**Methods for Estimating Streamflow to Chesapeake Bay**

The following is a description of how data presented on the website "Chesapeake Bay Estimated Streamflow" are computed.

Essentially, the methodology was published more than 51 years ago, and has been adapted for use in modern automated computing systems. Approaches for summarizing data and describing it using statistics follow standard practices. Discussion of methods is provided in four areas:

Summary of Bue (1968) Open-File Report 68-29 — Summary of the original report describing how to estimate monthly mean surface-water inflow to Chesapeake Bay, with a link to the full text of the original.

Automation of Bue (1968) for the World Wide Web — How an automated streamflow-estimation web application was developed using tables from Bue (1968) and continuous data from the USGS National Water Information System.

Calculation of monthly, annual, and period-of-record statistics — Definitions of the various maxima, minima, means, and percentiles used on this website.

Programming details for this website — Technical information about the programming behind the scenes of this we

### Summary of Bue (1968) Open-File Report 68-29 *by Brad Garner, Hydrologist, USGS*

This report presents a convenient and rapid means of estimating, on either a monthly or a yearly basis, the inflow from surface streams to Chesapeake Bay (hereafter, the Bay). The method was developed as a working base for the release entitled 'Estimated stream discharge entering Chesapeake Bay' prepared by the U.S. Geological Survey in cooperation with the States of Maryland, Pennsylvania, and Virginia.' In addition to the methodology used in making estimates of inflow, the report presents considerable data on drainage basins and on streamflow patterns. The report thus serves as a reference for those receiving the monthly release on current conditions. No account is taken of ground-water inflow or of rainfall on and evaporation from the water surface of the Bay.

**Basic Theory: **From 1951 through 1960, an intense streamflow monitoring program was undertaken by the USGS to determine streamflow entering the Bay, using over 50 gaging stations on rivers. Drainage-basin size monitored during this period varied widely: from less than 10 to over 25,000 square miles. In all, streamflow from about 80 percent of the total Bay drainage area was monitored; almost all of the unmonitored 20 percent was in the Coastal Plain lowlands.

Of the over 50 gaging stations, the three with the largest drainage areas were identified as reference gages: Susquehanna at Marietta, PA; Potomac River near Washington, D.C; and James River near Richmond, VA. The purpose of these reference gages was to provide a long-term, uninterrupted record of streamflow. Whereas it would be difficult to guarantee perpetual operation of over 50 gaging stations, it is more likely that three reference gages could be operated continually. Streamflow from about 67 percent of the total Bay drainage area of the Bay is monitored by the three reference gages.

Much of the analysis of Bue consisted of developing empirical relation curves that correlated streamflow at the reference gages to the best-possible estimate of streamflow entering the Bay using over 50 gages, for the period 1951 through 1960. Once developed, the relation curves allowed estimation of streamflow entering the Bay using only three streamflow values rather than needing over 50 streamflow values. In addition, because the three reference gages had long periods of record — back to at least 1937 — streamflow entering the Bay could be estimated even before the beginning of the study in 1951. In fact, monthly estimated streamflow entering the Bay has been computed for every month from October 1936 up to but not including the current month.

**Relation Curves: **In order to develop relation curves relating streamflow at each of the three reference gages to total streamflow entering the Bay, five cross sections were defined (Figure 2). To avoid multivariate analysis, most relation curves estimated incremental streamflow inputs between two cross sections rather than total cumulative streamflow up to that section. For example, one relation curve described incremental streamflow entering the Bay between Sections B and C; by adding the total streamflow at Section B to incremental streamflow entering between sections B and C, total streamflow at Section C can be calculated (see Figure 3, above). Ultimately, the key streamflow value is total streamflow passing Section E, which is a section across the mouth of the Bay between Cape Charles and Cape Henry. This value is what is reported most often throughout this website, although some plots and tables also report incremental streamflow data to indicate what proportion of streamflow various watersheds contribute to the Bay.

**Complicating Factors:** Several diversions of water upstream of the three reference gages complicated the analysis. Generally, these diversions were associated with municipal use of water. The assumption was made that all diverted water eventually enters the Bay, and these values mostly were included implicitly in the development of the relation curves and assumed to be constant.

Two groups of diversions, however, were not implicitly included in the relation curves and were not assumed to be constant. For diversions upstream from the Potomac and James River reference gages, diversion flow-rate values were added to the reported streamflow values before using the relation curves to estimate streamflow entering the Bay. Diversion data were obtained either from cooperating agencies or by direct measurement using USGS equipment.

Relation curves did not account for all water entering the Bay. Bue noted that the following are not quantified: rainfall directly onto the water surface of the Bay; evaporation from the water surface of the Bay; water flux into and out of the Bay at the Delaware Canal; and direct groundwater discharge into the Bay. Values for each of these were not known with high certainty at the time of publication, although each was described qualitatively. Their effect of the accuracy of streamflow estimates would be greatest when overall regional streamflow is low.

**Accuracy of Estimates: **The approach used by Bue did not allow for a "rigorous determination" of the error associated with these streamflow estimates. In part, this was because even the best-available estimates of streamflow entering the Bay (using over 50 gaging stations from 1951 through 1960) had unquantifiable error. In addition, relation curves were hand-drawn lines fitted through data points — a standard approach in 1968. A statistically rigorous approach, such as ordinary least squares regression, would be needed in order to define the error associated with the relation curves. Nevertheless, according the Bue, "as nearly as can be determined, the standard error of the monthly estimates of total inflow to the Bay is about 20 percent, and that of yearly estimates about 10 percent."

The original report has been converted into a web-based document *(link will be updated).*

### Automation of Bue (1968) for the World Wide Web *by Brad Garner, Hydrologist USGS*

The original method for estimating streamflow entering Chesapeake Bay (Bue, 1968) was a manual, pencil-and-paper process. The method has been adapted for automation using computers, using procedures outlined here.

**Convert Lookup Tables to Equations: **Like many scientific documents of its era, Bue (1968) presented its relation curves as reference (or "lookup") tables with only a limited number of values. For the purposes of automation, these tables were converted to equations by fitting a line through the table values in logarithmic space (Table 1; Figure 5).

Fitted Equation | Table in Bue (1968) |
---|---|

log(Q_{SecA}) = 0.6377 + 0.7511 log(Q_{Susq}) + 0.02493 (log(Q_{Susq}))^{2} |
2 |

log(Q_{SecB}) = 2.229 + 0.1517 log(Q_{Susq}) + 0.0831 (log(Q_{Susq}))^{2} |
3 |

log(Q_{SecB-C}) = 0.3996 + 0.9282 log(Q_{Poto}) + 0.0002986 (log(Q_{Poto}))^{2} |
4 |

log(Q_{SecC-D}) = -2.796 + 2.434 log(Q_{Jms}) - 0.1872 (log(Q_{Jms}))^{2} |
5 |

log(Q_{SecD-E}) = -0.9194 + 1.61 log(Q_{Jms}) - 0.08454 (log(Q_{Jms}))^{2} |
6 |

**Table 1.** Equations fitted through the lookup-table values Tables 2-6 of Bue (1968). Subscripts of Q variables indicate which streamflow value is being used or computed. [**Susq**, Susquehanna River at Marietta; **Poto**, Potomac River near Washington, D.C., Diversion-Adjusted;**Jms**, James River near Richmond, VA, Diversion-Adjusted; **SecA** and **SecB**, cumulative streamflow passing defined cross sections; and**SecB-C, C-D,** and **D-E**, incremental streamflow between cross sections]

**Adjustment of Potomac and James River Streamflow: **Water is diverted from the Potomac and James rivers upstream of the reference gages. Because these diversions are variable, they cannot be "built in" to the relation curves. Instead, the flow rates of these diversions are measured and used to adjust streamflow values, by adding the diversion flow rate to measured streamflow values. Conceptually, this "undoes" the effects of diverting this water. Adjusted values then are used as inputs for the relation curves. Mathematically, these operations are defined as:

- Q
_{Poto}= Q_{PotoUnadj}+ Q_{PotoDiv} - Q
_{Jms}= Q_{JmsUnadj}+ Q_{JmsCnl}

where **PotoUnadj** is unadjusted Potomac River streamflow (station 01646500), **PotoDiv** is reported Potomac diversion flow rate (station 01646501), **JmsUnadj** is unadjusted James River streamflow (station 02037500), and **JmsCnl** is measured diversion flow rate of the Kanawha Canal (station 02037000).

**Accumulate Incremental Inflows: **Of the five equations used to estimate streamflow to the Bay (Table 1), three describe incremental inflow between cross sections rather than cumulative flow. Addition and subtraction are used to obtain cumulative streamflow values:

- Q
_{SecA-B}= Q_{SecB}- Q_{SecA} - Q
_{SecC}= Q_{SecB-C}+ Q_{SecB} - Q
_{SecD}= Q_{SecC-D}+ Q_{SecC} - Q
_{SecE}= Q_{SecD-E}+ Q_{SecD}

**Temporary Estimation of Potomac River Diversions: **Diversion flow rates from the Potomac are not continuously measured by a USGS stream gage. Instead, daily diversion flow-rate values are obtained from cooperating agencies and entered manually into USGS database systems. Data typically are obtained one month at a time, and entered within a week after the beginning of each new month.

During the first week of a new month, therefore, not having diversion flow-rate values would preclude estimating streamflow entering the Bay for the just-completed month. As a placeholder only until true diversion data are entered, diversion flow-rate values are estimated using an empirical model:

- Q
_{PotoDiv}= 640 + 90 * sin(2π(D+250)/365)

where D is the day of the year, a number from 1 to 365 or 366. This model was developed by fitting a sinusoid curve through 10 years of data (Figure 6).

**Summary Data-Flow Diagram: **Procedures described on this page have been combined together to fully automate the computation of estimated streamflow to the Bay, and this can be represented by a data-flow diagram (Figure 7).

Calculation of monthly, annual, and period-of-record statistics *by Brad Garner, Hydrologist USGS*

The automated process for computing estimated streamflow to Chesapeake Bay involves calculation of statistics of the data. These summary values are used in plots and tables through this website.

**Monthly Mean Streamflow:** The relation curves of Bue (1968) for estimating streamflow entering Chesapeake Bay were developed using monthly mean streamflow data. However, streamflow is an instantaneous measurement, typically made every 15 minutes. To obtain monthly mean values:

- One day of instantaneous streamflow values are compiled (typically, 96 measurements at 15-minute intervals).
- The arithmetic mean of these 96 measurements is calculated to obtain a daily mean streamflow value.
- Steps 1 and 2 are repeated for all days.
- A hydrographer reviews daily mean values, adjusts or qualifies them as needed, and publishes and archives them.
- Daily mean values are grouped by month (28 to 31 values per group).
- The arithmetic mean is calculated for each monthly group to obtain monthly mean streamflow values.

Steps 1 through 4 are performed within one year of measurement of instantaneous values, with resulting daily mean values published in the USGS Annual Data Report. Steps 5 and 6 are fully automated and are performed periodically by software.

**Annual-Mean Summary Statistics: **Summary statistics of streamflow — minimum, maximum, arithmetic mean, 25th percentile, and 75th percentile — are calculated on an annual basis for daily mean or monthly mean streamflow data. When possible, daily mean streamflow data are used, and the calculations do not use any weighting.

When only monthly mean data are available — for example, estimated streamflow entering Chesapeake Bay — they are used to calculate annual summary statistics. For the arithmetic mean, each monthly value is weighted by the number of days in that month.

**Period-of-Record Summary Statistics: **Summary statistics of streamflow also are calculated for the full period of record using annual-mean streamflow data. The period of record ranges from October 1936 up to but not including the current month. These calculations are straightforward and do not use any weighting.

### Data Tables for Estimated Streamflow to Chesapeake Bay

All streamflow data presented on this website (measured streamflow and computed estimated streamflow to the Bay) are available in tabular format. Choose from:

- Monthly mean values
*(under revision)*— Values for each month, for all three reference gages, five cross sections, and three incremental inflows between cross sections, over the entire period of record. A very large table, please be patient waiting for this page to load. - Annual-mean values
*(under revision)*— Same as above, but only one value for each water year in the period of record. - Daily values for the three reference gages and associated diversions — Clicking a link will open a new window with data directly from USGS NWISWeb.
- 01576000 Susquehanna River at Marietta, PA
- 01646500 Potomac River (Unadjusted) near Washington, DC
- 01646502 Potomac River (Diversion-Adjusted) near Washington, DC — diversion data already added in to these data
- 02037000 James River and Kanawha Canal near Richmond, VA — diversions from James River
- 02037500 James River near Richmond, VA

**Methods for Estimating Streamflow to Chesapeake Bay**

The following is a description of how data presented on the website "Chesapeake Bay Estimated Streamflow" are computed.

Essentially, the methodology was published more than 51 years ago, and has been adapted for use in modern automated computing systems. Approaches for summarizing data and describing it using statistics follow standard practices. Discussion of methods is provided in four areas:

Summary of Bue (1968) Open-File Report 68-29 — Summary of the original report describing how to estimate monthly mean surface-water inflow to Chesapeake Bay, with a link to the full text of the original.

Automation of Bue (1968) for the World Wide Web — How an automated streamflow-estimation web application was developed using tables from Bue (1968) and continuous data from the USGS National Water Information System.

Calculation of monthly, annual, and period-of-record statistics — Definitions of the various maxima, minima, means, and percentiles used on this website.

Programming details for this website — Technical information about the programming behind the scenes of this we

### Summary of Bue (1968) Open-File Report 68-29 *by Brad Garner, Hydrologist, USGS*

This report presents a convenient and rapid means of estimating, on either a monthly or a yearly basis, the inflow from surface streams to Chesapeake Bay (hereafter, the Bay). The method was developed as a working base for the release entitled 'Estimated stream discharge entering Chesapeake Bay' prepared by the U.S. Geological Survey in cooperation with the States of Maryland, Pennsylvania, and Virginia.' In addition to the methodology used in making estimates of inflow, the report presents considerable data on drainage basins and on streamflow patterns. The report thus serves as a reference for those receiving the monthly release on current conditions. No account is taken of ground-water inflow or of rainfall on and evaporation from the water surface of the Bay.

**Basic Theory: **From 1951 through 1960, an intense streamflow monitoring program was undertaken by the USGS to determine streamflow entering the Bay, using over 50 gaging stations on rivers. Drainage-basin size monitored during this period varied widely: from less than 10 to over 25,000 square miles. In all, streamflow from about 80 percent of the total Bay drainage area was monitored; almost all of the unmonitored 20 percent was in the Coastal Plain lowlands.

Of the over 50 gaging stations, the three with the largest drainage areas were identified as reference gages: Susquehanna at Marietta, PA; Potomac River near Washington, D.C; and James River near Richmond, VA. The purpose of these reference gages was to provide a long-term, uninterrupted record of streamflow. Whereas it would be difficult to guarantee perpetual operation of over 50 gaging stations, it is more likely that three reference gages could be operated continually. Streamflow from about 67 percent of the total Bay drainage area of the Bay is monitored by the three reference gages.

Much of the analysis of Bue consisted of developing empirical relation curves that correlated streamflow at the reference gages to the best-possible estimate of streamflow entering the Bay using over 50 gages, for the period 1951 through 1960. Once developed, the relation curves allowed estimation of streamflow entering the Bay using only three streamflow values rather than needing over 50 streamflow values. In addition, because the three reference gages had long periods of record — back to at least 1937 — streamflow entering the Bay could be estimated even before the beginning of the study in 1951. In fact, monthly estimated streamflow entering the Bay has been computed for every month from October 1936 up to but not including the current month.

**Relation Curves: **In order to develop relation curves relating streamflow at each of the three reference gages to total streamflow entering the Bay, five cross sections were defined (Figure 2). To avoid multivariate analysis, most relation curves estimated incremental streamflow inputs between two cross sections rather than total cumulative streamflow up to that section. For example, one relation curve described incremental streamflow entering the Bay between Sections B and C; by adding the total streamflow at Section B to incremental streamflow entering between sections B and C, total streamflow at Section C can be calculated (see Figure 3, above). Ultimately, the key streamflow value is total streamflow passing Section E, which is a section across the mouth of the Bay between Cape Charles and Cape Henry. This value is what is reported most often throughout this website, although some plots and tables also report incremental streamflow data to indicate what proportion of streamflow various watersheds contribute to the Bay.

**Complicating Factors:** Several diversions of water upstream of the three reference gages complicated the analysis. Generally, these diversions were associated with municipal use of water. The assumption was made that all diverted water eventually enters the Bay, and these values mostly were included implicitly in the development of the relation curves and assumed to be constant.

Two groups of diversions, however, were not implicitly included in the relation curves and were not assumed to be constant. For diversions upstream from the Potomac and James River reference gages, diversion flow-rate values were added to the reported streamflow values before using the relation curves to estimate streamflow entering the Bay. Diversion data were obtained either from cooperating agencies or by direct measurement using USGS equipment.

Relation curves did not account for all water entering the Bay. Bue noted that the following are not quantified: rainfall directly onto the water surface of the Bay; evaporation from the water surface of the Bay; water flux into and out of the Bay at the Delaware Canal; and direct groundwater discharge into the Bay. Values for each of these were not known with high certainty at the time of publication, although each was described qualitatively. Their effect of the accuracy of streamflow estimates would be greatest when overall regional streamflow is low.

**Accuracy of Estimates: **The approach used by Bue did not allow for a "rigorous determination" of the error associated with these streamflow estimates. In part, this was because even the best-available estimates of streamflow entering the Bay (using over 50 gaging stations from 1951 through 1960) had unquantifiable error. In addition, relation curves were hand-drawn lines fitted through data points — a standard approach in 1968. A statistically rigorous approach, such as ordinary least squares regression, would be needed in order to define the error associated with the relation curves. Nevertheless, according the Bue, "as nearly as can be determined, the standard error of the monthly estimates of total inflow to the Bay is about 20 percent, and that of yearly estimates about 10 percent."

The original report has been converted into a web-based document *(link will be updated).*

### Automation of Bue (1968) for the World Wide Web *by Brad Garner, Hydrologist USGS*

The original method for estimating streamflow entering Chesapeake Bay (Bue, 1968) was a manual, pencil-and-paper process. The method has been adapted for automation using computers, using procedures outlined here.

**Convert Lookup Tables to Equations: **Like many scientific documents of its era, Bue (1968) presented its relation curves as reference (or "lookup") tables with only a limited number of values. For the purposes of automation, these tables were converted to equations by fitting a line through the table values in logarithmic space (Table 1; Figure 5).

Fitted Equation | Table in Bue (1968) |
---|---|

log(Q_{SecA}) = 0.6377 + 0.7511 log(Q_{Susq}) + 0.02493 (log(Q_{Susq}))^{2} |
2 |

log(Q_{SecB}) = 2.229 + 0.1517 log(Q_{Susq}) + 0.0831 (log(Q_{Susq}))^{2} |
3 |

log(Q_{SecB-C}) = 0.3996 + 0.9282 log(Q_{Poto}) + 0.0002986 (log(Q_{Poto}))^{2} |
4 |

log(Q_{SecC-D}) = -2.796 + 2.434 log(Q_{Jms}) - 0.1872 (log(Q_{Jms}))^{2} |
5 |

log(Q_{SecD-E}) = -0.9194 + 1.61 log(Q_{Jms}) - 0.08454 (log(Q_{Jms}))^{2} |
6 |

**Table 1.** Equations fitted through the lookup-table values Tables 2-6 of Bue (1968). Subscripts of Q variables indicate which streamflow value is being used or computed. [**Susq**, Susquehanna River at Marietta; **Poto**, Potomac River near Washington, D.C., Diversion-Adjusted;**Jms**, James River near Richmond, VA, Diversion-Adjusted; **SecA** and **SecB**, cumulative streamflow passing defined cross sections; and**SecB-C, C-D,** and **D-E**, incremental streamflow between cross sections]

**Adjustment of Potomac and James River Streamflow: **Water is diverted from the Potomac and James rivers upstream of the reference gages. Because these diversions are variable, they cannot be "built in" to the relation curves. Instead, the flow rates of these diversions are measured and used to adjust streamflow values, by adding the diversion flow rate to measured streamflow values. Conceptually, this "undoes" the effects of diverting this water. Adjusted values then are used as inputs for the relation curves. Mathematically, these operations are defined as:

- Q
_{Poto}= Q_{PotoUnadj}+ Q_{PotoDiv} - Q
_{Jms}= Q_{JmsUnadj}+ Q_{JmsCnl}

where **PotoUnadj** is unadjusted Potomac River streamflow (station 01646500), **PotoDiv** is reported Potomac diversion flow rate (station 01646501), **JmsUnadj** is unadjusted James River streamflow (station 02037500), and **JmsCnl** is measured diversion flow rate of the Kanawha Canal (station 02037000).

**Accumulate Incremental Inflows: **Of the five equations used to estimate streamflow to the Bay (Table 1), three describe incremental inflow between cross sections rather than cumulative flow. Addition and subtraction are used to obtain cumulative streamflow values:

- Q
_{SecA-B}= Q_{SecB}- Q_{SecA} - Q
_{SecC}= Q_{SecB-C}+ Q_{SecB} - Q
_{SecD}= Q_{SecC-D}+ Q_{SecC} - Q
_{SecE}= Q_{SecD-E}+ Q_{SecD}

**Temporary Estimation of Potomac River Diversions: **Diversion flow rates from the Potomac are not continuously measured by a USGS stream gage. Instead, daily diversion flow-rate values are obtained from cooperating agencies and entered manually into USGS database systems. Data typically are obtained one month at a time, and entered within a week after the beginning of each new month.

During the first week of a new month, therefore, not having diversion flow-rate values would preclude estimating streamflow entering the Bay for the just-completed month. As a placeholder only until true diversion data are entered, diversion flow-rate values are estimated using an empirical model:

- Q
_{PotoDiv}= 640 + 90 * sin(2π(D+250)/365)

where D is the day of the year, a number from 1 to 365 or 366. This model was developed by fitting a sinusoid curve through 10 years of data (Figure 6).

**Summary Data-Flow Diagram: **Procedures described on this page have been combined together to fully automate the computation of estimated streamflow to the Bay, and this can be represented by a data-flow diagram (Figure 7).

Calculation of monthly, annual, and period-of-record statistics *by Brad Garner, Hydrologist USGS*

The automated process for computing estimated streamflow to Chesapeake Bay involves calculation of statistics of the data. These summary values are used in plots and tables through this website.

**Monthly Mean Streamflow:** The relation curves of Bue (1968) for estimating streamflow entering Chesapeake Bay were developed using monthly mean streamflow data. However, streamflow is an instantaneous measurement, typically made every 15 minutes. To obtain monthly mean values:

- One day of instantaneous streamflow values are compiled (typically, 96 measurements at 15-minute intervals).
- The arithmetic mean of these 96 measurements is calculated to obtain a daily mean streamflow value.
- Steps 1 and 2 are repeated for all days.
- A hydrographer reviews daily mean values, adjusts or qualifies them as needed, and publishes and archives them.
- Daily mean values are grouped by month (28 to 31 values per group).
- The arithmetic mean is calculated for each monthly group to obtain monthly mean streamflow values.

Steps 1 through 4 are performed within one year of measurement of instantaneous values, with resulting daily mean values published in the USGS Annual Data Report. Steps 5 and 6 are fully automated and are performed periodically by software.

**Annual-Mean Summary Statistics: **Summary statistics of streamflow — minimum, maximum, arithmetic mean, 25th percentile, and 75th percentile — are calculated on an annual basis for daily mean or monthly mean streamflow data. When possible, daily mean streamflow data are used, and the calculations do not use any weighting.

When only monthly mean data are available — for example, estimated streamflow entering Chesapeake Bay — they are used to calculate annual summary statistics. For the arithmetic mean, each monthly value is weighted by the number of days in that month.

**Period-of-Record Summary Statistics: **Summary statistics of streamflow also are calculated for the full period of record using annual-mean streamflow data. The period of record ranges from October 1936 up to but not including the current month. These calculations are straightforward and do not use any weighting.

### Data Tables for Estimated Streamflow to Chesapeake Bay

All streamflow data presented on this website (measured streamflow and computed estimated streamflow to the Bay) are available in tabular format. Choose from:

- Monthly mean values
*(under revision)*— Values for each month, for all three reference gages, five cross sections, and three incremental inflows between cross sections, over the entire period of record. A very large table, please be patient waiting for this page to load. - Annual-mean values
*(under revision)*— Same as above, but only one value for each water year in the period of record. - Daily values for the three reference gages and associated diversions — Clicking a link will open a new window with data directly from USGS NWISWeb.
- 01576000 Susquehanna River at Marietta, PA
- 01646500 Potomac River (Unadjusted) near Washington, DC
- 01646502 Potomac River (Diversion-Adjusted) near Washington, DC — diversion data already added in to these data
- 02037000 James River and Kanawha Canal near Richmond, VA — diversions from James River
- 02037500 James River near Richmond, VA