The data in my quick opinion is insufficient to suggest they are related to each other.  You have no period to show higher springbok as compared higher streamflow.  The hydrology of springs do not always follow the local hydrologic boundaries.  I did not reanalyze the data, but the r value during recession curve is typically higher as both are declining.  Oct is the only month the river flow went up, and the spring was still declining.   Another year or two, or perhaps more data may be enough to answer your question., but you not only need the drying cycles, you need wetting cycles to determine interaction.

In fact, I am working on optimal operation of cascade hydro-power plants. Spring Y is considered as an inflow within the water balance equations. Since I am having hard time in obtaining the missing values from the source,  I tried to exploit the available information through the suggested method for generating spring flow.

Regarding your notes Mr. Hansen,

1- The spring can reach a flow of 4m^3/s  during the wet season.

2- During Oct. the opposite variation sense may be interpreted as: after long dry periods certain springs do not react to rainfall immediately but need considerable amounts of rain to increase discharge.

3- Data scarcity is the main challenge in this problem.

You might be somewhat interested in Spring water quality in Missouri 1975 on my researchgate.  The flow in the river is augmented by spring flow, but other than some possible connectivity to rainfall response with substantial differences, the springflow is not dependent on the river flow, but with more data, you may find a relationship nonetheless due to rainfall, but response markedly different.

Dear Bassam

With the limited data of spring flow, we hardly able to find the discharge for missing months.

Regression and best fit curve we can do for some extent but I am also not sure whether it gives appropriate answer or not. 

Do you have some other meteorological data? If then you can do it by water balancing.

 Dear Mr. Bassam,

You need more reading data across your basin all monitor springs flow variation for each month , and it would be great if your reading points are well distributed across the study area in order to create a contour map for spring flowing for each month. Furthermore,  these maps could be correlated with precipitation maps to find additional conclusions related to subsurface geology and topography. Maps could be drawn in several interpolation methods which may help you in predicting statistically the flow and precipitation within the river basin.

Best Regards

Your approach is purely statistical, but you may improve your interpretation if you can add physical knowledge. The number of observations is too small to draw conclusions about the relationship between both series.

Also I wonder how you can know the upper and lower limits of spring flow.

We really need to know where you came up with your values for the range of spring flow.  All the measured data is much lower than the maximum of 4 m3/s.  What makes you think it ever get above 2 m3/s?  You seem to be assuming facts not in evidence as part of your solution.

Having said all this, I don't like your solution because it doesn't reflect the lag in spring response in March.  If the spring doesn't immediately increase flow in September and October in response to increased precipitation, then the spring should have a similar lag, producing elevated flow even as precipitation decreases, in March and April.  I've attached my own little figure that displays both your measured and inferred values.  I've used separate vertical axes to emphasize my point.

How many years of data do you have?  If all you have is a single year, then I would suggest that all that your spring data is showing you is a decreasing spring flow in response to a longer term trend (i.e. a multi-year drought, or coming off a multi-year wet period), rather than just the annual wet-dry cycle.  Your data do not cause me to think that the spring flow is strongly coupled to the annual we-dry cycle reflected in the river flow (in this I'm just agreeing with what has been suggested by others above).

Is it possible that the elevated flows (i.e. > 2 m3/s) are a distinct, intense precipitation driven phenomena that is superimposed on the groundwater system that is represented in the data you've provided?

Dear, Mr. Mabey .

I will carefully treat your notes. In fact, my approach will give sense to our discussion on both mathematical and physical levels, bearing in mind the data limitation challenging issue.

I note here the bold numbers are the available data, that is evidence from the source (and not mine). The rest are obtained using them.

1- First, let us take a look on the River X.

•  It is highly variable with coefficient of variation: CV=1.075>1.
•  Almost dry during the months Jul., Aug., and Sep., then it starts rising up at the beginning of the wet season reaching its peak during Feb.
• Range = 23.38 m^3/s.

Based on that, we conclude that River X follows a pluvial regime.

Second, we switch to Spring Y.

• Yes, we have some information about the wet season. The spring can reach a flow of 4 m^3/s.
•  By simple calculation, we can determine the cumulative amount of water during the months Nov., Dec., Jan., Feb., Mar. and it is equal to 25.54 MCM. That is, more than 55.5 % of total volume (46 MCM) were accumulated during these five months. Mean flow for each month is 2 m^3/s.
• Using rule of thumb, the standard deviation of Spring Y is equal approximately to (4-0.4)/4=0.9 m ^3/s.

Apparently, precipitation is an influential factor. Thus, for a certain extent, Spring X is affected by the pluvial regime.

 Further, we had previously calculated the coefficient of determination between River X and Spring Y. As a result, 60.77% of the variation in the spring’s flow is explained by that of the river and they are correlated by 78%.  In circumstances of data scarcity, River X could be used as a boundary for Spring Y.

2-  Regarding not liking my solution. The issue here is not your judgement “liking or not liking”, it is a matter of providing a better solution or at least an advice that may help in the current situation.

The generated values by my problem-solution approach:

1.8          3.9          3.01        1.99        1.21        1.02        0.94        0.93        0.94        0.98        1.07        1.42

Performance:

• Standard deviation for the estimated series: 0.95 which is near the value obtained by rule of thumb previously.
• RMSE = 0.279 m^3 /s and Nash-Sutcliffe Efficiency NSE=0.567 for the validation period. Not bad, taking into account the limited data.
• The series satisfies the imposed constraints concerning total volume and flow limits.
•  I managed to obtain, from the internet, data regarding rainfall for region under study. I plotted both rainfall and spring flow (attached figure 1.). They show much of a similarity between the two signals except for Dec. and Jan, which could be explained due to snow accumulation (spring is located in the mountain region). However, when snow starts melting down during Feb. and Mar. it is reflected through the high flows during these two months.
• Correlation Analysis: The series shows a significant autocorrelation value at lag one (attached figure 2.). It means the present flow of the spring is affected by the previous month flow and as expected it fades as the number of lags ascends.

Concerning your note:  I quote, “If the spring doesn't immediately increase flow in September and October in response to increased precipitation, then the spring should have a similar lag, producing elevated flow even as precipitation decreases, in March and April.” The precipitation is 12 mm and 28 mm during Sep. and Oct. respectively, I don’t think these values will initiate an increase in the spring’s flow especially the soil is very dry and evaporation is still high. Regarding the second part, you may check months Feb. And Mar.(Figure 1. attached)   

3- Data was gathered from different sources. In fact, the challenge is data exploitation. It is a matter of serving our model: “best approximation for Spring flow X”.

4- Explained above

I hope that will do.

Bassam.

Thank you for the additional information.  I'm sure that it will help everyone better understand the parameters of the system you've asked about.

I apologize if my use of the term "like" was unclear in its meaning.  What I meant was that your solution deviated from the observed behavior of the physical world in other, possibly similar, situations.  There is much that isn't known by we readers about your drainage basin (i.e. soil and rock types, vegetative cover, topography, diversions, well pumping etc.) and so the degree of similarity can only be loosely inferred.

I'm still unclear about how and what you know about the spring flow rates during the November-March time period.  It seems that if one knows the cumulative and peak discharges during those months with the confidence that you seem to have, one would also know something about the average discharges in each month. 

In any event, the graph you provided of the precipitation shows a more typical symmetry of spring response lag than did the comparison to river flow.  In terms of mimicking data from similar data sets I've seen from other parts of the world, I'd still expect to see a little higher spring flow in March.  I would suggest that perhaps you might want to use the precipitation data in place of the river flow data as the input for the technique you've presented above. It appears that evapo-transpiration exceeds the precipitation in April. From April through October it seems that the spring water is draining out of storage, disconnected from precipitation because evapo-transpiration exceeds the precipitation during these months. Beginning in November the precipitation once again exceeds evapotranspiration and infiltration quickly reconnects with the groundwater system and once again the spring flows increase, driven by rising groundwater head.

Since the main provider of such type of  information is not helping us (data not accessible). So, we resorted to data published in different reports.

It is important to re-generate the spring's flow. It is a vital input when seeking an optimal solution for the operation of a hydropower-irrigation plant.

Dear Bassam,

Your last remark is very important because it gives us a clear view on the interest of your question. It's important that you have provided - in your  question - only two small time-series, of which we could not have an idea whether these numbers have any relation and also no idea why these time-series should be extended by estimation.

I think it's important to look to the solution in perspective of the consequences you will meet in case of a false (or at least inaccurate) prediction.

If inaccuracy of your prediction does not influence the operation of the plant too much, your proposed approach might be adequate. However, if this is not the case (and I fear it is), I think it is important to:

- Verify the possible influence of inaccuracies in the peak and average discharges that have been considered now as being 'the truth' and also try to ensure whether these values can be trusted. (I think that adapting these values will strongly influence the results so: if these numbers are not correct, your solution will also be wrong). So, it's very important to know the source.

- Besides that I would recommend to us all information you have: precipitation, land use (and evaporation), geological and geohydrological structure etc. etc. to verify your predictions. E.g. are groundwater monitoring data available?

For us, who try to help to find an optimal solution, it would help to know the location of your question and maybe even some of the mentioned data.

Good luck