10 APPENDIX D: UPPER MISSISSIPPI AND MISSOURI ASSESSMENT OF TREND. Introduction. Nicholas C. Matalas

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1 1 APPENDIX D: UPPER MISSISSIPPI AND MISSOURI ASSESSMENT OF TREND Introduction Nicholas C. Matalas Annual flood sequences have long been assumed to be realizations stationary, independent processes, such that the observed floods are variate values independent and identically distributed (iid) random variables. Studies devoted to improving methodology for flood frequency analysis continue to be based on the iid assumption. Current interest in climate change and it s potential impacts on hydrology in general and on floods in particular calls into question the iid assumption. Whether flood frequency analysis should continue to be pursued under the assumption or not is presently unsettled. A few studies have addressed the issue nonstationarity described as trend in flood flows over time. However, little attention has been given to whether or not the assumption temporal independence should continue to be accepted or if it should be rejected. The following discussions address both issues temporal trend and temporal dependence. A trend, positive or negative, has a beginning and an end. A sustained positive trend would in time become limited by the carrying capacity the stream s drainage area. And a sustained negative trend would in time render the stream dry. It is reasonable to assume that between these extreme hydrologic states, the slope a positive (negative) trend decreases (increases) as the flow regime approaches a new state equilibrium. It is also reasonable to assume that a linear trend begins as a nonlinear trend as the flow regime departs from a state equilibrium, and that in time, the linear trend will become nonlinear as the flow regime approaches a new state equilibrium. Such a pattern may be a trend rather than a trend. More specifically, the trend may be a segment an oscillatory wave. For hydrologic sequences, it is unlikely that an oscillatory wave would have a fixed periodicity. If the oscillatory wave is itself real, it may perhaps best be described as reflecting persistence short or long memory. Thus trend assessment is best pursued relative to persistence. Consequently, an assessment trend would be enhanced by taking into account the evolution the sequence. The account provides a focus on the time at which the assessment is made relative to the time the sequence began. By considering the evolution a sequence, it can be ascertained how trend assessment would have changed over time. This past to present view is complimented by a present to past view, i.e. by an assessment trend considering alternate dates at which the sequence began, where the alternate dates are within the historical time span the sequence. The two views serve to remind us that the future may contradict the past. Paleo-records allow the more remote past to be assessed relative to the historical record (the observed sequence flows), but the future remains unknown. Any trend/persistent assessment should consider the extent to which the observed sequences are correlated with one another. The greater the correlation, the greater the 139

2 redundancy in the information provided by the sequences. The extent to which the temporal pattern one sequence is reflected in another sequence varies directly with the degree redundancy in the information content the sequences. An evolutionary trend assessment is undertaken for annual flood sequences for selected sites in the Upper Mississippi and Missouri Basins. The sequences are examined to determine if they are characterized by statistically meaningful trends and if detected trends are reflective hydrologic persistence or whether persistence is a manifestation trend. Trends are limited to those in the mean defined by the linear regression time (year) on flow (flood). A statistically meaningful trend is taken to be a statistically significant regression at the 5% level and at the 1% level. As time is regressed on flow for only one sequence, the regression is said to be a simple regression. Each sequence is assessed under the null hypothesis that the regression coefficient is equal to zero. In the case a simple regression, the null hypothesis is equivalent to the null hypothesis that the coefficient correlation between time and flow is equal to zero. At a specific level significance, if the regression coefficient is significant or not, then the correlation coefficient is significant or not at the specified level significance. Herein, discussions are focused on the correlation coefficients. The issue temporal dependence is addressed in terms the estimates the first order autocorrelation coefficient and the Hurst coefficient. The assessment draws on selected flood sequences, 7 at sites in the Missouri Basin and 13 at sites in the Upper Mississippi Basin, where the time spans the sequences are all within the period 181 to See Table 1. 1

3 Selected Sequences The locations, drainage areas and lengths the selected sequences are presented in Table 1. Table 1: Selected Sites on the Missouri and Mississippi Rivers All the Missouri sequences span a common concurrent 1 year period, There are no years in which observations are missing at any the 7 sequences. The time spans the Mississippi sequences range from 59 to 135 years. The starting dates the sequences vary from 181 to 1937 (McGregor) and the terminal dates vary from 1995 to Of the 13 sequences, have a missing year observation. The drainage areas the Missouri sites vary from 31, sq. mi. to 58, sq. mi., whereas the Mississippi sites vary from 19, sq. mi. to 171,3 sq. mi. above the confluence the Missouri River to 713, sq. mi. below the confluence. Location Drainage Area (mi ) Missouri River Since all the Missouri sequences and all the Mississippi sequences relate to sites along the main stem the Missouri River and the Mississippi River, respectively, there is a fair degree redundancy information. A measure redundancy is provided by the correlations between sequences zero correlation implying zero redundancy and unit correlation, total redundancy. The correlations between sequences are for the longest concurrent period observation for each paired sequences. The inter-basin correlations are given in Tables and Time Span Missing Assessed for Trend Sioux City, Iowa 31, None Yes Omaha, Neb. 3, None Yes Nebraska City, Neb. 1, None Yes St. Joseph, Mo. 9, None Yes Kansas City, Mo. 89, None Yes Booneville, Mo. 55, None Yes Herman, Mo. 58, None Yes Mississippi River Annoka, Minn. 19, None Yes St. Paul, Minn. 3, Yes Winona, Minn. 59, Yes McGregor, Iowa 7, No Dubuque, Iowa 8, None Yes Clinton, Iowa 85, None Yes Keokuk, Iowa 119, None Yes Hannibal, Mo. 137, None Yes Louisiana, Mo. 1, No Alton/Grafton, Mo. 171, No St. Louis, Mo. 97, Yes Chester, Ill. 78, None Yes Thebes, Ill. 713, None Yes

4 Table : Inter-Basin (Missouri) s Sioux City, Iowa Omaha, Neb. Nebraska City, Neb. St. Joseph, Mo. Kansas City, Mo. Boonville, Mo. Sioux City, Iowa 1 Omaha, Neb Nebraska City, Neb St. Joseph, Mo Kansas City, Mo Boonville, Mo Hermann, Mo Hermann, Mo. The intra-basin correlations, i.e. the correlations between sequences in one basin with those in the other basin, are given in Table. Table 3: Inter-Basin (Mississippi) s Anoka, Minn. St. Paul, Minn. Winona, Minn. 1 Dubuque, Iowa Clinton, Iowa Keokuk. Iowa Hannibal, Mo. Anoka, Minn. 1 St. Paul, Minn Winona, Minn Dubuque, Iowa Clinton, Iowa Keokuk. Iowa Hannibal, Mo St. Louis, Mo Chester, Ill Thebes, Ill Table 3: Inter-Basin (Mississippi) s (Continued) St. Louis, Mo. St. Louis, Mo. 1 Chester, Ill Chester, Ill. Thebes, Ill.

5 Table : Intra-Basin (Missouri-Mississippi) s Sioux City, Iowa Omaha, Neb. Nebraska City, Neb. St. Joseph, Mo. Kansas City, Mo. Boonville, Mo. Hermann, Mo. The average inter- basin correlations are.59 and.39 for the Missouri and Mississippi basins, respectively. The intra-basin correlation is.7. The redundancy within and between basins may be measured by the effective information content at time T given approximately as Anoka, Minn St. Paul, Minn Winona, Minn Dubuque, Iowa Clinton, Iowa Keokuk. Iowa Hannibal, Mo St. Louis, Mo Chester, Ill Thebes, Ill N T T I T = N T + ρ m t m t t=1 ( ) (1) where N T denotes the number station-years data accumulated up to time T, m t denotes the gaging stations operating at time t, and ρ denotes the average correlation the paired sequences. The percent redundancy information is defined as R T = 1 I T 1% () N T If ρ =, then I T = N T, in which case R T =, i.e. there is no redundancy information. If ρ = 1 and m t = m t, then I T = T, in which case, R T = ( m ( m 1) )1%. And therefore as m, R T. For the Missouri basin, T = 1, N T = 7 station-years, m t = 7 t and ρ =.59. Therefore, I T 11.3 station-years. Thus, over the 1-year time span , the percent redundancy information is R T 8%. For the Mississippi Basin, it is assumed that the stations begin the year after the year missing observation and terminate in the year the last observation. Thus, N T = 985 and summation term in the denominator Eq. (1) is equal to,85. Given ρ =.39, then I T 1., in which case R T 83%. The redundancy is very nearly the same for the two basins. In the intra-basin 13

6 case, N T = 895 and the summation term in the denominator Eq. (1) is equal to 11,5. Given ρ =. 7, then I T,7, in which case, R T %. If the two basins are treated as a single basin, then N T = 8, 58 and the summation term in the denominator Eq. (1) is equal to 19,5. Given ρ =. 55, then I T,7, in which case R T 7%. In the trend assessment, all 7 Missouri sequences are considered, but not all the Mississippi sequences since some the sequences have a year missing observation. Of the 13 Mississippi sequences, 1 are assessed for trends. For the 3 sequences not used in the assessment McGregor, Louisiana and Alton/Grafton the missing year observation occurs late in the record. This is also the case with the St. Louis sequences, however because this is the longest the selected Mississippi sequences, spanning the period 181 to 1987, it is included in the assessment. The observations for the 3 sequences not used in the assessment are within the time span Refer to Table 1 above. The annual flood sequences for the 7 sites in the Missouri Basin and for the 13 sites in the Upper Mississippi Basin are shown graphically in Appendix C-A. To facilitate visual comparison one sequence with another, each set observations was standardized such that the means and standard deviations the observations equal and 1, respectively. Statistical characteristics the sequences are given in Appendix C-B. Trend Assessment In the following assessment trend, the evolution a flood sequence is taken into the account. This account provides a focus on the time at which we make the assessment relative to the time the sequence began. By considering the evolution the sequence, we can ascertain how our assessment would have changed over time as the length the sequence increases. This past to present view is complimented by a present to past view, i.e., an assessment trend considering alternate dates at which the record began, where the alternate dates are within the historical time span the sequence. The trend assessments under the two views are summarily given in Tables 5 and. Table 5: Assessment Trend in the Missouri Basin 1

7 Sioux City Omaha Nebraska City Past to Present 1 ( ) ( ).73 1 ( ) -.3 ( ) -.89 ( ).1 ( ).51* 3 ( ) -.35* 3 ( ) ( ) -.18 ( ) -.** ( ) -.1** ( ) -.515** 5 ( ) -.355* 5 ( ) -. 5 ( ) -.39** ( ) -.13 ( ).17 ( ) ( ) ( ).1 7 ( ) ( ) ( ) ( ) ( ) ( ).7 9 ( ) ( ) ( ) ( ) -.78 Present to Past 1 ( ).5 1 ( ).57 1 ( ).778** ( ).1 ( ).3 ( ).79 3 ( ) ( ).3 3 ( ).8 ( ) -. ( ).1 ( ). 5 ( ) -. 5 ( ) ( ) -.9 ( ) -.15 ( ) -.5 ( ). 7 ( ) ( ).5 7 ( ).1 8 ( ) ( ). 8 ( ) ( ) ( ).11 9 ( ) ( ) ( ) ( ) -.78 * 5% Level Significance; ** 1% Level Significance 15

8 Table 5: Assessment Trend in the Missouri Basin (continued) 1

9 St. Joseph Kansas City Booneville Past to Present 1 ( ).39 1 ( ).1 1 ( ).18 ( ).18 ( ). ( ).13 3 ( ) ( ) ( ) -.11 ( ) -. ( ) -.** ( ) -.357** 5 ( ) ( ) -.9* 5 ( ) -.15 ( ).83 ( ) -.13 ( ) ( ).19 7 ( ) ( ) ( ).11 8 ( ) ( ) ( ) ( ) ( ) -. 1 ( ).3* 1 ( ) ( ).11 Present to Past 1 ( ).5* 1 ( ). 1 ( ).11 ( ).77 ( ). ( ) ( ).5 3 ( ).33 3 ( ).1 ( ).1 ( ). ( ).397** 5 ( ).19 5 ( ).81 5 ( ).3 ( ).1 ( ).9 ( ). 7 ( ).5 7 ( ). 7 ( ).3* 8 ( ). 8 ( ) ( ).77* 9 ( ).17 9 ( ).53 9 ( ).11 1 ( ).3* 1 ( ) ( ).11 * 5% level significance; ** 1% level significance 17

10 Table 5: Assessment Trend in the Missouri Basin (continued) Herman Past to Present 1 ( ). ( ) ( ).1 ( ) ( ).51 ( ) ( ). 8 ( ) ( ).11 1 ( ) -.* Present to Past 1 ( ).95 ( ).37 3 ( ).35 ( ) ( ).37** ( ).3* 7 ( ).35** 8 ( ).9** 9 ( ).3 1 ( ) -.* * 5% Level Significance; ** 1% Level Significance 18

11 Anoka St. Paul Winona Past to Present Table : Assessment Trend in the Upper Mississippi Basin 7 ( ).717 ( ) ( ). 17 ( ).78** 1 ( ) -.19 (19-197).88 7 ( ).13** ( ) ( ).5** 37 ( ).** 3 ( ) -.55 (19-197).19** 7 ( ).315* ( ).13 5 ( ).11** 57 ( ).39 5 ( ) -.1 ( ).39** 5 ( ).113 ( ) -.8* 7 ( ).35** 7 ( ) ( ).59 9 ( ) ( ) ( ).19* 15 ( ).199* Present to Past 8 ( ).819* 9 ( ).58 8 ( ).5 18 ( ) ( ).1 18 ( ) ( ) ( ) ( ) ( ) ( ).3 38 ( ).3 8 ( ) ( ) -. 8 ( ) ( ) ( ).7 58 ( ).7 5 ( ) ( ).59* 7 ( ).35** 79 ( ).31** 89 ( ).7* 99 ( ).** 19 ( ).5** 119 ( ).5** 15 ( ).199* * 5% Level Significance; 1% Level Significance 19

12 Dubuque Clinton Keokuk Past to Present Table : Assessment Trend in the Upper Mississippi Basin (continued) 9 ( ) ( ).15 9 ( ) ( ) ( ).8 19 ( ) ( ) ( ) ( ) ( ) ( ) ( ) -. 9 ( ) ( ) ( ) ( ) ( ) ( ) -.3* 9 ( ) ( ) ( ) ( ) ( ) ( ) ( ).3* 93 ( ) ( ) ( ).8** 13 ( ) ( ). 19 ( ).33** 113 ( ).8 19 ( ) ( ).31** 1 ( ) ( ).17 Present to Past 9 ( ) ( ).3 9 ( ).3 19 ( ).9 19 ( ) ( ).19 9 ( ) ( ) ( ).1 39 ( ).1 39 ( ).7 39 ( ).13 9 ( ).78 9 ( ).98 9 ( ) ( ). 59 ( ) ( ). 9 ( ).9* 9 ( ).31 9 ( ).31** 79 ( ).8** 79 ( ). 79 ( ).39** 98 ( ).351** 89 ( ) ( ).31** 99 ( ).3** 99 ( ) ( ).7* 19 ( ).351** 19 ( ) ( ).3* 118 ( ).31** 1 ( ) ( ).17 * 5% Level Significance; ** 1% Level Significance 15

13 Hannibal St. Louis Chester Past to Present Table : Assessment Trend in the Upper Mississippi Basin (continued) 9 ( ) ( ).87 1 ( ).7 19 ( ) ( ).1 (19-197).7 9 ( ) ( ).8 3 ( ) ( ) ( ).11 (19-197) ( ).1 7 ( ).15 5 ( ) ( ) ( ). ( ).19 9 ( ).13 7 ( ) (19-199).59* 79 ( ) ( ) ( ).1* 87 ( ).9 99 ( ).39** 97 ( ). 19 ( ).5** 17 ( ). 118 ( ).7** 117 ( ). 17 ( ).139 Present to Past 9 ( ).5 1 ( ).51 9 ( ).* 19 ( ).8 ( ).7 19 ( ) ( ).18 3 ( ).9 9 ( ).35* 39 ( ).9 ( ) ( ).1** 9 ( ).3* 5 ( ).17 9 ( ).13** 59 ( ).31** ( ).8 59 ( ).3 9 ( ).5** 7 ( ). 71 (199-19).59* 79 ( ).58** 8 ( ) ( ).75** 9 ( ) ( ).75** 1 ( ) ( ).77** 11 ( ) ( ).7** 1 ( ).1 17 ( ).139 * 5% Level Significance; 1% Level Significance 151

14 Table : Assessment Trend in the Upper Mississippi Basin (Continued) Thebes Past to Present 5 ( ) ( ).589* 5 ( ) ( ) ( ).7 55 ( ).8* ( ).319* Present to Past 9 ( ).1 19 ( ).7 9 ( ) ( ).399* 9 ( ).** 59 ( ).7* ( ).319* * 5% Level Significance; 1% Level Significance 15

15 Missouri Basin None the 7 sequences indicate significant trends at the 1% level. For the 7 sequences, those for St. Joseph and Herman, there are significant trends at the 5% level. In both cases, the significance the trends attains with the past to present view and with the present to past view with the inclusion the observations for the period 1988 through 1997 and for the period 1898 through 197, respectively. Had the assessment been made in 1988, the past to present view would not have revealed a significant trend at the 5% level. Moreover, had there been no observations for the period 1898 through 197, the present to past view would not have indicated a significant trend at 5% level. There are not strong indications trends in the Missouri Basin. Whether the trends in the sub-basins, St. Joseph and Herman, can be accounted for by climate change or by land use change or by some other kinds change remains to be determined. In any case, it is changes that have occurred in the most recent years or changes that have occurred in the past but are just now manifesting themselves that reflect trends for the sequences. The trends may be reflections segments oscillatory movements that may themselves be reflections segments persistence. The fact that the estimates the first order autocorrelation coefficients for the sequences vary from. to.18 indicate that the effects persistence may be weak. In any case, persistence does not seem to be long memory given that the estimates the Hurst coefficient vary from.58 to.8. Refer to Table A-1 in Appendix A. Mississippi Basin Of the 13 selected sequences for the Mississippi Basin, 1 were assessed for trends. Over the entire periods record, the ten sequences indicate significant trends 3 (St. Paul, Chester, Thebes) at the 5% level and 3 (Winona, Dubuque, Hannibal) at the 1% level. With respect to the 15-year St. Paul sequence, had the assessment been made in 1978, there would have been no indication a significant trend at the 5% level. Thus, it is the inclusion the most current 19 years observations that results in a significant trend for the entire record. Had the observed record extended from 199 back to 1938, there would have been no indication a significant trend at the 5% level. However, as the record extended further into the past, the indications significant trend would oscillate about the levels 5% and 1%. With respect to the 71-year Chester sequence, it is the most current 9 years observations that bring about a 5% significant trend for the entire record. With respect to the -year Thebes sequence, it is the inclusion the most current 19 years observations that bring about a 5% significant trend. With respect to the 7-year Winona sequence, the past to present view suggests that the significant trend at the 1% level is well substantiated as the levels significance persist as the most current observation increases from 1958 to However, the present to past view indicates that if the observations for the period 19 through 1937 were not available, the there would not be an indication trend at the 5% level. The Winona sequence begins in 1885 and extends to 1995 with the observation for 193 being 153

16 missing. If the missing year observation is ignored and the sequence is treated as a continuos 11-year sequence, then the correlation between time and flow is.157 indicating a significant trend near the 5% level. Thus, it seems as the sequence extends further into the past, the indication a significant trend weakens. Whether the pattern would persist further into the past can not be addressed at this level analysis. With respect to the 118-year Dubuque sequence and the 118-year Hannibal sequence, significant trends at the 1% level are strongly indicated with both the past to present and the present to past views. These two sequences are in strong contrast to the other 8 sequences. Why this is so is an open question. With the exception the St. Louis sequence, the sequences include the most current years observation. The St. Louis sequence extends from 181 to 199, with the 1988 observation being missing. For the period 181 through 1987, there is no indication a significant trend at the 5% level. If the missing year observation is ignored and the sequence is treated as a continuos 17-year sequence, then the correlation between time and flow is. indicating a significant trend at the 5% level. Thus, it is the most current 9 years observations that bring about a significant trend at the 5% level. 15

17 Appendix D-A Selected Sequences The sequences relating to the seven sites in the Missouri Basin and to 1 sites in the Mississippi Basin are depicted in Figures 1 through 17. Missouri Basin Standardized Units Flow Figure A 1: Sequence Standardized Annual Floods on the Missouri River at Sioux City, Iowa Standardized Units Flow Figure A : Sequence Standardized Annual Floods on the Missouri River at Omaha, Nebraska 155

18 Standardized Units Flow Figure A3: Sequence Standardized Annual Floods on the Missouri River at Nebraska City, Nebraska Standardized Units Flow Figure A: Sequence Standardized Annual Floods on the Missouri River at St. Joseph, Missouri Standardized Units Flow Figure A5: Sequence Standardized Annual Floods on the Missouri River at Kansas City, Missouri 15

19 Standardized Units Flow Figure A: Sequence Standardized Annual Floods on the Missopuri River at Booneville, Missouri Standardized Units Flow Figure A7: Sequence Standardized Annual Floods on the Missouri River at Hermann, Missouri 157

20 Mississippi Basin Standardized Units Flow Figure A 8: Sequence Standardized Annual Floods on the Mississippi River at Amoka, Minnisota Standardized Units Flow Figure A 9: Sequence Standardized Annual Floods on the Mississippi River at St. Paul, Minnisota Standardized Units Flow Figure A 1: Sequence Standardized Annual Floods on the Mississippi River at Winona, Minnisota 158

21 Standardized Units Flow Figure A 11: Sequence Standardized Floods on the Mississippi River at McGregor, Iowa Standardized Units Flow Figure A 1: Sequence Standardized Floods on the Mississippi River at Dubuque, Iowa Standardized Units Flow Figure A 13: Sequence Standardized Annual Floods on the Mississippi River at Clinton, Iowa 159

22 Standardized Units Flow Figure A 1: Sequence Standardized Annual Floods on the Mississippi River at Keokuk, Iowa Standardized Units Flow Figure A 15: Sequence Standardized Annual Floods on the Mississippi River at Hannibal, Missouri Standardized Units Flow Figure A 1: Sequence Standardized Annual Floods on the Mississippi River at Louisiana, Missouri 1

23 Standardized Units Flow Figure A 17: Sequence Standardized Annual Floods on the Mississippi River at Alton, Missouri Standardized Units Flow Figure A 18: Sequence Standardized Annual Floods on the Mississippi River, at St. Louis, Missouri Standardized Units Flow Figure A 19: Sequence Standardized Flows on the Mississippi River at Chester, Illinois 11

24 Standardized Units Flow Figure A : Sequence Standardized Annual Floods on the Mississippi River at Thebes, Illinois 1

25 Appendix D-B Statistical Characteristics Sequences used in the trend assessment are characterized by the absolute measures the mean, standard deviation, maximum, minimum and range, and by the relative measures the coefficients variation, skewness, kurtosis, first order autocorrelation and Hurst. See Tables B-1 and B-. Table B-1: Statistical Characteristics Missouri Sequences Statistic Sioux City Omaha Nebraska City St. Joseph Mean 158, 15,5 179,1 178, Std. Dev. 8,89 5,3 73,5 7,8 Maximum 51, 9, 98, 9, Minimum 3,, 5, 5, Range 78,, 8, 3, Coeff. Var Skewness Kurtosis Autocorr, Hurst Coeff Table B-1: Statistical Characteristics Missouri Sequences Continued) Statistic Kansas City Booneville Herman Mean 9, 8,99 3,87 Std. Dev. 1,575 15,98 1,1 Maximum 713, 917, 97, Minimum 9, 8, 1, Range, 835, 88, Coeff. Var Skewness Kurtosis Autocorr, Hurst Coeff

26 Table B-: Statistical Characteristics Mississippi Sequences Statistic Anoka St. Paul Winona Dubuque ( ) ( ) ( ) ( ) Mean 3,8 3,57 93, 13, Std. Dev. 1,3,71,5 7,17 Maximum 91, 171, 8, 98, Minimum 5,97 7, 1,3 3,9 Range 85,3 13,5 57,7 7,3 Coeff. Var Skewness Kurtosis Autocorr, Hurst Coeff Table B-: Statistical Characteristics Mississippi Sequences (Continued) Statistic Clinton Keokuk Hannibal St. Louis ( ) ( ) ( ) ( ) Mean 11,89 188,5 11,5 59,81 Std. Dev. 8,17,31 8,553 19,1 Maximum 37,,717 51,93 875, Minimum,7 5,5, 13, Range,3 388,17 79,53 739, Coeff. Var Skewness Kurtosis Autocorr, Hurst Coeff Table B-: Statistical Characteristics 1 Mississippi Sequences (Continued)

27 15