&C��&�:���Ad�0�#���i)L D�3� D���U� ����*),+7��&�+�6"��@X\t!���@�:��f� �3&�:� +8 ��d1��%�# ��%�#��6G��+��j+�����+��$��� �a�#��4�&^��_`=h� ������#+.v$�3A3�#��+�&(���j�K+��$���#��1��8�R&���42�.� ��1��O�WY��# ���&Z�# ����*)�+!��3@�:���� �3&�:t� +H ��d1���� ����#��68�8�R+�����+��#�X� �O��#��4�&^Q

����AM I�������/-¦)L���#�L!�@X� ���$��:T�/�M�0�#�f���,���$�#+;A����0� �3:")L���# I��&T�0&�+�68��@�\H���$+.v$��A9� ��+�&^Yb=R+���# ��f�$�d���$+�&� ��0�g����AM �!�@�� �3�#�3:%�$� �0� �����?��0�#��:%[��K+�&��*)�+f:��XJc�3�#��&2�e?����$��A68�36�?K�3�#�e��&�:a� ��3�#���*)�+f6G��6%?K�3�#��k��&�+�6"��@X\I!���@�:��f ��O1��H�"A��3�$� ����&C�0&�+�68��@�\I�D�#+.v$�3A3� ��+�&^QS�� D���f���S�"�$��@��310��&2�f=q��A9�U?���Ad�0[��#�%�# ����$����$�j�$[�4�4��3�$�#��+�&���� ��0��� ��j�K�3�$=R+��$6"�0&�A��L+�=c�d��AM G!�@��#���#�3:"�/�M�0�#�j���h� +a�#+�6G�i�WV;� ��&2�,�a=R[�&DA3� ��+�&H+�=���#�>��&D+�6"�0@�\G���#+wv$��A3�#��+�&81��0@�[��0Qh�e+5�$���S)L ��3�#�S� D���#�U�$[�4�4��3�$�#��+�&��L�0�#���$�f�$��ob[����$���>�# ��S=R+�@�@�+O)L��&�4��&��0@�\;�#����Q��g���#�$��YB=R+��L�0&b\T4���1���&t��&��$��6�?D@���Y��# �� Ê�Ê !�@��#���$��:t�$� �0� �3�i�0�#���#+��/� �3:7=R�$+�6½�$6"��@�@��3�$�,�#+@P�0�#4��3�$����&T� ���$68��+0=e�$68�$��QX�j�9V;�dY��# ��U!�@��#���#�3:I�$� �0� �U)L���# T� ��U�$6"�0@�@��3�$�,�#6G�#�U���L�0�#�#��4�&��3:7�#��&�£��YD�# ��U!�@��#���$��:7�$�M�.� �S)L���# I� ����#�3A�+�&D:Z�#68��@�@����/���#6G�#�U���L���#�$��4�&D��:7� ��&�£ � Y���&�:t�$+G=R+��/� ^YD[�&2� ��@� D�>!�@��#���#�3:5�$� �0� �>)L�X� �� D��@P���$4����/�����$�#+��h���h���$�#��4�&���:5�#��&�£ Ê�Ê Qg�� ��3&^Y2� ��>��&�+�68��@�\a���$+.v$��A9� ��+�&A�+��#�#�3�#��+�&�:���&�4L� +S�L4��X1���&a!�@��#���$��:a�$�M�.� ����m���#�$��4�&D��:��# ��� ��6G���#��&�£U�0�m�# ��!�@�� �3�#�3:%�$� �0� �0Qg�� ��.�

Page 23: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

��x

���3Y�� ��i��&�+�6"��@X\%�D�#+.v$�3A3� ��+�&HA�+��$�#�3�#��+�&�:���&�4a�#+U� ��L!�@��#���$��:H�$� �0� �L)L���# G�#��&�£T�j���h�D[D���&G�a�#��&�£�U´�?���&�´�Y;� �����&D+�6"�0@�\H���$+.v$��A9� ��+�&IA�+��$�#���$�K+�&�:���&D4H�#+�� ��i!�@��#���#�3:t�$�M�.� �S)L���# T� �0&�£ � ���>��[D�L��&�(�#��&�£ � ´�?���&�´DYe�9� A�Q7�� ��T�D�#��A3��:���&�4Z�/� �3���5���$�"Ad�0�#�#����:]+�[D�a=R+����91��3�$\'�O10����@P��?D@��5��&D�#��6%?�@��0Q�>�S�# ���A�[D@�6G��&��0�#��+�&7+�=�� D���f�9Jc+��$�dY^�"�#�3A�+��$:r���f+�?B�M����&���:F+0=�# ��5��&D+�6"�0@�\t���$+.v$��A9� ��+�&F1��0@�[��3����$�#+;A��P�.� ��:7)L���# (����AM F+�=��# �� Ê�Ê �#��&�£b��QL_*&(+�� ��3�j)�+��$:���YÉ)L���# (� D���#��68�D@���+�= x �B���3&��#�36�?�@��3��d10����@���?�@��G=R+��%�# �����)�+��#£ÉY� �� x �B�(�0&�+�68��@�\r���$+.v$��A9� ��+�&E10��@�[����G���$�#+;A����0� �3:E)L�X� N�#��&�£��7���$�+�?D� ����&D��:^Yb� �� x �D�S�0&�+�68��@�\5�D�#+.v$�3A3� ��+�&"10��@�[����,���#�$+BA3�P�0�#��:")L�X� "�#��&�£ � ���$�f+�?D� ����&D��:^YB��&�:T�$+=R+��/� ^QG�� ������$��A�+��#:'A���&r?K�G[��#�3:]�#+Z���$�#�3�#��)L D�3� D���a�#����A��X!�AH=R+��#�3Ad���/�%�#��&�£b��+0=,��&b�#���$���$�%���$�AM ����#��A3�#���$���$�#��A���@�@X\"��+���[�@��0� �3:t?b\"!�@��#���#�3:(�/�M�0�#����)L���# Z�5A3���$� ����&I�*\;���a+0=����$+.v$��A9� ��+�&t10��@�[���Q

��+��h� ��LA3[��#�$��&2�,)�+��#£ÉYb� ��L�$��A�+��#:8),�����M��?D[�@P�0�#��:G[��#��&�4U� ��j��&2�#���#�L�#��68�D@��L+�= x �D�j��&��$��65-?�@�����Q��g��4�[D�#�a���i:D���#�D@P�d\;�,�#+�68�j�$� ��&�:����$:T�$� �0� ���$�#��A���+�=c� ��i���#+.v$�3A3�#��+�&G1��0@�[��3�,���$�#+;A����0� �3:8)L���# �d�0AM <+�=,�# �� Ê�Ê �#��&�£b��Yh�0��:��3�#�X1���:]=R�#+�6 �# ����a�#�3A�+��$:^Qt�g���#�$��YeA3+�&��$��:��3�a�# ��8�#��&�£b���0�#�#+;A����0� �3:)L���# 8�# ��j)�+��$�$�>�$68�$��Y;� ��&D£;��� Ê - Ê�Ê Q�§�+�� G� ��i68����&"�0&�:T6G��:�����&"�0&�+�68��@�\%���#+.v$�3A3�#��+�&G1��0@�[��3�=R+��L�d�0AM F+�=g�# ����$�%�#��&�£b�S���$���$[�?��/�M��&2� ����@�@X\I6G+��#����+��#�X� ��1��U� ���&7� ��a+O1��3� ��@�@m6G�d�0&(���$+.v$��A9� ��+�&10��@�[D��YK��&�:F?�+�� (� ���68����&C��&�:Z68�3:��P��&(���#+wv$��A3�#��+�&(1��0@�[��3�j=R+��i�d��AM C+0=�� ��%� ��&�£b� Ê�« - Ê�Ê ���$���+��#�X� ��1���Qh�� ����/�M��&�:����#:t:��91B���0� ��+�&I+�=m���$+.v$��A9� ��+�&t10��@�[��f=R+�������AM 7+�=�� ��S=R+��$��A����$�j� �0&�£b��� Ê - Ê�Ê�����@��#+��#�3@P�0�#��1���@�\%@�+O)�Y2��&D:���A��0� ��&�4S� ��.��# ��j��&�+�68��@X\����$+.v$��A9� ��+�&��h=R+���� ��3�#�i� ��&�£b��:�+a&�+���� �3&�:� +I:D�������/�S=q���S��&2� +"�# ���&��342�0� �X1��5���0�$�U+�=h�# ��5�#����A9� �$[�6(Q��m+t��:�:C�368�� ����#���f� +T�# ����$�5�K+���&2� ��Y�������a=R+�[�&�:]� ��0��=R[�@�@X\ x2� �N�R+������D� v ��+�=�� �� x �D�T�0&�+�68��@�\����$+.v$��A9� ��+�&'10��@�[D�����0�#�#+;A����0� �3:)L���# F� �0&�£ Ê�Ê ���$�%��+��#�X� ��1��I�R&�+��S�$ �+O)L&É�9QU�� b[��3Yc� D���#�����#��A�@������f��&�:���Ad�0�#��+�&����# ��0�f�# ����#��&�£b����$�#+;A��P�.� ��:I)L�X� T� D�S),+��$�$�L�$68�$�G�q�¦Ql��Qh�#��&�£b��� Ê - Ê�Ê �h� ��&D:7�#+5?K�U��+���[�@��0� �3:I?2\"!�@X� ���$��:t�$� �0� �3�)L���# N��&�+�6"��@X\C���$+.v$��A9� ��+�&��%��&'+��a1����/\<&��������# ��"��+��$��� �X1��8�����$��+�=��# ��"�$�K�3A3� �$[�67Q�¥,+�&��$��:��3���@��#+C� ��t�#��&�£b�"�0�#�#+;A����0� �3:N)L��� E� D�7?����/�G�#6G�#��Y� �0&�£b�(�3- Ê QE§,+��# ��# ��t68����&���&�:�68�3:��P�0&��&�+�6"��@X\7���#+.v$�3A3�#��+�&F10��@�[����S=R+��f�d��AM r+�=�� ��3�#�H� �0&�£b�����#�5&��342�0�#��1��0Yp��&�:D��Ad�.� ��&D4"� ��.�S� ��3�#�5����#+�68�L��&DA�@���&��0�#��+�&%=R+���� ��3�#�j� ��&�£b��� +a?K�L��+���[�@��0� �3:H?b\5!�@�� �3�#�3:8�/�M�0�#����)L��� H&��342�0�#��1��i��&�+�6"��@X\���$+.v$��A9� ��+�&���Qh_*&�:D����:cYD�������=R+�[�&�:8�# ��0� � � Ê �q+��r�ç�*),+0-¦� ����#:�� ��+0=c� D� x �D�i��&�+�6"��@X\����$+.v$��A9� ��+�&10��@�[D���L���$�#+;A��P�.� ��:t)L���# t� ��&�£Z���0�#�U&��342�0� �X1��0Q�ªj+O),�91����3Yc�e��4�Q>�����0@��#+5�$�31��d��@��L�# ��0�L� D�U68����&

Page 24: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� �

��&�+�6"��@X\5���$+.v$��A9� ��+�&"10��@�[��L=R+����d��AM I+�=c� ��i� ��&�£b�S�9- Ê ����ob[��X� �iA�@�+��#�i�#+%� D�f+O1��3� ��@�@K6G�d��&"���#+0-v$��A9� ��+�&710��@�[���Y��#�0� D���L� ���&7�#[D?��$� ��&2� �P�0@�@�\T@����$�j�# ���&7����Q�_*&Z��:�:��X� ��+�&^YD� D���/�M��&�:����#:(:��31;�P�.� ��+�&+�=���$+.v$��A9� ��+�&Z10��@�[���=R+��S�d�0AM r+0=h� ��5�#��&�£b�U���i�#��@��0� �X1��3@�\7 ���4� ^Yc��&�:���A��0� ��&�48�# ��0�S�# ��H��&�+�6"��@X\���$+.v$��A9� ��+�&��i=R+��j�d�0AM C� ��&�£t4��3&����#��@�@X\I ��d1�����&2\;)L D���#��=R�$+�6�@P�0�#4��a&���4��0� �X1����#+"�$6"��@�@��K+��$���#��1��10��@�[D����Q5�� b[��3Ye�X�UAd�0&r?��H��&D=R�3�#�$��:C=R�#+�6±� ��G��?�+O1��H+�?��$���/1��.� ��+�&��S� ��.�U� ��H� �0&�£b���0�#�#+;A����0� �3:)L���# C� ��5?����/���$68�$�H���#�0Y^=R+��S� D�54��#���0� �3�U���#+���+��/� ��+�&C+�=��#��6G��Y^��+���[D@P�0�#��:C?2\F!�@X� ���$��:��$� �0� �3�)L���# 8&D��42�0�#��1��f��&D+�6"�0@�\5���$+.v$��A9� ��+�&G10��@�[����3YD�0&�:T���$�L�K+��D[�@P�0�#��:H=R+����a@����#�$����?�[D��&�+����&��$��4�&D��!D-Ad�0&b�e���$+��K+��$� ��+�&U+�=B� ��6G�h?2\U!�@��#���$��:a�$�M�.� ����)L���# ��#��@��0� �X1��3@�\S�$6"�0@�@���+��#�X� �X1����&�+�68��@X\f���$+.v$��A9� ��+�&10��@�[D����Qe_`��A���&5��@��#+f?��,��&D=R���$�#�3:�� ��0�g!�@X� �3�#��:%�$�M�.� ���e)L��� �68+;:��3� �0�#��#+S@P���$4����K+��#���#��1��>��&�+�6"��@X\���$+.v$��A9� ��+�&���:D+%&�+�����+���[�@��0� ��� D���#�f� ��&�£b�)L��� 8��&2\G�#[�?D�$�M�0&b�#�P��@É=R�#�3ob[���&�A9\�Q�¥,+�@�@���A9� �X1���@X\�Y2� D�+�?��$���/1��.� ��+�&���=R�$+�6Ñ�g��4DQ,���i�#��4����#:���&�4a� ��&�£b�S�9- Ê ��&�:T� Ê - Ê�Ê �#[�4�4����/�,� ��0��!�@�� �3�#�3:T�$� �0� �3��)L���# 6G+B:��3� �0�#�5+���4��#���0� �3����+��$��� �X1��H��&D+�6"�0@�\F�D�#+.v$�3A3� ��+�&�1��0@�[��3���#��&�:]&�+���� +7�����$=R+��#6°���U)���@�@��0�!�@X� ���$��:r�$� �0� �3��)L�X� r&���4��0� �X1��H+��U�#68��@�@g��+��#�X� ��1��H��&�+�68��@X\Z���$+.v$��A9� ��+�&r10��@�[����3QH�� ����$�8������@��$+�#+�68���$[�4�4��3�$� ��+�&�� ��0�h!�@��#���$��:H�$� �0� �3��)L��� H��+��#�X� �X1��>���$+.v$��A9� ��+�&�10��@�[��3��� �3&�:G�#+U�K�3�$=R+��$6Ï)�+��#�$�� ���&(!�@X� �3�#��:7�$� �0� �3�i)L�X� 7&���42�.� ��1����D�#+.v$�3A3� ��+�&t1��0@�[��3��Q,�� D���j@P���/�i�$[�4�4��3�$�#��+�&7���L?�+�@��/� ���$��:(?b\+�?��$���/1��.� ��+�&���+�=��# ��I��&�+�6"��@X\����#+wv$��A3�#��+�&D�����$�#+;A��P�.� ��:<)L�X� '�# �+��$�T?K�3�$�/-*�����$=R+��#6G��&�47!�@��#���#�3:�$� �0� �3�f� ��0�U�0JK+��$:Æ��� «-v �#��@��0� �X1�����6G���#+O1��368�3&2�dQf_*& x ��+�=h�# �� x �8Ad���$���U��&7)L ���AM F�# ��%?����/�!�@X� ���$��:7�$� �0� ���0JK+��#:��b�¢� «-v ��68�D�#+O1��368�3&b�j+O1��3�j�# ��a?K�3�$�i+w1����#��@�@p+������#�0� ��+�&���@c68�36�?K�3��Y�� D�?����$��!�@��#���$��:'�/�M�0�#�"�����0�#�#+;A����0� �3:])L�X� <�7&���4��0� �X1��8��&�+�68��@�\F���#+.v$�3A3�#��+�&^QI�� D���a���z��¨�� v +�=� D���#��6G��YBA�@������#@X\"�# ��U6�[�AM 74��#���0� �3�L���#+���+��/� ��+�&^Q�_*&r��� Ê +�=m� ��H� � �GA����#�3�i��&T)L ���AM t� D�a?����/�!�@X� ���$��:T�/�M�0�#�S�0JK+��$:��!� � «-v ��6G���$+w1���6G��&2��+w1����,�# ��S?����/�>?����$��Ai��&��$��6�?�@��f6G��6�?����3Y;�# ��f?����/�!�@X� ���$��:E�$� �0� �I���5���$�#+;A��P�.� ��:N)L�X� E�C&���4��0� �X1��t��&�+�6"��@X\r���#+.v$�3A3�#��+�&^Qr�� ����5���¬� � x�v +�=j� D�� ��68�0Yb��42����&8A3@������#@X\�� ��j6�[�AM 84��#���0� �3���D�#+���+��$�#��+�&cQg�f:D:����#��+�&��0@���&���@X\;�#���i�R&�+����# �+O)L&É���#�31��d��@��+�&��H+��# ����U�$[�4�4��3�$�#��+�&^Q5_*&r�����/� ��A�[�@�����YK�9VD��6G��&��0�#��+�&C+0=��# ��5���#+wv$��A3�#��+�&D���# ��0����+���[�@��0� �5�#��&�£����&�:���A��0� �3�j�# ��0�i� ��3�#�����$+.v$��A9� ��+�&��U���$�%&�+0�S4��3&����#��@�@X\7��&2\(+0=h� �+��$�%)L���# (�# ���@P�0�#4��3�$�j1��0@�[��3��q�3���# ����L��+��#�X� ��1���+��i&D��42�0�#��1��O�>��&(��&2\74���1���&7�#�3�f+�=��D�#+.v$�3A3� ��+�&t1��0@�[��3��Q>�� b[��3Y�)L ���@����������/-¦)L���#�!�@X� ���$��:I�/�M�0�#���>� ��0�L ��d1��U@����#4��i�K+��#���#��1��S���#+wv$��A3�#��+�&T10��@�[������ ��&D:t�#+5�K�3�$=R+��$6Î�# ��S)�+��#�/�dY����0���/-

Page 25: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

��Ê

)L���$�i!�@X� �3�#��:"�$�M�.� ����� ��0�, ��d1��S@����#4��i&��342�0�#��1��i���#+wv$��A3�#��+�&G10��@�[��3�,:�+%&�+��,�#��&�:8� +������/=R+��#6¢� D�?����$��Q5�g�0£;��&�4I� ����U�#[�4�4����/� ��+�&��#+�4��9� ��3�S)L��� C�# ��H+�� ��3�S!�&�:���&�4��3Y��# ��5+O1��3� ��@�@���6G���#�3�#�$��+�&C���� ��0��� D�j?����/���K�3�$=R+��$68��&�4S���0���/-¦)L���$��!�@X� ���$��:H�$� �0� �3�h� ��&D:H�#+U?K�>�# �+��$�L)L�X� 56G+B:D��� �.� �,�#+U�#68��@�@&��342�0� �X1��U�0&�+�68��@�\H���$+.v$��A9� ��+�&T10��@�[��3��Qh�� ��f��� ��A9� ��Ad��@K��&2� �3�#���$�3�M�.� ��+�&8� ��0�>=R+�@�@�+O)L�,����� ��0�>� D��������/-¦)L���#��!�@��#���#��&�4U���$+BA3���#����h6G+��/���3JK��A3�#��1��L)L D��&8�# ��>�*)�+%?����$��A�6G��6�?����$�h+�=c�U!�@�� �3�#�3:8�/�M�0�#� ��d1��H�0&�+�68��@�\T!���@�:D�f� ��.�U���$�%+�&�@X\768+;:����/� @X\(�$��6G��@�������&F���0�#�#���#&F��&�:F� ��0�U���$�%+�=�4���&��3� ��@�@�\+�����+��$��� �U�$��4�&cQ

Ç^z Öt��Ød�2�mØdØd�R��|©"�|m�ÉÈ��K|m�-�q�eØd�R��|eØ

�� ����h���������h ����h�*)�+�6"����&5+�?;v$�3A3� �X1��3��Q¬S&������e� +���&2� �$+B:D[�A��L��&H�3&��#�36�?�@�����+��/�$-*�D�#+;A����$�#��&�46G�3� D+B:�+�@�+�4�\�:D���#��4�&��3:�� +UA�+�[�&2� �����# ��>�9Jc�3A3�h+�=��/� +;AM ����$�#��A>6G+;:���@;���$�#+��$��+�&5��&D�#��6%?�@��>A3+O1��0�#�X-��&�A3��Qh�� ��S!��#�$�>�$� �3�I��&"�# ��S6G�3�# �+B:D+�@�+�40\5���� +������/=R+��#6��%�#�3�#�����>+�=p!�@�� �3�#��&�4%�WVD�����$��6G��&2� �,+�&� D�,+������#�0� ��+�&���@��3&��#�36�?�@��,6G��6�?����$��Y0)L�X� �� ��,����6�+�=�+�?D� ����&D��&�4j�i�$�3�g+�=��/�M�0�#���e)L�X� ��#��:D[�A��3:B-��6G��@��X� [�:D�,�$�#+BAM ����$�#��A>�3�#�#+��hA�+�6G�K+�&���&2� �3Q�sB��&�A��>��&�ob[����/\%��&2� +S� D��68�9� �+;:�+�@�+�4�\a�#�3o;[D���#�3��� ��.��#+�68�j=R+��#6Ñ+�=p!�@X� �3�#��&D4%�#AM D��6G�U?K�i���$� ��?�@����# D��:^Y;� ��S+0� ��3�,68����&8+�?;v$��A9� �X1��S+�=p�# ��������������������#+:��9!�&��U��&D:I�310��@�[��0� �i�%���$+�� +��*\;���f!�@X� ���$��&�4a�#AM D��6G��Q��� ����,�#AM ��368�0Y��#�9=R���#�$��:"�#+5���>´��������Â-`)L���#��´!�@X� ���$��&�4DYc��&21�+�@�1����S=R+��$68��&�4T��@�@e��+��$�#��?D@��%�������$�S+�=�+������#�0� ��+�&���@m68�36�?����#�S��&�:F� D��&r�d1����#��4���&�4� D�86G��6�?����$����&]����AM <���0����Q8_*&]�# ��"�91��0@�[��0�#��+�&�+�=�� ��G�#AM D��6G��Y��368�� ����#�������a�D@P��A3��:'+�&]� D�+�[D�#�$� ��&�:���&�4I���$�#[��5+�=�)L ��9� ��3�S� ��5�������Â-`)L���$��!�@X� �3�#��:C�/�M�0�#���a���$��Y^��&C4���&��3� ��@�Yc@����#�UA3+��#�$[��D�#��:?2\H���$�#+��$��+�=K�$�#+BAM ����$�#��Aj+��#��4���&��# ���&H� ��j+������ �.� ��+�&���@D6G��6%?K�3�#����#�0Qh_`��������$4�[���:cYB?��0�#��:8[���+�&��@���6G��&2�M�0�$\����#+�?���?���@����/� ��AA�+�&�A��3�D� �3Yb�# ��0���# ����������/-¦)L���#�!�@��#���$��:H�$� �0� �3�h6�[��$��4���&��3� ��@�@�\a?K��@����$�A�+��#�#[D�D� �3:(�X=m�# ���@�+O)�����?�+�[�&�:t+�=m� ��3�����#��&�4��U+�=e�#6G�#�U���>1��3�$\IA3+�&��#���$�#��&2� @X\I�#68��@�@����� ���&t� D�@�+O)����i?K+�[D&�:F+�=h�# ��%+O1��3� ��@�@e�3&��#�36�?�@���kl�S� ��&D4��%+�=��#6G�#��Qj­¡���# (� D���f��:��d�8���S�84�[���:D��@���&���Y x �B�:��XJc�3�#��&2�§�f¥>��<Óa�hs «�«�«�«�� �j¥E��&����#�P��@���©d�.� ��+�& � ä Ê�2 -*?2\2- � ä Ê*2 �#�3�#+�@�[D� ��+�&T���9-*6G��6�?����e��&��$��6�?�@��=R+��$��Ad�0�$� �i+�=��O� � Z@��d��:D�#��6G� Ê�«�« ��g�"4���+���+�� �3&2� �P�0@m D����4� b�f���$����&��0@�\;©��3:^QL�� ����j@P���$4��a� ��6G��@��+�=��3&��#�36�?�@����%�$����&���� ��8�K�3�#��+;: «�«�«�«�� �j¥ � �"�S��A��36�?���� �0«�«2� � + «�«�«�«�� �j¥ � �T�S�3A���6%?K�3���«�«�x QI�f@�@j� Ê�Ê �U��+��#�$��?�@��8���0���/-¦)L���$�H!�@�� �3�#�3:'�$� �0� �3�����$�":��3�#�X1���:]=R+����d��AM N+�=,� �� x �D�8��&��$��65-

Page 26: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� �

?�@�����Yc��&�:F�# ��%�����/=R+��#68��&�A3�%+�=�����AM ��#�3�S+�=�!�@X� ���$��:C�/�M�0�#���S���f�#[D?��#�3o;[D��&2� @X\F1��3�#��!���:^Q�ÐL���$[�@��#�+�=�� D�5��&���@X\;�#���f�# �+O) � ��0�f=R+��f=R[�@�@�\ x�x2� �R+��S� �*v �j+�=�� �� x �B����&��$��6�?D@������WV��068��&���:F�# ����#�9�+�=�������/-¦)L���#��!�@��#���#�3:��$�M�.� ����A3+�&2�M����&��U�0��@��d���/�U+�&��5�/�M�0�#�H�# ��0�U+�[D� �����/=R+��#6G�f� D�5?K�3�$�a+O1����#��@�@+������#�0� ��+�&���@h6G��6�?����T�q��&�A�@�[�:���&D47�# ��"A3+�&2� �#+�@¦Ye�# ��T�����/� [��$?K�3:<68�36�?����#�3Yh��&�:]� ��8��&��$��6�?�@��6G�d��&É���&T� �3�#6G�>+�=m�#6G�#�0Qh_*&I=q��A3��YB=R+��>����AM t+�=�� �� x �D�U��&��$��6�?D@����>�# ����$�����#�0Y�+�&t�d1��3� ��4���Y��0- Ê!�@X� ���$��:��$� �0� �3�S� ��.�U ��d1��H@�+O)����U�$68�$��� ���&F�# ��5?����$��+O1���� �0@�@�+������#�0� ��+�&���@m6G��6�?����3Q%��[��/� ��3���&��0@�\;�#���g�# D+w)L�h�# ��0���X������&�+��g[�&�A�+�686G+�&%=R+��e� ��>�$�3��+�=�!�@�� �3�#�3:5�$�M�.� �����#+S��&DA�@�[D:����0�g@��d�0�$�h+�&D��$� �0� �L)L ���AM T�0JK+��#:D�)L ��.�,6G��4� 2�?��j�#���#6G��:G&�+��#�3)�+��$�# 2\"��68���$+O1���6G��&2�f�q��Q³�0Q���6G���$+w1���6G��&2�+�=Ê�v +��>6G+��#�d����&I�$68�$�S�#�3@P�0�#��1��i� +��# ��U?����$�L+O1���� �0@�@^+������#�0� ��+�&���@�6G��6�?����3Q�sB����A��X!�Ad�0@�@�\�YD�0?K+�[B�+�&��"+�=L�31����$\]�*),+C�$�3�#�5+�=�!�@��#���$��:N�$� �0� �3�%\;���3@�:����#[�AM ��Z�/�M�0�#��QC�f:D:����#��+�&��0@�@�\�Yg�0?K+�[B��+�&��8+�=�31����/\�� « �#�3�#��+�=>!�@�� �3�#�3:<�$� �0� �3��\;���3@�:��%�(�$� �0� �G� ��.�5�0JK+��#:D�N�±� «-v ��6G���$+w1���6G��&2����&'�$68�$��#�3@P�0�#��1��>� +U� ��j?����$�+O1��3� ��@�@�+������ �.� ��+�&���@D6G��6%?K�3��Qg�� D�i68�.VB��6�[D6Ï+�?D�#���/1��3:T��6G���#+O1���6G��&2��+�=�5!�@��#���#�3:(�/�M�0�#���$��@��0� �X1��S� +H� D�a?����/�f+O1���� �0@�@p+������#�0� ��+�&���@^6G��6%?K�3����� ��« Q «-v Q��m+G�#[�6G6"�0�#��©3��Y� D�"��&���@�\;�#����+0=,� D� x �B�"��&D�#��6%?�@��3�%��&D:���A��0� �3���# ��0�%�t10���/�%68�wv$+��$���*\C+0=>� ��H� ��68���# ��"�$�3��+�=�������/-¦)L���#��!�@X� �3�#��:8�$� �0� �3�,A�+�&b� ����&D��6�[D@�� ����@��,�$�M�.� ���� ��.�, ��d1��f@��3�#���#6G�#�L� ���&G� ��j?����$�,+O1����#��@�@+������#�0� ��+�&���@�6G��6�?����3Yc��&D:C��@��$+8�# ��0�f�# ����$�3�U+�=Ë�#��&FA3+�&2�M����&��S�.�S@������$�f+�&����$�M�.� �%)L �+��#���$68�$����L�5&�+0� �3)�+��/� 2\7��6G���#+O1��368�3&2�L�#��@��0� �X1��S�#+H�# ��0�L+�=m� D�a?����/�f+O1���� �0@�@^+������#�0� ��+�&���@K68�36�?�����Q

ÐL��Ad�0@�@���&�4L� ����0�#4�[�6G��&2�#��+�=�sB�3A3�#��+�&5��:cY0� D����?�+w1��,!�&�:D��&�4��g�#[�����+��/��� ���A�+�&DA�@�[D�#��+�&%� ��.�=R+�����&2\H4��X1��3&G�3&��#�36�?�@��j=R+��$��A����$��� D�j�������Â-`)L���$�>!�@��#���$��:H�$� �0� �3�,���$��Yb��&H4��3&����#��@¦Y2@����$�A3+��#�$[��D�#��:?2\����$�#+��$��+�=,�$�#+BAM ��0�$� ��A8+��#��4���&C� ��0&r�# ��8+��K�3� �0�#��+�&��0@�68�36�?����#�a���$��QT�� D���#�G!�&�:���&�4����0nG�#6� D��[D� ��@����*\�+0=��# ����#��68�D@��,�������Â-`)L���$��!�@X� �3�#��&D4S���#+;A��3:�[��$�L��&�:H ���&DA�����&DA�+�[��#��4������$��@���6G��&����/\��#�W-�#�����#AM G�#�342���$:���� ��,�#��A3+�&�:G�����/�h+�=É� ����D�#+���+��#�3:H�K+��/�$-`���#+;A��3�#�#��&�4�68�9� �+;:�+�@�+�4�\�Y0)L ���AM G���g� D�4��3&����#�0� ��+�&T+�=��� b\;?��$��:I�3&��#�36�?�@��f[��#��&�4��#+�68�i�#[�?��$�3��+0=p� ��f!�@�� �3�#�3:T�$� �0� �3��Qh­¡ ��3�# ����,+��,&�+0��� b\;?��$��:G��&��$��6�?D@��LAd��&G�9VB ���?�������68�D�#+O1��3:8A3+w10���$�P��&DA��j��&�:G���#+O1;��:D�j?��3�$� �3�6%[�@��#�X-*:D��6G��&��$��+�&��0@���$+�?���?���@����$� ��A�=R+��#�3Ad���/� ������� ��U���#�#[D�f� ��.��&�+O)Ô� ��£��3�����#�3A��3:���&�A3��Q>�(Ad���d\I��&D:Z�(���$�#��&�� ��«�«2Ê ���&21����$�#��42�.� �S� D�U b\;?��$��:t��&D�#��6%?�@���4��3&����#�0� ��+�&I���#+�?�@��36�1;�P����:��d��@���©3��:T�WVD�����$��6G��&2� �L�0&�:t���#+�?;-��?���@����$� ��Ai=R+��$��A����$�j�310��@�[��0�#��+�&��3Q

Page 27: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� �

sB+�6G�>�.�#� �3&2� ��+�&����g��@��$+S:��31�+�� �3:H��&��# ��>���$���$��&2�����������g�#+f� D�>ob[����/� ��+�&5+�=�)L ��3�# ����gA����/�M����&�������/-¦)L���#��!�@�� �3�#�3:'�$� �0� �3�%�0�#�G�$\;�$�#��68�0� ��Ad��@�@�\F6G+��$�G�9Jc�3A3�#��1��H� ���&'+��# ����$��QT_*&D:����3:^Ye�# ��#+�[D4� ���$+�?���?���@����$� ��A�6G+B:��3@�@���&�4�Y��*),+(�#[�?D�#�3�#����6G���$4��G���a?K�3��&�4t���$�3=R���$�#�3:<�#+�[D�#A��3�%+�=�� �+��#�H!�@��#���#�3:�$� �0� �3�G�# ��0�5\;���3@�:N� D�76G+��/�H&�+�� �9),+��$� 2\E�$��@P�.� ��1��I��6G���#+O1���6G��&2�G��&E�#6G�#�r�D�u� «-v �9QN�� ��3�#��#[D?��#�9� �a��&�A3@�[�:����# ��5+�&��HA3+�68�D�#���$��:�+�=�!�@X� ���$��:r�$� �0� �3�a?����#�3:][���+�&�+�&��5+��U68+��$�5+�=�� ��5�#+��=R+�[��G+������#�0� ��+�&���@�6G��6�?����$�"6G+��$�8:����$�M�0&b�G=R�#+�6 �# ��Z�3&��#�36�?�@��Z6G�d��&ç�1�������:�9�WYL��&�:��# ��Z+�&D�A�+�68���$���$��:I+�=�!�@X� ���$��:t�$� �0� �3��?����$��:7[��K+�&7+�&��S+���6G+��#�f+�=�� ��S�#+��T!�1���+������#�0� ��+�&���@�6G��6�?����$�6G+��$�U:D���$� ��&2�S=R�#+�6 � D����&��$��6�?D@��H68����&��1�����L��M�9Qa�� ����$�8�$[�?��#�9� �a���$�5´����#�9=R���$�#��:D´7�$+�[��$A�������&� D�a�$��&��$���# ��0�L� ��U!�@X� �3�#��:7�$� �0� �3�f�0�#�#+;A����0� �3:7)L�X� I� D��6G+��$�L&D+��M��?D@��U�#�3@P�0�#��1��U��6G���#+O1��368�3&2�+��$��4���&��0� ��=R�$+�6¢� ��3�#�U�#[D?��#�9� ���#��4�&��X!�Ad��&2�#@�\H6G+��#�i+�=Ë� �3&"� ��0&IAd��&T?��S�WVD����A9� �3:t�# ��#+�[�4� T6G���$�AM ���&DA���Q�� ��U!�&�:D��&�4����$��@��0� �3:t�#+¬�������:�S�0&�:����������f���$�U��&2� �3�#�3�$� ��&�4�)L ��3&Z���>���>A3+�&��#��:����$��:t� ��.�� D�Z�������/-¦)L���#�I!�@X� �3�#��:��$� �0� �3�8� ��0�"A3+�6G���#���#���������:�t��&�:¹���������I���$�(?��0�#��:�[��K+�&�+������#�0� ��+�&���@6G��6�?����$�,�#�3@P�0�#��1���@�\H:D���$� ��&2��=R�$+�6¢� ��i��&��$��6�?D@��S6G�d��&cYD��&D:I���,�$[�AM t�#�3���#�3�#��&2�LA3+�6�?���&��0� ��+�&��+�=,68�36�?K�3�#��� ��.�%�0�#�H� ��G68+��$�a:���JK���$��&2��=R�#+�6°+�&��"�0&�+�� D������&]�7�$+B+0�$-*6G�d�0&B-*�$o;[����#�H�#�3&��#�0Q�� ��>��68�D@���A��0� ��+�&^Yd�# ���&^Y2���e�# ��0���������Â-`)L���$��!�@X� �3�#��&D4f���e68+��$���9Jc�3A3�#��1��>�0���3@���6G��&��0� ��&�4L�$� +;AM ����/� ��A���$�#+��fA�+�6G�K+�&���&2� �i)L ���&C� D�%�*),+I+��K�3� �0�#��+�&��0@e6G��6�?����$�f+�=��"4���1��3&F!�@��#���$��:C�$� �0� �����#�%=q�0���#@X\:����#�#��68��@P���3Qp_`�A�+�[�@�:H?K�L�#�����#+�&���:H� ��.�h� ��������� +�?��j�WVB�K�3A3� �3:^Y;+�&5� ��L4��$+�[�&�:��h�# ��0��� ��L6G+��#�:����#�#��68��@P���S�*)�+C6G��6�?����$�����$��Yg�# ��T6G+��#�8@���£0��@X\����������# ��0��� ��T�/� +;AM ����/� ��A8���$�#+��$����&'�# ��8�*)�+6G��6�?����$�>���$�f:��XJc�3�#��&2�>��&�:^Y;� ��3�#�3=R+��#��Y;� ��i68+��#�f@���£0��@�\H�X�,����# ��0��� ��f�O1����#��4���&D4����#+;A��3�#�,)L��@�@���$+B:�[DA��a�G�/�M�0�#�U)L��� t:���68��&����$ ���:"�$� +;AM ����/� ��Aa�3�#�$+��#�3Q

¬f� ��3�,�$[�4�4��3�$� ��+�&��+�=p�/\B�/� �36"�0�#��Aj�K�3�$=R+��$6"�0&�A��L:��XJc�3�#��&2�#�P��@����0�#�j42����&D��:"?2\G���$�#�3�#�#��&�4�� D��������/-¦)L���#�I!�@X� �3�#��:��$�M�.� ���3k,�$68�$�(�����/=R+��#68��&�A3�Z���"�r=R[�&DA3� ��+�&�+0=��#+0-`Ad��@�@���:���&�+�6"��@X\N���$+.v$��AW-� ��+�&�1��0@�[��0QI����AM '!�@X� ���$��:]�$�M�.� �G��&]�(4��X1���&]�#�9�%���%���$�#+;A����0� �3:])L���# '�0&<��&�+�6"��@X\F���$+.v$��A9� ��+�&10��@�[D��Yp?���Ad�0[��#�G�d��AM r+�=� ����*)�+7?����#��A568�36�?����#��[���+�&C)L ���AM r�# ���!�@��#���$��:r�$�M�.� �5���U?����#�3:]������$�#+;A��P�.� ��:I)L�X� t��&7��&�+�6"��@X\H!���@�:7�$��@��0� �X1��f�#+�� ��U�3&��#�36�?�@���6G�d��&t�0&�:I� ��S�*)�+8?����#��AU68�36H-?����#�3kh��&�+�6"��@X\�!���@�:��� ��d1��7�FA3���$� ����&N�D�#+.v$�3A3� ��+�&N+�&2� +C+�&��I��&�+0� ��3��Qr�>),+��#[D4�4����/� ��+�&��H���$���&D4���&�:D���#�3:E?b\]� ��T��&�+�68��@�\F���#+.v$�3A3�#��+�&<10��@�[��T��&���@�\;�#����QF¬U&D�T���a� ��0��!�@��#���$��:N�$� �0� �3�%)L���#

Page 28: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

� ¨

6G+B:��3� �0�#�5+���4��#���0� �3����+��$��� �X1��H��&D+�6"�0@�\F�D�#+.v$�3A3� ��+�&�1��0@�[��3���#��&�:]&�+���� +7�����$=R+��#6°���U)���@�@��0�!�@X� ���$��:]�$�M�.� ���a)L���# '&��342�0� �X1��8+����$6"�0@�@h��+��#�X� ��1��G��&�+�68��@�\F���#+.v$�3A3�#��+�&r1��0@�[��3��Qt�� ��"+0� ��3������ ��0�>!�@X� ���$��:I�/�M�0�#���>)L���# I&���4��0� �X1��U��&D+�6"�0@�\H���$+.v$��A9� ��+�&T1��0@�[��3�,� �3&�:I� +������/=R+��#6Î?K�9�#�#���>� ���&!�@X� ���$��:<�/�M�0�#���a)L��� '�K+��$���#��1��8��&�+�6"��@X\C���$+.v$��A9� ��+�&]10��@�[D����Ye� �+�[D4� '&�+0��+O1��3�$)L ��3@�6G��&�4�@X\��#+�Q�� ��I���#��A3�#��A���@,��&b�#���$���#�9�M�0�#��+�&<+0=�� ��3�#�I�*),+C�$[�4�4��3�$�#��+�&��5���%� ��0�����0���/-¦)L���$�8!�@X� ���$��&�4()L��@�@?��6G+��$���9Jc�3A3� �X1��L)L ���&H� D�j+��K�3� �0�#��+�&��0@�68�36�?����#�h+0=^��4���1���&H�������h ��d1��i��&�+�68��@X\�!��3@�:��h�# ��0����$�?�+�� r�#+�6G�3)L ��0�%�$��6G��@����f��&]���.�#� �3�#&]��&�:]4���&��3� ��@�@�\Z+�=>+��D�K+��$���#�G�$��4�&cQ"_*&'�$[�AM N�tAd���$�G�# ����$�)L��@�@É?K�i�#+�6G�SAd�0&�A��3@�@P�.� ��+�&8+�=m�0&�+�68��@�\�!���@�:"A�+�6G�K+�&���&2� �,:�[��#��&�4%�# ��U�d1��3� ��4���&�4����$+BA3���$��Y�?�[B�� D�IAd��&�A3��@�@P�0�#��+�&r)L��@�@�=q��@�@��$ �+��$��+�=j?K�3��&�4Z)L D+�@��3� ��@���Q��� �����)L��@�@���@�@�+O)¢=R+����$+�68�"�$� +;AM ����/� ��A���$�#+��LA3+�68��+�&��3&2� ��� +G?K���3@���6G��&��0� �3:^YD?�[B�f�.�j�# ���� ��6G�U� ��68�f)L��@�@^�3&��#[D�#���# ��0�j�5=q�����L�068+�[D&b�+�=m��&2\"@���4���� ��6"�.� ����&B=R+��#68�0� ��+�&H=R+�[�&�:I��&"� ��S��&�+�68��@�\5!���@�:D�,)L��@�@�?��S�$�3�M�0��&��3:^Qh�� �����&�+�6"��@X\���$+.v$��A9� ��+�&]�$���#[D@�� �a���$�HA�+�&��$���/� ��&2�a)L���# r�# �+��#�G42����&���:C=R�$+�6 � D�G�D�#+�?��0?���@����$�#��A�6G+;:���@�@���&D4�Yc��&� D�T�#�3&��#�T�# ��0���# ��3\'��@��$+F��&D:���A��0� �G� ��0�%�������/-`)L���#�H!�@��#���$��:N�$� �0� �3�5���$�T68+��$���9Jc�3A3�#��1��T)L D��&?����$��:([��K+�&7+��K�3� �0�#��+�&��0@c6G��6�?����$�L�# ��0�L:�+�&^kl�j ��d1���4��$�d�0�L�$��6G��@����#�X� ���3����&t���0�$� ���$&^Q

_`�H���%&�+�� �3:^Y�!�&���@�@�\�Ye� ��0�5)L D��@��G� ��I�/\;�$� �36"�.� ��AI�����/=R+��#68��&�A3�I:���JK���$��&2� ����@��� ���4� �@���4� 2� ��:1;�P�t� ��8���#+�?���?���@����/� ��A%68+;:���@�@���&�4t��&�:N�0&�+�68��@�\Z���#+.v$�3A3�#��+�&N�0&���@�\;�$���� ���@��'� +(�#�91��d�0@�# ��"A3���/-A�[D68�/�M��&�A3������&])L D��AM '�������Â-`)L���$�5!�@��#���#��&�47����68+��$���3JK��A3�#��1���Yg�# ��3\r68�O\r�0@��#+7?��8+�=,��� �0A3� ��Ad��@A�+�&��#�3o;[D��&�A3��Q��� ����h���e=R+��e� ��>�$�d���$+�&5� ��.�h£b&�+O)L@���:D4���+�=�� D���K�3�$=R+��$6"��&DA��>:��XJc�3�#�3&b�#�P��@��e6G��4� b�?����9VB��@�+����#��:C��&Z�3JK+��$�#�f� +I�$��@��3A3�U�$[�?��$�3� ��+�=��# ����������/-¦)L���#�a!�@��#���$��:C�$� �0� �3�S=R+��f��&�A�+��#��+�� �0�#��+�&��&2�#+G 2\;?��$��:7��&��$��6�?�@����3Q

Page 29: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

29Acknowledgments

This research was supported by a grant from the University of Wisconsin-Madison.

Page 30: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

30References

Alhamed, A., S. Lakshmivarahan, and D. J. Stensrud, 2002: Cluster analysis of multimodel

ensemble data from SAMEX. Mon. Wea. Rev., 130, 226-256.

Atger, F., 1999: Tubing: An alternative to clustering for the classification of ensemble

forecasts. Wea. Forecasting, 14, 741-757.

Barkmeijer, J., T. Iversen, and T. N. Palmer, 2003: Forcing singular vectors and other sensitive

model structures. Quart. J. Roy. Meteor. Soc., 129, 2401-2423.

Bright, D. R., and S. L. Mullen, 2002: Short-range ensemble forecasts of precipitation during the

southwest monsoon. Wea. Forecasting, 17, 1080-1100.

Buizza, R., M. Miller, and T. N. Palmer, 1999: Stochastic representation of model uncertainties

in the ECMWF Ensemble Prediction System. Quart. J. Roy. Meteor. Soc., 125, 2887-

2908.

Colucci, S. J., and D. P. Baumhefner, 1998: Numerical prediction of the onset of blocking: A

case study with forecast ensembles. Mon. Wea. Rev., 126, 773-784.

Eckel, F. A., and M. K. Walters, 1998: Calibrated probabilistic quantitative precipitation

forecasts based on the MRF ensemble. Wea. Forecasting, 13, 1132-1147.

Evans, R. E., M. S. J. Harrison, R. J. Graham, and K. R. Mylne, 2000: Joint medium-range

ensembles from the Met. Office and ECMWF systems. Mon. Wea. Rev., 128, 3104-3127.

Hamill, T. M., 2001: Interpretation of rank histograms for verifying ensemble forecasts. Mon.

Wea. Rev., 129, 550-560.

Hamill, T. M., and S. J. Colucci, 1998: Evaluation of Eta-RSM ensemble probabilistic

precipitation forecasts. Mon. Wea. Rev., 126, 711-724.

Page 31: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

31Harrison, M. S. J., T. N. Palmer, D. S. Richardson, and R. Buizza, 1999: Analysis and model

dependencies in medium-range ensembles: Two transplant case studies. Quart. J. Roy.

Meteor. Soc., 125, 2487-2515.

Houtekamer, P. L., L. Lefaivre, J. Derome, H. Ritchie, and H. L. Mitchell, 1996: A system

simulation approach to ensemble prediction. Mon. Wea. Rev., 124, 1225-1242.

Leith, C. E., 1974: Theoretical skill of Monte Carlo forecasts. Mon. Wea. Rev., 102, 409-418.

McLay, J. G., and J. E. Martin, 2005: Using filtering to mitigate stochastic model errors’ effect

on ensemble covariance. Part II: Use of filtered states in hybrid ensembles. Mon. Wea.

Rev., submitted.

Molteni, F., R. Buizza, T. N. Palmer, and T. Petroliagis, 1996: The ECMWF Ensemble

Prediction System: Methodology and validation. Quart. J. Roy. Meteor. Soc., 122, 73-

119.

Mylne, K. R., R. E. Evans, and R. T. Clark, 2002: Multi-model multi-analysis ensembles in

quasi-operational medium-range forecasting. Quart. J. Roy. Meteor. Soc., 128, 361-384.

Orrell, D., 2003: Model error and predictability over different timescales in the Lorenz ’96

systems. J. Atmos. Sci., 60, 2219-2228.

Orrell, D., L. Smith, J. Barkmeijer, and T. N. Palmer, 2001: Model error in weather forecasting.

Nonlinear Proc. Geoph., 8, 357-371.

Palmer, T. N., 2000: Predicting uncertainty in forecasts of weather and climate. Rep. Prog.

Phys., 63, 71-116.

Palmer, T. N., 2001: A nonlinear dynamical perspective on model error: A proposal for non-

local stochastic-dynamic parameterization in weather and climate prediction models.

Quart. J. Roy. Meteor. Soc., 127, 279-304.

Page 32: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

32Palmer, T. N., R. Buizza, and F. Lalaurette, 1999: Performance of the ECMWF ensemble

prediction system. Proc. ECMWF Workshop on Predictability, ECMWF, Reading,

United Kingdom, 203-214. [Available from ECMWF, Shinfield Park, Reading,

Berkshire, RG2 9AX, United Kingdom.]

Ross, S. M., 1998: A First Course in Probability. Prentice-Hall, 514 pp.

Roulston, M. S., and L. A. Smith, 2003: Combining statistical and dynamical ensembles. Tellus,

55A, 16-30.

Stensrud, D. J., H. E. Brooks, J. Du, M. S. Tracton, and E. Rogers, 1999: Using ensembles for

short-range forecasting. Mon. Wea. Rev., 127, 433-446.

Stensrud, D. J., J.-W. Bao, and T. T. Warner, 2000: Using initial-condition and model physics

perturbations in short-range ensemble simulations of mesoscale convective systems.

Mon. Wea. Rev., 128, 2077-2107.

Toth, Z., and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon.

Wea. Rev., 125, 3297-3319.

Wandishin, M. S., S. L. Mullen, D. J. Stensrud, and H. E. Brooks, 2001: Evaluation of a short-

range multimodel ensemble system. Mon. Wea. Rev., 129, 729-747.

Wang, X., and C. H. Bishop, 2004: Ensemble augmentation with a new dressing kernel.

Preprints, 20th conference on weather analysis and forecasting/16th conference on

numerical weather prediction, Seattle, WA, Amer. Meteor. Soc., J6.4.

Page 33: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

33Figure Captions Figure 1. Conceptual depiction of forecast trajectories evolved with the dynamics of the

real atmosphere and of a numerical model. The light grey striped shading represents an

initial PDF, and the grey dot represents an initial state sampled from the PDF. Black

solid arrow (dashed arrow) is the forecast trajectory evolved from the initial state using

the dynamics of the real atmosphere (of a numerical model). Time increases from left to

right, as indicated by the axis at the bottom of the figure.

Figure 2. Conceptual depiction of errors introduced by a numerical model. Same as for

Fig. 1, except that the open circle (black dot) represents the state at time t on the forecast

trajectory evolved with the dynamics of the real atmosphere (of a numerical model). The

difference between the two states, represented as the thick solid grey arrow, defines the

errors introduced into the forecast as of time t by the numerical model.

Figure 3. Illustration of the nature of the error component associated with systematic

processes. The box encompasses a two-dimensional forecast sample space for some time

t. Each open circle mi (black dot ni) represents the forecast state obtained by evolving

some initial state si forward in time using the dynamics of the real atmosphere (of a

numerical model). A grey arrow represents the error associated with each ni.

Figure 4. Illustration of the nature of the error component associated with stochastic

processes. Symbology the same as for Figure 3.

Page 34: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

34Figure 5. Two simple illustrations of the effect of averaging two dissimilar vectors.

Each panel represents a two-dimensional space, with coordinate axes given as thin solid

lines. In each panel, two dissimilar vectors are represented as solid black arrows. The

average of the two dissimilar vectors is represented as a dotted black arrow. Note that the

average vector has smaller magnitude than either of the two vectors it derives from.

Figure 6. Several conceptualizations of the scenario wherein the possible true states are

all grouped near a small proportion of the filtered states. Each panel encompasses the

sample space for some ensemble forecast. In each panel the forecast state associated with

a given ensemble member is represented as a large dot, and the pair-wise samples

associated with each pair of ensemble members are represented as X’s. The continuum

of possible true states is defined with grey shading. In each of these examples the event

E1 is very likely to occur just because the possible true states are all grouped near the

pair-wise samples with asterisks.

Figure 7. Composite range of 192h 500 hPa geopotential height rmse (m) for a) the basic

operational ensemble and b) the distribution of pair-wise filtered states. In each case the

composite range of rmse is delimited by a grey shaded box and the composite median

rmse is depicted with a dashed line. The composite lower bound of the overall

operational ensemble’s range of rmse is depicted with the dotted line.

Figure 8. Number of pair-wise filtered states whose 192h 500 hPa geopotential height

rmse is less than that of the best overall operational member, for each of the 361

Page 35: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

35ensembles available between 21 December 2002 and 21 December 2003. Each bar

represents the number of filtered states for a specific ensemble. The bars are arranged in

chronological order, beginning with the bar for the 21 December 2002 ensemble in the

upper left corner and proceeding left-to-right and top-to-bottom. The labels ‘J’, ‘F’, ‘M’,

etc. identify the bars associated with the ensembles on the first day of January, February,

March, and so forth. The number of filtered states associated with a given bar is

determined by the scale on the ordinate of each rectangular plot. The bold horizontal line

indicates the average over all 361 ensembles of the number of pair-wise filtered states

whose rmse is less than that of the best overall operational member.

Figure 9. The best pair-wise filtered state’s percentage improvement in 192h 500 hPa

geopotential height rmse relative to the rmse of the best overall operational member, for

each of the 361 ensembles available between 21 December 2002 and 21 December 2003.

Each bar represents the percentage improvement for a specific ensemble. The percentage

improvement associated with a given bar is indicated by the scale on the ordinate of each

rectangular plot. The bold horizontal line indicates the average over all 361 ensembles of

the best filtered state’s percentage improvement in rmse. Layout otherwise the same as

for Figure 8.

Figure 10. Number of best filtered states whose percentage improvement, α, in 192h 500

hPa geopotential height rmse relative to the rmse of the best overall operational member

lies within a specified range. The abscissa indicates the number of best filtered states.

The ranges of percentage relative improvement are given along the ordinate. Bars with

Page 36: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

36light (dark) grey shading are associated with best filtered states that yield positive

(negative) percentage relative improvement.

Figure 11. Mean and median anomaly projection value, and mean anomaly projection

value plus or minus one standard deviation, as a function of forecast rank. Forecast rank

is given on the abscissa and anomaly projection value is given on the ordinate. The “+”

(“•”) symbols indicate the mean (median) anomaly projection value associated with each

forecast rank. The upper (lower) sequence of diamond symbols denotes for each forecast

rank the mean anomaly projection value plus (minus) one standard deviation. The grey

horizontal dashed line indicates the mean anomaly projection value for all forecast ranks.

Page 37: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Figu

re 1

. C

once

ptua

l dep

ictio

n of

fore

cast

traj

ecto

ries e

volv

edw

ith th

e dy

nam

ics o

f the

real

atm

osph

ere

and

of a

num

eric

al

mod

el.

The

light

gre

y st

riped

shad

ing

repr

esen

ts a

n in

itial

PD

F, a

nd th

e gr

ey d

ot re

pres

ents

an

initi

al st

ate

sam

pled

from

the

PDF.

Bla

ck so

lid a

rrow

(das

hed

arro

w) i

s the

fore

cast

traj

ecto

ry e

volv

ed fr

om th

e in

itial

stat

e us

ing

the

dyna

mic

s of t

he re

al

atm

osph

ere

(of a

num

eric

al m

odel

). T

ime

incr

ease

s fro

m le

ft to

righ

t, as

indi

cate

d by

the

axis

at t

he b

otto

m o

f the

figu

re.

t 0tim

e

37

Page 38: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Figu

re 2

. C

once

ptua

l dep

ictio

n of

err

ors i

ntro

duce

d by

a n

umer

ical

mod

el.

Sam

e as

for F

ig. 1

, exc

ept t

hat t

he o

pen

circ

le (b

lack

dot

)re

pres

ents

the

stat

e at

tim

e t o

n th

e fo

reca

st tr

ajec

tory

evo

lved

with

the

dyna

mic

s of t

he re

alat

mos

pher

e (o

f a n

umer

ical

mod

el).

The

diff

eren

ce b

etw

een

the

two

stat

es, r

epre

sent

ed a

s the

thic

k so

lid g

rey

arro

w, d

efin

es th

e er

rors

intro

duce

d in

to th

e fo

reca

st a

sof t

ime

tby

the

num

eric

al m

odel

. t 0t

time

38

Page 39: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Figu

re 3

. Ill

ustra

tion

of th

e na

ture

of t

he e

rror

com

pone

nt a

ssoc

iate

d w

ith sy

stem

atic

pro

cess

es.

The

box

enco

mpa

sses

a

two-

dim

ensi

onal

fore

cast

sam

ple

spac

e fo

r som

e tim

e t.

Eac

h op

en c

ircle

mi

(bla

ck d

ot n

i) re

pres

ents

the

fore

cast

stat

e ob

tain

ed b

y ev

olvi

ng so

me

initi

al st

ate

s ifo

rwar

d in

tim

e us

ing

the

dyna

mic

s of t

he re

al a

tmos

pher

e (o

f anu

mer

ical

mod

el).

A

gre

y ar

row

repr

esen

ts th

e er

ror a

ssoc

iate

d w

ith e

ach

n i.

m4

n 4

m3

n 3

m2

n 2

m1

n 1

39

Page 40: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Figu

re 4

. Ill

ustra

tion

of th

e na

ture

of t

he e

rror

com

pone

nt a

ssoc

iate

d w

ith st

ocha

stic

pro

cess

es.

Sym

bolo

gyth

e sa

me

as fo

r Fi

gure

3.

m1

n 1

n 5m

5

m2

n 2m

4 n 4

m3

n 3

40

Page 41: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Figu

re 5

. Tw

o si

mpl

e ill

ustra

tions

of t

he e

ffec

t of a

vera

ging

two

diss

imila

r vec

tors

. Ea

ch p

anel

repr

esen

ts a

two-

dim

ensi

onal

spac

e, w

ith c

oord

inat

e ax

es g

iven

as t

hin

solid

line

s. In

eac

h pa

nel,

two

diss

imila

r vec

tors

are

repr

esen

ted

as

solid

bla

ck a

rrow

s. T

he a

vera

ge o

f the

two

diss

imila

r vec

tors

is re

pres

ente

d as

a d

otte

d bl

ack

arro

w.

Not

e th

at th

e av

erag

e ve

ctor

has

smal

ler m

agni

tude

than

eith

er o

f the

two

vect

ors i

t der

ives

from

.

b)a)

41

Page 42: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Figu

re 6

. Se

vera

l con

cept

ualiz

atio

ns o

f the

scen

ario

w

here

in th

e po

ssib

le tr

ue st

ates

are

all

grou

ped

near

a

smal

l pro

porti

on o

f the

filte

red

stat

es.

Each

pan

el

enco

mpa

sses

the

sam

ple

spac

e fo

r som

e en

sem

ble

fore

cast

. In

eac

h pa

nel t

he fo

reca

st st

ate

asso

ciat

ed w

ith

a gi

ven

ense

mbl

e m

embe

r is r

epre

sent

ed a

s a la

rge

dot,

and

the

pair-

wis

e sa

mpl

es a

ssoc

iate

d w

ith e

ach

pair

of

ense

mbl

e m

embe

rs a

re re

pres

ente

d as

X’s

. Th

e co

ntin

uum

of p

ossi

ble

true

stat

es is

def

ined

with

gre

y sh

adin

g. I

n ea

ch o

f the

se e

xam

ples

the

even

t E1

is v

ery

likel

y to

occ

ur ju

st b

ecau

se th

e po

ssib

le tr

ue st

ates

are

all

grou

ped

near

the

pair-

wis

e sa

mpl

es w

ith a

ster

isks

.

X

X*

XX*

X

X

X*X*X*

XXX

XX

X

X*X

X

42

Page 43: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Figu

re 7

. C

ompo

site

rang

e of

192

h 50

0 hP

age

opot

entia

lhei

ght r

mse

(m) f

or a

) the

bas

ic o

pera

tiona

l ens

embl

e an

d b)

the

dist

ribut

ion

of p

air-

wis

e fil

tere

d st

ates

. In

eac

h ca

se th

e co

mpo

site

rang

e of

rmse

is d

elim

ited

by a

gre

y sh

aded

box

and

the

com

posi

te m

edia

n rm

seis

dep

icte

d w

ith a

das

hed

line.

The

com

posi

te lo

wer

bou

nd o

f the

ove

rall

oper

atio

nal e

nsem

ble’

s ran

ge o

f rm

seis

dep

icte

d w

ith th

e do

tted

line.

120

110

100 90 80 70 60

a)b)

rmse

43

Page 44: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Figu

re 8

. N

umbe

r of p

air-

wis

e fil

tere

d st

ates

who

se 1

92h

500

hPa

geop

oten

tialh

eigh

t rm

seis

less

than

that

of t

he b

est o

vera

ll op

erat

iona

l mem

ber,

for e

ach

of

the

361

ense

mbl

es a

vaila

ble

betw

een

21 D

ecem

ber 2

002

and

21 D

ecem

ber 2

003.

Eac

h ba

r rep

rese

nts t

he n

umbe

r of f

ilter

ed st

ates

for a

spec

ific

ense

mbl

e.

The

bars

are

arr

ange

d in

chr

onol

ogic

al o

rder

, beg

inni

ng w

ith th

eba

r for

the

21 D

ecem

ber 2

002

ense

mbl

e in

the

uppe

r lef

t cor

ner a

nd p

roce

edin

g le

ft-to

-rig

ht

and

top-

to-b

otto

m.

The

labe

ls ‘J

’, ‘F

’, ‘M

’, et

c. id

entif

y th

e ba

rs a

ssoc

iate

d w

ith th

e en

sem

bles

on

the

first

day

of J

anua

ry, F

ebru

ary,

Mar

ch, a

nd so

forth

. Th

e nu

mbe

r of f

ilter

ed st

ates

ass

ocia

ted

with

a g

iven

bar

is d

eter

min

ed b

y th

e sc

ale

on th

e or

dina

te o

f eac

h re

ctan

gula

r plo

t. T

he b

old

horiz

onta

l lin

e in

dica

tes t

he

aver

age

over

all

361

ense

mbl

es o

f the

num

ber o

f pai

r-w

ise

filte

red

stat

es w

hose

rmse

is le

ss th

an th

at o

f the

bes

t ove

rall

oper

atio

nal m

embe

r.

44

Page 45: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Figu

re 9

. Th

e be

st p

air-

wis

e fil

tere

d st

ate’

s per

cent

age

impr

ovem

ent i

n 19

2h 5

00 h

Page

opot

entia

lhei

ght r

mse

rela

tive

to th

e rm

seof

th

e be

st o

vera

ll op

erat

iona

l mem

ber,

for e

ach

of th

e 36

1 en

sem

bles

ava

ilabl

e be

twee

n 21

Dec

embe

r 200

2 an

d 21

Dec

embe

r 200

3.

Each

bar

repr

esen

ts th

e pe

rcen

tage

impr

ovem

ent f

or a

spec

ific

ense

mbl

e. T

he p

erce

ntag

e im

prov

emen

t ass

ocia

ted

with

a g

iven

bar

is

indi

cate

d by

the

scal

e on

the

ordi

nate

of e

ach

rect

angu

lar p

lot.

The

bold

hor

izon

tal l

ine

indi

cate

s the

ave

rage

ove

r all

361

ense

mbl

es

of th

e be

st fi

ltere

d st

ate’

s per

cent

age

impr

ovem

ent i

n rm

se.

Layo

ut o

ther

wis

e th

e sa

me

as fo

r Fig

ure

8.

45

Page 46: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

Figu

re 1

0. N

umbe

r of b

est f

ilter

ed st

ates

who

se p

erce

ntag

e im

prov

emen

t, α,

in 1

92h

500

hPa

geop

oten

tialh

eigh

t rm

sere

lativ

e to

th

e rm

seof

the

best

ove

rall

oper

atio

nal m

embe

r lie

s with

in a

spec

ified

rang

e. T

he a

bsci

ssa

indi

cate

s the

num

ber o

f bes

t filt

ered

st

ates

. Th

e ra

nges

of p

erce

ntag

e re

lativ

e im

prov

emen

t are

giv

enal

ong

the

ordi

nate

. B

ars w

ith li

ght (

dark

) gre

y sh

adin

g ar

e as

soci

ated

with

bes

t filt

ered

stat

es th

at y

ield

pos

itive

(neg

ativ

e) p

erce

ntag

e re

lativ

e im

prov

emen

t.

020

4060

8010

012

014

016

0

range of percentage improvement

α≤

-10%

-10%

< α

≤-5

%

-5%

< α

< 0

%

0% <

α<

5%

5% ≤

α<

10%

10%

≤α

num

ber o

f bes

t filt

ered

sta

tes

46

Page 47: Jonathan E. Martinmarrella.aos.wisc.edu/McLay_and_Martin2005I.pdf · 1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part I: Evaluation of

fore

cast

rank

anom

aly

proj

ectio

nva

lue

Figu

re 1

1. M

ean

and

med

ian

anom

aly

proj

ectio

n va

lue,

and

mea

n an

omal

y pr

ojec

tion

valu

e pl

us o

r min

us o

ne st

anda

rd d

evia

tion,

as

a fu

nctio

n of

fore

cast

rank

. Fo

reca

st ra

nk is

giv

en o

n th

e ab

scis

sa a

nd a

nom

aly

proj

ectio

n va

lue

is g

iven

on

the

ordi

nate

. Th

e “+

”(“

•”) s

ymbo

ls in

dica

te th

e m

ean

(med

ian)

ano

mal

y pr

ojec

tion

valu

e as

soci

ated

with

eac

h fo

reca

st ra

nk.

The

uppe

r (lo

wer

) se

quen

ce o

f dia

mon

d sy

mbo

ls d

enot

es fo

r eac

h fo

reca

st ra

nk th

e m

ean

anom

aly

proj

ectio

n va

lue

plus

(min

us) o

ne st

anda

rd

devi

atio

n. T

he g

rey

horiz

onta

l das

hed

line

indi

cate

s the

mea

n an

omal

y pr

ojec

tion

valu

e fo

r all

fore

cast

rank

s.

47