Technical Article: An Improvement to Node Positioning Algorithms

An Improvement to the Node Positioning Algorithms of the Method of Fundamental Solutions and Complex Variable Boundary Element Method

Authors: 

Sebastian A. Neumann1, Saleem A. Ali2, Theodore V. Hromadka II3

1Cadet, United States Military Academy, United States of America
2Cadet, United States Military Academy, United States of America
3Distinguished Professor, United States Military Academy, United States of America

1 Abstract

In this work we examine use of various geometries and schemes for defining computational node positions on the complex plane. Circular patterns may be of interest for problems involving groundwater well fields or studies of well placement spacing and the like. This contrasts with existing methods, which used candidate nodes and refined them based on performance. One of the most pertinent use-cases we discussed when working on this method was the case of pinpointing the source water contamination in a suburban/urban setting. With existing methods, it is challenging to accurately model the flow of drinking water as it moves from supplier to consumer. This can become an issue if it gets contaminated at some point along the pipe. With some refinement, we believe this new method can be used to efficiently model fluid flow through potable water pipes, allowing investigators to generate a more accurate picture of possible contamination sources, allowing for better resolution of the crisis. 

2 Literature Review

The Complex Variable Boundary Element Method (CVBEM) was used for modeling groundwater transmission by Wilkins et al.,[1]. The Wilkins method generates an initial distribution of node candidates, and gradually introduces these candidates into the final model, optimizing for the lowest error on the boundary. Also of note is the work of DeMoes et al.,[2], who detail in their paper the method iterated upon by Wilkins et al.

3 Methods

The CVBEM method for solving 2-dimensional boundary value problems positions a set of candidate nodes in a grad pattern outside the Study Domain. As more collocation points are added to the approximation function, nodes are chosen from this set and refined to give the best possible approximation for each given node (best being minimum difference between the known boundary function and the approximation function). In this work, we replace the existing method for positioning nodes with a less accurate, but more computationally efficient method.

In this method, rather than selecting candidate nodes and refining them, we simply generate the nodes based on a pre-determined probability density function. For our examples, we used a circle as the basic shape, then added variance to each point’s distance from the boundary (according to the normal distribution). This led to a band of points with some closer to the Study Domain and some farther away.

Now, rather than going through the much-laborious process of refining each candidate node each iteration of the algorithm (computationally, O(n2) time), we simply generate a pseudo-random point in accordance with the predefined probability distribution, which still proves to be quite effective and has a computational time of O(n).

4 Results

Overall, this method decreases accuracy by a factor of 102 while speeding up computation by a factor of 104. The charts below show a comparison between the two methods on some of the benchmark problems shown in the original CVBEM paper. Figure 1 shows a plot of the log of the error as the number of points is increased for our first benchmark – the 90-degree bend. Figure 2 shows the same plot but for our second benchmark – the 90-degree bend with a hump.

Figure 1
Figure 2

Also included are a few graphics of the benchmark problems to show how the points are arranged with respect to the boundary. Figure 3 depicts the 90-degree bend problem and Figure 4 depicts the 90-degree bend with hump.

Figure 3
Figure 4

We noticed that the irregularity of the surface negatively contributes to the accuracy of the model. In Figures 3 and 4, we demonstrate the use of 150 node and collocation point pairs to generate our results and, in these specific problems, these were sufficient in lowering the maximum error to under 10−14. However, in the problem only concerning the 90-degree bend, this method was able to progress to a further 10−16, a significant improvement.

5 Conclusions

With the initial methods, we attempted to define a pure algorithm which has a basis in logic and mathematical principles. While this algorithm was effective, and could be shown to converge, it was impractical to implement for real-world simulations. The real world is messy, and the margin for error of measurements taken is far higher than any of our models introduce. For that reason, even though this new method loses some fidelity, this error is vastly outweighed by the errors created by measurement devices. By creating a faster, less computationally demanding algorithm, we can make it easier to implement in real-world applications.

6 Post-Script

Sebastian Neumann is a cadet at the United States Military Academy studying mathematics and computer science. In his free time, he enjoys tutoring his friends in various computer science classes and is working on an introductory video for all prospective CS majors at West Point.

References

[1] B. D. Wilkins, T. V. H. II, W. Nevils, B. Siegel, and P. Yonzon, “Using a node positioning algorithm to improve models of groundwater flow based on meshless methods,” 2014.

[2] N. J. DeMoes, “Optimization algorithm to locate nodal points for the method of fundamental solutions,” 2014.

Technical Article: The Enhanced Flow Duration Curve

The Enhanced Flow Duration Curve

Author: 

Richard Koehler, PhD, PH
CEO, Visual Data Analytics, LLC
visual.data.analytics@outlook.com

Abstract

Flow duration curves (FDCs) are a mainstay analysis technique examining the composition of streamflow using a ranking system approach. But, despite the multiple versions developed over the past 100+ years, FDCs have never been able to display day-to-day discharge information – until now. Presented is a technique to add temporal sequence information to the fundamental FDC by incorporating a lag(1) autocorrelation scatterplot. This modification shows regions of daily discharge increases (dQ/dt > 0), discharge persistence (dQ/dt = 0), discharge decreases (dQ/dt < 0), and the number and degree of daily discharge changes. The result is a more informative visual representation of streamflow.

Key words: enhanced flow duration curves, lag(1) autocorrelation, daily discharge variation, discharge permutations

Background

Flow duration curves (FDCs) have a long history as a hydrological analysis tool with a straightforward graphical display and ease of construction. In general use since 1915, the FDC is essentially a cumulative frequency curve showing the percentage of the period of record where specific discharge levels were equaled or exceeded (Searcy, 1959). FDCs are computed by ranking daily-value data and assigning exceedance probabilities or percentages to each value by means of a plotting position formula, such as the Weibel equation used in this study (Ziegeweid et al., 2015).

Limitations

A serious FDC issue deals with probability plots. Such plots assume that the data used are independent, identically distributed (IID) random data points. However, stream discharge is often highly autocorrelated (flows followed by similar flows), which violates this assumption.

Given that FDCs do not show the chronological sequence of flow (Searcy, 1959), it means all FDCs represent just the dataset composition. Unless the time-based order of the source data is somehow included, it is not possible to know if the IID assumption is valid. Therefore, until this assumption is validated, the flow duration calculations must be treated as percentages and not probabilities. A descriptive example follows.

Figure 1 shows the observed hydrograph from WY 2015 to 2020 for the Merced River at Happy Isles Bridge near Yosemite, California (USGS site 11264500) along with two reordered hydrographs – one ranked (extreme persistence, extremely stable with autocorrelation (r) ≈ 1) and one randomized (no persistence, extremely flashy with autocorrelation (r) ≈ 0). All three plots have identical FDCs as the data composition is identical. Only the data timing is different.

Figure 1. WY 2016 to 2020 observed, ranked, and randomized hydrographs, Merced River (a), with common FDC for all three hydrographs (b).

A single FDC, in fact, represents all permutations of the source dataset since the composition is the same regardless of the data time-based order. The permutation value is the number of ways to choose a sample of elements (r) from a set of distinct objects (n) when order matters and replacements are not allowed.

P(n,r) = n!/(n-r)! Eq (1)
where n = total number of objects, r = number of objects selected, and 0! = 1

Given a daily FDC with a one-year period for the source data, the number of possible arrangements is estimated to be P(365, 365) = 2.5 · 10778. For comparison, astrophysicists estimate the number of protons in the known universe, the Eddington number, as 1.5 · 1080 (Abramowicz, 2008). The challenge is clear – how is it possible to make an FDC unique to a specific dataset and to know if the source data meets the IID condition?

Enhanced Flow Duration Curve

There is a surprisingly simple solution that also provides additional information about the source dataset and addresses the IID issue. The answer is to use a lag(1) autocorrelation scatterplot in conjunction with an FDC and calculate the lag(1) autocorrelation coefficient, r(1).

For this study, day-to-day pairings use a Qt, (flow for day “t”) and Qt+1 (flow for following day “t+1” ) system, where “1” represents one-day lag between the discharges. By using the Qt exceedance percentage for the x-axis and the Qt+1 discharge value for the y-axis, a unique overlay of next-day discharge points is possible for the FDC. Table 1 summarizes these values. Figure 2 shows the enhancement (compare to Figure 1b).

Table 1. Date, FDC exceedance probability values, and lag(1) autocorrelation data order.
Figure 2. Enhanced FDC, WY 2016 to 2020 for the Merced River, USGS site 11264500.

The spread of Qt+1 datapoints is key to visualizing the autocorrelation within the dataset. Points clustered along the FDC line indicate more persistent, stable conditions with higher autocorrelation, while increasingly scattered points indicate more random, flashier conditions with smaller autocorrelation.

This is consistent with Heckert and Filliben (2003) for describing a lag(1) autocorrelation scatterplot. They state that a tight clustering of data points along the (Qt = Qt+1) line is a signature of positive autocorrelation, indicating highly persistent conditions. Conversely, if the data points have a loose, dispersed pattern, it is a signature of lower autocorrelation and that the data are more random.

Conclusion

This short article is an introduction to the concept of an enhanced flow duration curve (EFDC) and some of its benefits. By combining two common hydrologic tools, the flow duration curve and the lag(1) autocorrelation scatterplot, a more informative graphic is now available.

Multiple applications include enhanced water resources management, improved detailed information about a river system, more targeted ecohydrology research projects, and new digital model evaluation methods, among other possibilities.

Remarks

The author thanks AIH for the opportunity to share this self-funded research.

Abramowicz, M. A., 1988, Eddington number and Eddington mass, The Observatory.
https://adsabs.harvard.edu/full/1988Obs…108…19A/0000019.000.html

Heckert, N., & Filliben, J., 2003. NIST/SEMATECH e-Handbook of Statistical Methods; Chapter 1: Exploratory Data Analysis.
https://doi.org/10.18434/M32189

Helsel, D.R., Hirsch, R. M., Ryberg, K. R., Archfield, S. A., & Gilroy, E. J. (2020).
Statistical Methods in Water Resources: U.S. Geological Survey Techniques and Methods, book 4.
https://doi.org/10.3133/tm4A3

Searcy, J.K., 1959. Flow-Duration Curves (Report No. 1542). US Government Printing Office.
https://pubs.usgs.gov/publication/wsp1542A

AIH Membership and Certification Database Manager

Membership Services

  • Customer service support and help to AIH members and key stakeholders by answering phones, emails and managing all AIH inquiries.
  • In collaboration with the Executive Director, develop and execute an overall marketing plan for membership recruitment and retention; evaluate for effectiveness.
  • Manage and execute monthly webinars (Zoom), including PowerPoint slides and post-webinar communications.
  • Coordinate AIH Board member attendance at occasional partner conferences
  • In collaboration with the Executive Director, support AIH training and non-conference meetings such as pre-meeting logistics, annual awards preparation, booth materials coordination, and other duties as assigned.
  • Provide governance support for the Board of Directors and Committee meetings, including assembling the final agenda, Zoom support, meeting minutes, calendar and Doodle scheduling, board roster, and annual calendar.
  • Staff liaison to the Diversity, Equity, and Inclusion Committee, Webinar Committee, Communications Committee, and Board of Registration.
  • Assist AIH Technical Committees with meeting logistics (Zoom) and SharePoint folders
  • Manage the member life cycle of AIH Certification, Recertification, and recruitment toward certification.
  • Administer multiple choice online exams several times a year for certification, using questions prepared by the Board of Registration.
  • Email membership renewal notices and mail notices annually.
  • Coordinate with Website and Marketing Consultant to publish announcements for awards, calls for nominations, social media, and various AIH announcements.

Certification Database Management

  • Manage all aspects of the Association Management Software, including Certemy and AIH website.
  • Update records and maintain data quality; maintain membership reports (IQAs) and statistics for reporting.
  • Process all payments, including membership renewals, membership recertifications, and Board member expenses.
  • Prepare monthly and quarterly reports for the Board of Directors, CEO, Marketing Consultant, and CFO Services/auditors.

Requirements – The first three points are required

  • Bachelor’s degree – Required
  • Membership Management and Certification Experience – Required
  • Association Management Software Experience – Preferred
  • Certemy Database Preferred
  • Team player

Benefits

  • Virtual position
  • 100% employer-paid health insurance upon start
  • 401K match after two year of service
  • 10 vacation days, accrual of sick days, 11 paid holidays, and paid time off between Christmas and New Years Eve
  • 2 paid time off days for volunteer work of your choice
  • Flexible holiday exchange program

Please send a short cover letter and resume to dresden@awra.org with the subject line: AIH Membership and Website Manager/Director

AIH Examinations – Get Involved

AIH is seeking assistance with examination processes. You can review the scopes for these projects below.

 

Scope of Work to Review AIH Examination Questions

More Information

Need:


The American Institute of Hydrology (AIH) is proposing to hire two or three qualified hydrologists to review a compiled database of exam questions for the AIH certification examinations for professional hydrologists in the disciplines of Water Quality, Surface Water and Groundwater exams and hydrologic technician exam.

Overview:


Currently the board of the examination committee (BOE) maintain four sets of examination database by disciplines with a total of approximately 1,100 questions:

  • Professional Hydrologist – Groundwater (300 questions)
  • Professional Hydrologist – Surface Water (300 questions)
  • Professional Hydrologist – Water Quality (300 questions)
  • Hydrologic Technician (200 questions)

Most of these questions were newly developed by eight hydrologists in 2022. Each question was developed with four multiple-choice answers, where only one is supposed to be correct. However, in order to finalize the exam questions used for the AIH certification examinations, these exam questions need to be reviewed . Here is proposed scope of work for the review:


1- Review the appropriateness of each exam question. Each exam question has four multiple choice answers, one of which is supposed to be the correct one. The reviewer needs to confirm the appropriateness of each questions and answer. If the exam is not appropriate or out of date, these questions need to be removed from the database.
2- Rate the difficulty of each question. Exam questions were rated with three levels of difficulty. These difficulty levels need to be checked and make sure they are correctly rated. If they are not appropriately rated, the reviewer need to correct the level of difficulty.
3- Divide the examination questions under sub-categories. Each question was categorized as one or multiple categories. The review need to check if they are appropriately categorized, otherwise need to be corrected.

 

Deliverables:

Updated database for Water Quality, Surface Water and Groundwater exams and Hydrologic Technician examination questions. Each database needs to include
1- A column note and indicate if the question is valid.
2- A column with update level of difficulty
3- A column with update sub-category
4- A column with any changes made to each question.

Fee:

Each reviewer will be paid $1000 for one set of examination database (SW, GW, WQ, HT) described above. Total budget for the examination question review is $4000.00.

Deadline:

The deadline to respond to this task is August 15, 2023, and finish this task is October 15, 2023.

Scope of Work to Develop a User-Friendly AIH Examination Tool

More Information

Need:

The American Institute of Hydrology (AIH) is proposing to hire an expert to develop a user-friendly examination tool to automatically create required exam questions in the disciplines of Water Quality, Surface Water and Groundwater exams and hydrologic technician exam from current exam question database.

Overview:

Currently the board of the examination committee (BOE) maintain four sets of examination database by disciplines with a total of approximately 1,100 questions:

    • Professional Hydrologist – Groundwater (300 questions)

    • Professional Hydrologist – Surface Water (300 questions)

    • Professional Hydrologist – Water Quality (300 questions)

    • Hydrologic Technician (200 questions)

All of exam questions are multiple choices and currently stored in different sheets in MS Excel files. Each exam database includes three level of difficulties and several categories listed below.

No Category
Surface Water (SW)
1 Watershed and Hydrologic Cycle
2 Precipitation
3 Losses and runoff
4 Hydraulics/Flow/Routing/Storage
5 Prediction/Modeling/Design
6 Peripheral Topics: Climate change and economics, stream restoration, etc.
   
Groundwater (GW)
1 Water budget
2 Site exploration/aquifer properties
3 Groundwater Flow and model simulation
4 Groundwater quality and solute transport
5 Well Hydraulics, well development, testing, maintenance
6 Groundwater recharge, vadose zone, infiltration, MAR, SW/GW interaction
7 Groundwater management, laws, regulations, well head protection, subsidence
8 Peripheral Topics: climate change, economics, water supply, social
   
Water Quality (WQ)
1 Wastewater
2 Environmental
3 Monitoring and Measuring
4 Limnology
5 Aquatic chemistry
6 Microbiology
7 Soils and sediment
8 Source pollution and control/treatment
9 Pollutant fate and transport
10 Peripheral Topics
   
Hydrologic Technician (HT)
1 Data Collection and measurement
2 Information/dissemination/outreach
3 Analysis/records
4 Methods and Instrumentation
5 Database management
6 Gage and other monitoring system
7 Quality assurance/control (QA/QC)
8 Knowledge (hydrologic system)
9 Peripheral Topics

AIH manages following five (5) examinations:

    • Professional Examination in Surface Water

    • Professional Examination in Groundwater

    • Professional Examination in Water Quality

    • Hydrologic In Training (Fundamental)

    • Hydrologic Technician

The AIH examination tool is expected to have following features:

 

    1. A dropdown menu to select what AIH examination questions (SW, GW, WQ, HIT, HT) are created.

    1. Specify percentage questions from each category and level of difficulty under SW, GW, WQ, HT examination database.

    1. Automatically create an AIH exam with 100 questions with correct answers, and

    1. Export an exam into a csv format file.

Deliverables:

A completed and fully tested software package and a user manual (MS word file) documenting how to use the tool.

Fee:

Total budget for the examination tool development and update is $4000.0. $3,000 is used for the tool development and $1,000 will be used for bug fixes and future updates.

Deadline:

The deadline to respond to this task is August 15, 2023, and finish this task is October 15, 2023.

President’s Corner – Summer 2023

The future of the American Institute of Hydrology is bright! We’ve experienced some amazing membership growth in 2023 and now we’re up to nearly 500 members. One of our most promising statistics is our growth in student membership! I can’t wait to engage our students as they explore career options and make sustainable contributions to hydrology, and I look forward to the continued growth of our organization.

In this issue of our Bulletin, you’ll be learning more about our growing webinar series, our first ever Diversity, Equity and Inclusion Scholarship Program, our on-going collaboration with sister organizations like SEDHYD, and some cutting-edge studies conducted by prominent hydrologists.

AIH thrives because of volunteer members like you! Here’s the latest on our committee activities, and some specific ways your participation is critical to our success:

        Get involved with our DEI Committee and help launch our first AIH Scholarship Program. We need your help developing the scholarship application process and criteria, as well as developing a robust outreach program to students interested in pursuing higher education in hydrology. With so many new student members, the competition will be fierce, but exciting (Contact co-chair Matt Naftaly at mnaftaly@dudek.com to join).

        On June 29, please attend our 4th webinar of the year, titled “Investing in People to Achieve Clean Water Goals”, by Jenny Seifert! Join the Webinar Committee and help guide the future of our webinar series, including new opportunities to sponsor webinars (Contact Committee Chair Luciana Cunha at lcunha@westconsultants.com to join).

        Join the Board of Registration and help us administer our certification process (Contact BOR Chair Nick Textor at nick.textor@austin.rr.com to join).

        Join the Communications Committee and help curate articles for our Bulletin, and content for our website (Contact Brennon Schaefer at brennon.schaefer@state.mn.us to join).

 

Still not sure where to get started or have an idea for a new committee? Drop me a line at president@aihydrology.org. I’d love to hear your ideas on how to make AIH your professional home for hydrology!

2023 Webinars: the latest and greatest

In 2023, AIH continues to serve its members with free webinars. Hot topics include the state of science research, operational hydrology, the history of hydrology and AIH, and trends in the field. This year, Glenn Moglen presented the Sensitivity of the NRCS Curve Number to Calibration Methods, Luciana Cunha discussed the Next Generation Water Resources Modeling Framework being developed by NOAA, and Roman Kanivetsky told us stories about founding AIH and exposed AIH members to an in-depth and enlightening discussion on Water Sustainability. We have much more to come. In June, Jenny Seifert will discuss the need to invest in People to Achieve Clean Water Goals. In July, we will start a series on flood forecasting, monitoring, and mitigation.

AIH is happy to offer those in good standing the opportunity to sponsor AIH webinars. For more information, please contact Luciana Cunha at lcunha@westconsultants.com.

AIH DEI Scholarship Fund

The AIH Diversity, Equity, and Inclusivity (DEI) Committee has landed on an exciting priority initiative: the DEI Scholarship Fund. Since beginning its work early in 2022, the DEI Committee has developed several initiatives to further its goal of extending educational and professional opportunities to those not commonly represented in hydrology-related fields. These initiatives include:

  • An International Mentorship Program
  • Minority Serving Institutions Outreach
  • An AIH Speakers Bureau
  • A Hydrology Job and Career Portal

Approved by the AIH Board of Directors at the May 23rd Meeting, the DEI Scholarship Fund will make funding available to selected individuals for hydrology-related training or education. As part of its action, the Board approved seed money for the Fund from revenue collected by the AIH Career Center, an on-line AIH service for advertising job opportunities. It is anticipated that additional revenue for the Fund will come from voluntary AIH membership contributions and corporate sponsors.

There is much to be done!

The DEI committee invites your participation in committee activities including development of the Scholarship Fund and the other DEI initiatives. With the Scholarship Fund being approved by the AIH Board of Directors, it’s time for the DEI Committee to get busy developing the details of implementation including program criteria and management, recipient selection, funding sources, and program outreach. The goal of the DEI Committee is to start awarding funds in 2026.

To join the AIH DEI Committee, or for additional information on the Scholarship Fund and other DEI Committee initiatives, contact Committee Co-Chairs Matt Naftaly at mnaftaly@Dudek.com or 805.308.8529 or John Ramirez at jramirez@cee.msstate.edu or 662.325.9885.

Help support this fund and the future of a student by making a donation. Donations are not tax-deductible.

SEDHYD

AIH was a proud Silver-Level Sponsor and Exhibitor at the 2023 Federal Interagency Sedimentation and Hydrology Modeling Conference (SEDHYD), May 8-12, 2023, in St. Louis, Missouri. The conference is held every 4 to 5 years and brings together engineers and scientists from federal agencies, universities, and consultants with a focus on watersheds, stream channels, reservoirs, and related infrastructure. Jamil Ibrahim (Immediate Past-President) represented AIH at the conference to staff the AIH booth, discuss benefits of students earning early certification as Hydrologists-in-Training (HITs) at the Student Luncheon, and share organizational and individual perspectives during a Diversity, Equity, Inclusion, Accessibility breakout session. There were several AIH members in attendance, including: Laura Keefer, PH (Member, Board of Registration); Dr. Jim Selegean, PH; Dr. Marty Teal, PH, Dr. David Williams, PH (Former Chair, Board of Registration), and. Dr. Zhong Zhang, PH (Director, Academic Affairs).

This was the first time SEDHYD had sponsors for their conference. In 2019, SEDHYD organizers graciously provided AIH an exhibit booth and dedicated time during the conference’s opening plenary to introduce AIH and benefits of certification for hydrologists and hydrologic technicians, and present AIH’s 2019 awards. Over 400 hydrologists, engineers, and scientists from across the U.S. attended the conference. AIH looks to maintain a strong connection to the SEDHYD community and actively support future SEDHYD conferences.

A Climate Condition Analysis Using Palmer Hydrologic Drought Index (PHDI) values

A Climate Condition Analysis Using Palmer Hydrologic Drought Index (PHDI) values

Richard Koehler, PhD, PH, CEO, Visual Data Analytics, LLC  visual.data.analytics@outlook.com

 

Abstract

Drought and climate change are important factors to include in any hydrologic analysis. Current weather-related events in California, such as the extended drought and recent multiple atmospheric rivers, demonstrate how quickly hydrologic conditions can change. A lag(1) autocorrelation analysis of California Climate Division 2 (Sacramento Drainage) using monthly Palmer Hydrologic Drought Index (PHDI) values was conducted to find data ranges, persistence of conditions, along with seasonal and historical drought patterns. Results show distinct conditions within the hydrologic-climatic system which include periods of (a) persistent drought, (b) persistent wet, (c) transition from drought to wet, and (d) transition from wet to drought. Month-to month PHDI changes are quantified using a summation infographic based on the autocorrelation scatterplot.

Key words: PHDI, drought, lag(1) autocorrelation

 

Background

California is divided into seven climate divisions, each with various types of climate indices (Figure 1). For this study, Climate Division 2, Sacramento River drainage (NOAA, 2023a) is used as it contains the Lake Shasta reservoir, an important component of California’s water resources system.

This study examined PHDI information produced by NOAA’s National Centers for Environmental Information (NCEI). The NOAA website for drought data states that the PHDI “measures hydrological impacts of drought (e.g., reservoir levels, groundwater levels, etc.) which take longer to develop and longer to recover from. This long-term drought index was developed to quantify these hydrological effects, and it responds more slowly to changing conditions than the Palmer Drought Severity Index” (NOAA, 2023b). Table 1 describes the different PHDI levels (Hayes, 2007) and Figure 2 shows the 1895 to 2023 timeline plot of this monthly data  (NOAA, 2023c).

Some of the most severe drought values have occurred in recent years, with June 2021 through September 2021 all exhibiting PHDI values in the extreme drought range (-5 or more negative).

Figure 1: California Climate Divisions,

Climate Division 2, Sacramento River drainage (source: NOAA).

Table 1: PHDI level descriptions (Hayes, 2007).

Figure 2: California Climate Division 2,  monthly PHDI values, 1895 to 2023 (source: NOAA).

A histogram (Figure 3) shows the number of months for each PDHI value. There is a distinct bimodal distribution, with a total of 804 months of drought conditions (negative PHDI) and 661 months of wet conditions (positive PHDI). The 0 condition (near normal) represents only 15 months, indicating that this rarely occurred.

Figure 3: California Climate Division 2,  PHDI histogram – months per PHDI condition.

Analysis technique

A lag (1) temporal autocorrelation scatterplot was used to examine the PHDI data. Table 2 shows the one-month data shift used to create an x-y system with PHDI at time “t” for the x coordinate,  and PHDI at time “t+1” for the y coordinate.

Table 2: PHDI one-month data shift example.

Results

Extremes can be represented by large positive and large negative monthly PHDI changes, shown in the following tables (Tables 3 and 4).

Table 3: Ranked largest positive monthly PHDI changes, California Climate Division 2.

Table 4: Ranked largest negative monthly PHDI changes, California Climate Division 2.

However, this simple approach provides an incomplete view of how the hydrologic-climatic system operates. A more expansive approach is to graph all monthly changes with a lag(1) autocorrelation scatterplot (Figure 4), where each point represents the monthly PHDI change.

Any point on the dashed diagonal line (y = 1x) indicates no PHDI change from month-to-month. Any point above the diagonal line signifies positive PHDI changes, while any point below the diagonal line signifies negative PHDI changes. Additionally, the overlay shows groups of points that represent four components of the hydrologic-climatic system, (a) persistent drought, (b) persistent wet, (c) transition from drought to wet, and (d) transition from wet to drought. The Table 1 description for “near normal” (-0.49 to +0.49) rarely happened, as the hydrologic system constantly oscillates between persistent drought and persistent wet conditions.

Figure 4: PHDI lag(1) autocorrelation scatterplot with four conditions.

As each point can be identified by month, a breakout of seasonal scatterplots is possible as shown in Figure 5.

Spring and summer have fewer transitions. These two seasons also have less overall scatter, indicating more persistent wet and dry conditions. Interestingly, summer has both the driest and some of the wettest PHDI values. Winter and fall show more randomness, as data points are more scattered. These two seasons also have more transition points. Atmospheric rivers typically occur during winter (NASA 2023) but, with transition points seen in all four seasons, other mechanisms are likely in play.

Figure 5: Seasonal autocorrelation scatterplots, (a) winter, (b) spring, (c) summer, (d) fall.

PHDI values were rounded to the nearest 0.5 to provide a consistent way to compare all month-to-month pairings, allowing for a summation of all changes for the period of record: 1895 to 2023 (Figure 6).

The PHDI value of -3 shows the single greatest range of change, -4 to +2.5 (Figure 6a). This display helps identify the most and least common changes that have taken place. The most common value is the monthly PHDI value of -1.5 followed by -1.5, which occurred 81 times (Figure 6b). Summation values of 1 indicate unusual conditions as these specific monthly changes occurred only once in the 128-year record.

Figure 6: Historical summation of all PHDI monthly changes for California Climate Division 2:

(a) single largest change, (b) most common month-to-month occurrence.

Coordinates for count values are based on categorized PHDI values.

Conclusions

The lag(1) autocorrelation scatterplot provides a basis for additional information about climatic datasets not possible with other methods. The identification of four distinct components of the hydrologic-climatic system provides new opportunities for planning and management activities by water resource organizations. The success of this approach suggests that more research should be directed to looking into mechanisms that enable large PHDI changes.

For more information about the Lag-1 autocorrelation, please read Dr. Koehler’s previous article, titled “The Lag-12 Hydrograph – Alternate Way to Plot Streamflow Time-Series Data”, AIH Bulletin, Fall 2022.

 

References

Hayes, M. J., 2007. Drought Indices.

https://wwa.colorado.edu/sites/default/files/2021-09/IWCS_2007_July_feature.pdf

NOAA, 2023a. Location of US Climate Divisions. https://psl.noaa.gov/data/usclimdivs/data/map.html

NOAA, 2023b. Historical Palmer Drought Indices.

https://www.ncei.noaa.gov/access/monitoring/historical-palmers/overview

NOAA, 2023c. Climate at a Glance Divisional Time Series. https://www.ncei.noaa.gov/access/monitoring/climate-at-a-glance/divisional/time-series/0402/phdi/all/3/1895-2023?base_prd=true&begbaseyear=1901&endbaseyear=2000

NASA, 2023. Atmospheric Rivers

https://ghrc.nsstc.nasa.gov/home/micro-articles/atmospheric-rivers

 

About the author

Dr. Koehler is the CEO of Visual Data Analytics and a certified professional hydrologist with over 40-years’ experience.

Previously he was the National Hydrologic and Geospatial Sciences Training Coordinator for NOAA’s National Weather Service and is a retired NOAA Corps lieutenant commander. Assignments included navigation and operations officer for two NOAA oceanographic research ships, the Colorado Basin River Forecast Center and the Northwest River Forecast Center, where he oversaw the implementation of an operational dynamic wave model for Lower Columbia River stage forecasts. Other positions include Director of Water Resources for an Arizona consulting company and the water resources hydrologist for Cochise County, Arizona.

He is also a member of the science department faculty at Front Range Community College and is instructor for astronomy, geology, geography, GIS and geodesy courses. He is also an FAA certified professional drone operator.

He has a PhD, MS and BS in Watershed Management from the University of Arizona and an additional MS in Hydrographic Sciences from the US Naval Postgraduate School. The focus of his research are alternate methods of analyzing environmental time-series data along with associated data visualizations.