I'm going to give you an unorthodox idea. I'm sure other people can give you ideas about actual earth moving kind of landscaping.
The way I see it, first step is to identify where all the water is coming from. If the water is coming from your own roof rain gutter down spout, you could do several things to redirect that water, simplest is to just attach an extender.
But what I have decided to do is to contain all the water that falls on my property and PREVENT runoff from reaching the street and the system. So I have been building raingardens at strategic areas where the rain gutter drains and where water from my and neighbors property pools as you have shown. They are planted with bog plants, shrubs and thirsty trees to soak up the water. I also dig shallow trenches called swales to redirect and sequester the water for vegetable and flower beds. I have two rain barrels that fill with rainwater, then overflow is redirected. One has two way split overflow so the rain water is distributed to the garden more efficiently, including supplying extra water to the new raspberry bed. The other one has a soaker hose connected to it to supply extra water to a planting bed and fill the swale for the bed.
Here is one of my raingardens planted with native bog and wetland plants:
- This area is planted as a butterfly garden and is a
portion of our Monarch Butterfly Waystation
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